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Oscillatory and streaming flow due to small-amplitude vibrations in spherical geometry

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Oscillatory and streaming flow due to small-amplitude vibrations in spherical geometry

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Content OSCILLATORY AND STREAMING FLOW DUE TO SMALL-AMPLITUDE VIBRATIONS IN SPHERICAL GEOMETRY by Dejuan Kong A Dissertation Presented to the FACULTY OF UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (AEROSPACE and MECHANICAL ENGINEERING) August 2016 Copyright 2016 Dejuan Kong With love for my parents and Leana ACKNOWLEDGEMENTS I owe my deepest gratitude to my advisor, Professor Satwindar Singh Sadhal, for his constant guidance, inspiration and encouragement. Without him, my PhD dissertation would never have become a reality. He offered me an opportunity and helped me to rebuild my confidence when I almost gave up. His precious advice enabled me to overcome all obstacles in research. His inspiration and warm personality have won my highest respect. I would like to thank my thesis committee members, Professor Veronica Eliasson, Professor Larry G. Redekopp, Professor Katherine Shing and Professor Paul Newton for providing valuable comments on my research during my qualify exam. I also appreciate Dr. Anita Penkova for the inspirational discussions. A special thank goes to Ms. Samantha Graves for her help during my graduate study. I acknowledge the close friendships at USC with Fanhui Xu, Hang Song, Yangyang Huang, Gabriel Acuna and many other friends. I am also grateful to the members of our research group, Dr. Hao-Kun Chu, Dr. Mohammed Alhamli and Dr. Komsan Rattanakijsuntorn. I cherish the enjoyable swimming and surfing moments with Coach Sarah Hayes, Coach Ian Culbertson and Coach Stephanie Eggert. I have enjoyed working with Dr. Yann Staelens and Tailai Ye as a teaching assistant in AME 308 class. Thanks also go to the Department of Aerospace and Mechanical Engineering at USC for the teaching assistantship while I was in the PhD program. Finally, I thank my father, Chang’an Kong, and my mother, Xiaoxia Shen for their endless love, support and understanding. TABLE OF CONTENTS Page LIST OF FIGURES v NOMENCLATURE ix ABSTRACT xi CHAPTER 1 Introduction 1 1.1 Protein aggregation 1 1.2 Ultrasound contrast agents 2 1.3 Steady streaming 4 2 Torsional oscillation of hemisphere 7 2.1 Introduction 7 2.2 Model and theoretical development 8 2.2.1 The first order solutions 9 2.2.2 The second order solutions 10 2.3 Theoretical results and discussion 14 3 Transverse oscillation of hemisphere 21 3.1 Introduction 21 3.2 Model and theoretical development 22 3.2.1 The first order solutions 23 i 3.2.2 The second order solutions 25 3.3 Theoretical results and discussion 28 4 Combined oscillations of sphere 35 4.1 Introduction 35 4.2 Model and theoretical development 36 4.2.1 The leading order solutions 38 4.2.2 The solutions to order 𝜀𝜀 39 4.2.3 The low-frequency limit, 𝜀𝜀 ≪ 1 ≪ | 𝑀𝑀 | − 2 43 4.2.4 The high-frequency limit, 𝜀𝜀 ≪ 1 ≪ | 𝑀𝑀 | 2 47 4.2.4.1 The oscillating field 48 4.2.4.2 Steady streaming 52 4.3 Theoretical results and discussion 61 5 Torsional oscillation of spherical cavity filled with porous medium 75 5.1 Introduction 75 5.2 Model and theoretical development 75 5.2.1 The first order solutions 75 5.2.2 The second order solutions 77 5.3 Theoretical results and discussion 81 6 Transverse oscillation of sphere in porous medium 89 6.1 Introduction 89 6.2 Model and theoretical development 89 6.2.1 The first order solutions 89 ii 6.2.2 The second order solutions 92 6.3 Theoretical results and discussion 96 7 Discussion and future directions 105 7.1 Summary 105 7.2 Future work 106 BIBLOGRAPHY 109 iii iv LIST OF FIGURES Page Figure 2.1: Schematic of the proposed experiment. The circular rod can be vibrated rotationally. 7 Figure 2.2: First order shear stress profiles [Eqn. (2.11)] on the equatorial plane over one time period for the torsional oscillation with 𝛾𝛾 =0.5. The dashed curve shows the time average of absolute shear stress values over one period. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=5, (c) | 𝑀𝑀 |=20. 18 Figure 2.3: The streaming flow pattern for torsional oscillation with 𝛾𝛾 =0.5. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=20. 19 Figure 2.4: Streaming intensity as a function of Womersley number | 𝑀𝑀 | at various polar angle 𝜃𝜃 for the torsional oscillation with 𝛾𝛾 =0.5. 20 Figure 3.1: Schematic of the proposed experiment. The circular rod can be vibrated laterally. 21 Figure 3.2: First order shear stress profiles [Eqn. (3.14)] on the equatorial plane over one time period for the transverse oscillation with 𝛾𝛾 =0.5. The dashed curve shows the time average of absolute shear stress values over one period. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=5, (c) | 𝑀𝑀 |=20. 32 Figure 3.3: The streaming flow pattern for the transverse oscillation with 𝛾𝛾 =0.5. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=20. 33 Figure 3.4: Streaming intensity as a function of Womersley number | 𝑀𝑀 | at various polar angle 𝜃𝜃 for the transverse oscillation with 𝛾𝛾 =0.5. 34 Figure 4.1: Schematic of the combined oscillations case. The inner sphere can oscillate both rotationally and laterally. 36 Figure 4.2: A schematic of the streaming flow regions at high frequency. 47 v Figure 4.3: Leading order shear stress profiles [Eqns.(4.14) and (4.32)] on the equatorial plane over one time period with 𝛾𝛾 =0.5, 𝛼𝛼 =1 and 𝛽𝛽 = 𝜋𝜋 4 ⁄ . The dashed curve shows the time average of absolute shear stress values over one period. (a) low frequency, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=5, (d) | 𝑀𝑀 |=20. 65 Figure 4.4: The streaming flow pattern on 𝑟𝑟 - 𝜃𝜃 plane for low frequency [Eqn.(4.36)] under different amplitude ratio with 𝛾𝛾 =0.5. (a) 𝛼𝛼 =0, (b) 𝛼𝛼 =1, (c) 𝛼𝛼 =3. 66 Figure 4.5: The streaming flow pattern on 𝑟𝑟 - 𝜃𝜃 plane for | 𝑀𝑀 |=10 [Eqn.(4.28)] under different amplitude ratio with 𝛾𝛾 =0.5. (a) 𝛼𝛼 =0, (b) 𝛼𝛼 =1, (c) 𝛼𝛼 =5. 67 Figure 4.6: The streaming flow pattern on 𝑟𝑟 - 𝜃𝜃 plane for high frequency (| 𝑀𝑀 |=25) [Eqn.(4.102)] under different amplitude ratio with 𝛾𝛾 =0.5. (a) 𝛼𝛼 =0, (b) 𝛼𝛼 =1, (c) 𝛼𝛼 =8. 68 Figure 4.7: Example of the streaming flow pattern change on 𝑟𝑟 - 𝜃𝜃 plane with increasing amplitude ratio (| 𝑀𝑀 |=10, 𝛾𝛾 =0.5). (a) 𝛼𝛼 =3.9, (b) 𝛼𝛼 =4.0, (c) 𝛼𝛼 =4.15. 69 Figure 4.8: Velocity profiles of steady streaming for (a) low frequency, (b) | 𝑀𝑀 |=10 and (c) high frequency (| 𝑀𝑀 |=25) with 𝛾𝛾 =0.5, 𝛼𝛼 =1 and 𝛽𝛽 =0. 70 Figure 4.9: Steady part of first order azimuthal velocity profiles on the plane 𝜃𝜃 = 𝜋𝜋 4 ⁄ under different phase difference 𝛽𝛽 with 𝛾𝛾 =0.5, 𝛼𝛼 =1 for (a) low frequency, (b) | 𝑀𝑀 |=10 and (c) high frequency (| 𝑀𝑀 |=25). 71 Figure 4.10: Steady part of first order velocity profiles with 𝛾𝛾 =0.5, 𝛼𝛼 =1, 𝛽𝛽 =0 and | 𝑀𝑀 |=25. (a) asymptotic analysis [Eqn. (4.103)], (b) Eqns. (4.24)-(4.27) 72 Figure 4.11: Steady part of first order velocity profiles with 𝛾𝛾 =0.5, 𝛼𝛼 =1, 𝛽𝛽 =0 and | 𝑀𝑀 |=50. (a) asymptotic analysis [Eqn. (4.103)], (b) Eqns. (4.24)-(4.27) 73 Figure 4.12: Steady part of first order velocity profiles with 𝛾𝛾 =0.5, 𝛼𝛼 =1, 𝛽𝛽 =0 and | 𝑀𝑀 |=100. (a) asymptotic analysis [Eqn. (4.103)], (b) Eqns. (4.24)-(4.27) 74 Figure 5.1: Leading order velocity profiles [Eqn.(5.12)] on the equatorial plane over one time period with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=0.1, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=10. 83 Figure 5.2: Leading order shear stress profiles [Eqn.(5.12)] on the equatorial plane over one time period for different Darcy number with | 𝑀𝑀 |=5. (a) 𝐷𝐷𝐷𝐷 =0.1, (b) 𝐷𝐷𝐷𝐷 =1, (c) 𝐷𝐷𝐷𝐷 =10, (d) 𝐷𝐷𝐷𝐷 ≫1. 84 vi Figure 5.3: The streaming flow pattern [Eqn.(5.24)] with 𝐷𝐷𝐷𝐷 =0.1. (a) | 𝑀𝑀 |=4.5, (b) | 𝑀𝑀 |=5, (c) | 𝑀𝑀 |=5.5. 85 Figure 5.4: The streaming flow pattern [Eqn.(5.24)] with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=2.7, (b) | 𝑀𝑀 |=2.85, (c) | 𝑀𝑀 |=3.2. 86 Figure 5.5: The streaming flow pattern [Eqn.(5.24)] with 𝐷𝐷𝐷𝐷 =7.5. (a) | 𝑀𝑀 |=1.25, (b) | 𝑀𝑀 |=1.6, (c) | 𝑀𝑀 |=2.1. 87 Figure 6.1: Leading order velocity profiles [Eqn.(6.12)] on the 𝜃𝜃 = 𝜋𝜋 4 ⁄ plane over one time period with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=0.1, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=10. 98 Figure 6.2: Leading order shear stress profiles [Eqn.(6.13)] on the equatorial plane over one time period with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=0.1, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=10. 99 Figure 6.3: Leading order velocity profiles [Eqn.(6.12)] on the 𝜃𝜃 = 𝜋𝜋 4 ⁄ plane over one time period for different Darcy number with | 𝑀𝑀 |=5. (a) 𝐷𝐷𝐷𝐷 =0.01, (b) 𝐷𝐷𝐷𝐷 =0.1, (c) 𝐷𝐷𝐷𝐷 =1, (d) 𝐷𝐷𝐷𝐷 ≫1. 100 Figure 6.4: Leading order shear stress profiles [Eqn. (6.13)] on the equatorial plane over one time period for different Darcy number with | 𝑀𝑀 |=5. (a) 𝐷𝐷𝐷𝐷 =0.01, (b) 𝐷𝐷𝐷𝐷 =0.1, (c) 𝐷𝐷𝐷𝐷 =1, (d) 𝐷𝐷𝐷𝐷 ≫1. 101 Figure 6.5: The streaming flow pattern [Eqn.(6.24)] with 𝐷𝐷𝐷𝐷 =0.1. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=15, (c) | 𝑀𝑀 |=50. 102 Figure 6.6: The streaming flow pattern [Eqn.(6.24)] with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=10, (c) | 𝑀𝑀 |=45. 103 Figure 6.7: The streaming flow pattern [Eqn.(6.24)] with 𝐷𝐷𝐷𝐷 =10. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=7.5, (c) | 𝑀𝑀 |=40. 104 vii viii NOMENCLATURE 𝐷𝐷 radius of the container or the sphere in porous medium cases 𝑅𝑅 𝑖𝑖 radius of the inner hemisphere or sphere 𝜔𝜔 angular frequency of oscillation 𝑈𝑈 0 velocity amplitude of oscillation 𝑟𝑟 radial coordinate 𝜃𝜃 polar angle 𝜙𝜙 azimuthal angle 𝑡𝑡 time 𝜈𝜈 𝜌𝜌 kinematic viscosity of the fluid density | 𝑀𝑀 | Womersley number of the flow 𝜀𝜀 perturbation parameter 𝛾𝛾 ratio of boundary radii 𝛼𝛼 ratio of amplitude ix 𝛽𝛽 𝛽𝛽 ’ 𝜂𝜂 phase difference phase lag inner variable 𝑅𝑅 𝑠𝑠 streaming Reynolds number 𝐷𝐷 𝐷𝐷 Darcy number 𝜓𝜓 stream function Ω angular circulation 𝑢𝑢 𝑝𝑝 Velocity pressure 𝜏𝜏 𝜅𝜅 shear stress permeability x ABSTRACT In this dissertation, some fundamental fluid flow problems, due to oscillations in spherical geometry have been studied. It consists of three parts. In the first part, the flow in the annular region between two rigid hemispheres induced by the transverse and torsional oscillations of the inner solid hemispherical boundary has been studied. This work was originally motivated by the need to design an experiment system in which we can effectively apply and control the shear stress and correlate it to the protein aggregation rate. In the second part, we consider combined transverse and torsional oscillations in the annular region between two spheres, as a fundamental development in streaming phenomenon. In the third part, we aim to mathematically model the flow around bubbles in porous media as ultrasound contrast agents in ultrasonography. However, we consider only solid particulates in the present work. We also consider the basic problem of torsional oscillation within a porous fluid-filled sphere. First, the flow properties of fluid between two concentric hemispheres with inner hemisphere undergoing torsional oscillation and transverse oscillation are explored in detail in Chapters 2 and 3 respectively, using perturbation method. The Womersley number | 𝑀𝑀 |, which expresses oscillation inertia forces in relation to the shear forces is introduced to determine the flow, along with perturbation parameter 𝜀𝜀 , which is amplitude of the oscillation in radians, scaled with the oscillation frequency. With mathematical analysis, the analytical solutions for the velocity field, shear rate, and the flow pattern of steady streaming are obtained, which can be applied to unrestricted Womersley number | 𝑀𝑀 | values. In Chapter 4, we consider the same xi system with combined oscillations with phase difference 𝛽𝛽 and amplitude ratio 𝛼𝛼 . The leading order velocity field and shear stress profiles, and the steady streaming are discussed not only for unrestricted | 𝑀𝑀 | values, but also in the low frequency (| 𝑀𝑀 | ≪ 1) and high frequency (| 𝑀𝑀 | ≫ 1) limits. Especially in high frequency limit, the flow field has been divided into three regions, two boundary layers and the outer region. The streaming flow field in determined for the limiting case of the streaming Reynolds number 𝑅𝑅 𝑠𝑠 ≪ 1. In Chapter 5, a mathematical model of flow for fluid through porous medium in a sphere with torsional oscillation is described. Darcy number 𝐷𝐷𝐷𝐷 is defined to represent the relative effect of the permeability 𝜅𝜅 of the medium versus its cross-sectional area. The mathematical analysis of both the oscillatory flow and the steady streaming is performed. Finally, the flow outside a sphere in porous medium due to transverse oscillation is analyzed in Chapter 6. The effects of Darcy number 𝐷𝐷𝐷𝐷 and Womersley number | 𝑀𝑀 | on the flow are provided. xii CHAPTER 1 INTRODUCTION The study of oscillatory flow is of fundamental interest for many biological processes, such as pulsating blood flow, ultrasound visualization, saccadic motion of the eyes, and also potential drug delivery application. In this research, the possibility of using oscillatory flows to protein aggregation has been explored. The aim is to provide analytical solutions for selected hemispherical geometer and apply to protein aggregation. Additionally, the oscillatory flow in a bioporous medium has been studied for ultrasound contrast agents. 1.1 Protein aggregation Protein aggregation is found to be one of the common molecular mechanisms for neurodegenerative diseases, for example, Parkinson’s disease, Huntington’s disease, Alzheimer disease, Creutzfeldt-Jakob disease, to name a few[1]. Usually, proteins can fold into three- dimensional conformation, which is called the native state. This native state is the only functional state of proteins. However, under some circumstances, some proteins may not fold correctly. The misfolding of proteins can lead to aggregation. In this process, molecules of those proteins interact with each other to generate dimers, oligomers, and eventually fibrils. Those insoluble fibrillar deposits can cause neurodegenerative diseases[2]. Protein aggregation is also a significant issue during development of biotherapeutics manufacture[3]. It may occur during several steps in the process of manufacture, such as 1 fermentation, purification, formulation, filling, shipment and storage. Attributed to protein aggregation, the small aggregates may cause an immunogenic reaction, and particulates can result in adverse events on administration. Shear stress can enhance protein aggregation, which has numerous mechanisms. Five common mechanisms of protein aggregation are explained in details by Philo and Arakawa[4]. One mechanism is aggregation of conformationally-altered monomer. For this mechanism, shear stress can trigger the initial conformational change and thus promote first step of aggregation, which is conformational change to a non-native state. For other two mechanisms, nucleation- controlled aggregation and surface-induced aggregation, stress may also induce aggregation, due to conformational stress involved. Therefore, shear rate is a key factor either to determine or to influence the properties of the protein solution when it forms a nucleus or aggregates. Many attempts have been made to understand the mechanisms of protein aggregation. However, the mechanism of protein aggregation is still poorly understood, according to Frieden [5]. Also, in many works of shear effects on protein aggregation [6-8], protein aggregation is actually investigated as a function of stirring speed, instead of shear stress values and distribution in the flow field. In this respect, experimental apparatus in which we can effectively apply shear with predictable flow characteristics that can be calculated and control the aggregation is needed to understand detailed mechanisms of protein aggregation. 1.2 Ultrasound contrast agents Medical ultrasound is a widely used medical imaging modality since it is safe, fast, simple and relatively inexpensive[9]. It is based on the pulse-echo principle[10]. The soundwaves with high 2 frequencies are produced and sent into the body. When meeting different tissues, they are reflected to varying degrees. The echoes are received with information about transmission time and echo intensity, to produce greyscale images. Ultrasound contrast agents are used to enhance echogenicity for better medical image quality. In 1968, Gramiak et al[11, 12] first reported the echocardiographic contrast effect, using saline injection within the lumen of aortic root. They speculated the generated microbubbles of gas are the reason. Since then, the research and use of ultrasound contrast agents has been developed. Generations of the microbubble agents are discussed by Kaul[13]. Researchers have developed various models to study the flow induced by microbubble oscillations responding to the ultrasound pulse, based on early simple models by Rayleigh[14], Plesset[15] and Vokurka[16]. Brujan[17-19] did a series of work regarding liquid compressibility. Brujan[19], Allen and Roy[20, 21] also developed models of non-Newtonian fluid to investigate the blood flow in vessels. The effects of gas diffusion on bubble equilibrium size are theoretically studied by Payvar[22]. The bounded flow has also been investigated to describe flow in vessels, such as flow near a surface[23-27], and within a tube[28-30]. A more detailed review on the models of microbubble dynamics is given by Qin[31]. The permeability for blood flow in vessel[32] has not been considered in the above models will be studied. The flow in porous medium is governed by Darcy equation. While bubble dynamics is an important problem with ultrasound, in the present analysis we shall stay restricted to rigid spheres. The compressibility effects in bubbles bring about additional complexities that we will not address in this phase of the work. 3 1.3 Steady streaming When periodic oscillations are imposed on a body within an incompressible fluid at rest, steady background appears in addition to the oscillatory flow. This type of non-zero time-average flow is referred to as steady streaming. Mathematically, the leading order solution is oscillatory, and the steady non-zero component exists in the higher order terms with higher harmonics due to the existence of the nonlinear terms in Navier-Stokes equation. It was first pointed out by Rayleigh[33] and later studied by Schlichting[34]. Riley[35, 36] reviewed several examples of acoustic streaming, such as the quartz wind introduced by an ultra-high-frequency beam penetrating a fluid, Rayleigh streaming, and torsional oscillations, as well as free surface flow. More recently, Sadhal[37-39], Rednikov and Sadhal[40], Dual et al[41], Bruus[42, 43], Wiklund[44], Wiklund et al[45], and Green et al[46] also reviewed several cases of steady streaming. The principles of acoustic streaming have also been presented by Lighthill[47] and Nyborg[27, 48, 49]. Many researchers have studied steady streaming generated by oscillations. The steady streaming introduced by torsional oscillations of a spherical cell containing a fluid drop was examined by Zapryanov and Chervenivanova[50]. Three standing vortices in every quadrant between the drop and container, as well as steady streaming inside the drop, were found. In connection with saccadic motion of eyes, Repetto and his collaborators [51-55] carried out a series of studies on steady streaming in a spherical and non-spherical chamber undergoing periodic torsional oscillation. They conducted several experiments of vitreous motion induced by saccadic eye movements, compared the experimental results with analytical results from 4 models they proposed, based on the assumption of Newtonian fluid. However, the vitreous is a viscoelastic fluid. Recently, they extended the work, by considering the effects of viscoelastic fluid on steady streaming[56, 57]. The flow generated by small-amplitude torsional oscillation of a sphere in a viscous fluid was presented by Riley[58], Gopinath[59, 60] and Mei[61]. These above -mentioned studies provide descriptions of various cases of steady streaming with torsional and transverse oscillations of a solid sphere and spherical container, corresponding to flows in the regions outside a sphere and within a shell. The steady streaming induced by a cylinder performing different types of oscillation was also studied in several works[62-66], as well as by an oscillating plate[67, 68]. Both the low- and high-frequency limits were examined in those works. Especially, in high-frequency limit[67, 69], a thin Stokes layer (also called shear-wave region/layer) of thickness 𝑂𝑂 � � 𝜈𝜈 𝜔𝜔 �, exists on the boundary. The first order fluctuation vorticity is confined to this Stokes layer. The steady streaming outside the Stokes layer is analyzed by introducing the streaming Reynolds number 𝑅𝑅 𝑠𝑠 under three situations: 𝑅𝑅 𝑠𝑠 ≪ 1 [40, 58, 70, 71], 𝑅𝑅 𝑠𝑠 = 𝑂𝑂 (1) [72-74], and 𝑅𝑅 𝑠𝑠 ≫ 1 [75, 76]. The streaming Reynolds number 𝑅𝑅 𝑠𝑠 was first identified by Stuart[77]. The steady streaming flow generated by combined small-amplitude oscillations have been examined by Kelly[62], Panagopoulos et al[78] and Riley[79] for the case of a circular cylinder and by Gopinath[80] for the case of a sphere. The streaming phenomenon has been the subject of many recent investigations in connection with manipulation microparticle in microstreaming generated by ultrasound on ssessile bubbles[81, 82]. However, little has been considered before about flow induced by combined oscillations for the fluid between two concentric hemispheres, or spheres for that matter. 5 In the present investigation, we have examined the case of the fluid region between two concentric hemispheres/spheres with the inner hemisphere/sphere undergoing torsional and transverse oscillations. Also, the cases of flow in a sphere as porous medium undergoing torsional oscillation, and flow outside a sphere in porous medium are studied. 6 CHAPTER 2 TORSIONAL OSCILLATION OF HEMISPHERE 2.1 Introduction In this chapter we consider flow in a hemispherical bowl with a rounded rod undergoing torsional oscillation. The proposed experiment system is shown in Figure 2.1. In Section 2.2, the model is introduced. The details of deriving and solving the governing equations to get first order solutions and steady streaming are given. In Section 2.3, the theoretical results are discussed. Figure 2.1 Schematic of the proposed experiment. The circular rod can be vibrated rotationally. 7 2.2 Model and theoretical development We consider a hemispherical container of radius 𝐷𝐷 , with a concentric inner hemisphere of radius 𝑅𝑅 𝑖𝑖 ( 𝑅𝑅 𝑖𝑖 < 𝐷𝐷 ) performing torsional oscillations about its axis which is in the 𝑧𝑧 -direction. Let us suppose that the oscillations have frequency of 𝜔𝜔 and velocity amplitude 𝑈𝑈 0 . The spherical coordinates ( 𝑟𝑟 , 𝜃𝜃 , 𝜙𝜙 ) is used to study the motion, with the origin at the center of the hemispheres. The space between the two hemispheres is filled with an incompressible Newtonian fluid. Since the fluid is viscous and the amplitude of the oscillation is very small, we ignore the free surface development. The flow field is given by ∂ 𝒖𝒖 ∂ 𝑡𝑡 + 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 1 𝜌𝜌 𝛁𝛁 𝑝𝑝 + ν ∇ 2 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0, (2.1) with the boundary conditions 𝑟𝑟 = 𝑅𝑅 𝑖𝑖 𝑢𝑢 𝜙𝜙 = 𝑈𝑈 0 sin 𝑡𝑡 sin 𝜃𝜃 , 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 0, 𝑟𝑟 = 𝐷𝐷 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 𝑢𝑢 𝜙𝜙 = 0. (2.2) Now with the scales of the velocities, time, radial distance and pressure chosen as 𝜔𝜔 𝐷𝐷 , 𝜔𝜔 − 1 , 𝐷𝐷 and 𝜇𝜇 𝜔𝜔 , respectively, the following dimensionless quantities are obtained: 𝒖𝒖 ∗ = 𝒖𝒖 𝜔𝜔𝜔𝜔 , 𝑡𝑡 ∗ = 𝜔𝜔 𝑡𝑡 , 𝑟𝑟 ∗ = 𝑟𝑟 𝜔𝜔 , 𝑝𝑝 ∗ = 𝑝𝑝 𝜌𝜌 𝜈𝜈 𝜔𝜔 , 𝛁𝛁 ∗ = 𝐷𝐷 𝛁𝛁 . (2.3) As a result, the dimensionless form of Navier-Stokes and continuity equations, in which the asterisks are dropped, for describing the flow fields are | 𝑀𝑀 | 2 ∂ 𝒖𝒖 ∂ t + | 𝑀𝑀 | 2 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 𝛁𝛁 𝑝𝑝 + ∇ 2 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0, (2.4) 8 together with the boundary conditions 𝑟𝑟 = 𝑅𝑅 i 𝜔𝜔 𝑢𝑢 𝜙𝜙 = 𝜀𝜀 sin 𝑡𝑡 sin 𝜃𝜃 , 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 0, 𝑟𝑟 = 1 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 𝑢𝑢 𝜙𝜙 = 0, (2.5) where | 𝑀𝑀 | = � 𝜔𝜔 𝜔𝜔 2 𝜈𝜈 and 𝜀𝜀 = 𝑈𝑈 0 𝜔𝜔𝜔𝜔 are the Womersley number of the flow and perturbation parameter, respectively. The Womersley number | 𝑀𝑀 | expresses oscillation inertia forces in relation to the shear forces. Normally, the Womersley number values in regular vessels of man and dog are smaller than 20 [83]. The perturbation parameter 𝜀𝜀 we introduced is the amplitude of the oscillation (𝑈𝑈 0 /𝐷𝐷 ) in radians, scaled with the oscillation frequency. These two dimensionless parameters determine the flow. Taking 𝜀𝜀 as a small parameter, the perturbation method is applicable and the following form of solution is written: 𝑝𝑝 = 𝜀𝜀 𝑝𝑝 1 + 𝜀𝜀 2 𝑝𝑝 2 + 𝑂𝑂 (𝜀𝜀 3 ) , 𝒖𝒖 = 𝜀𝜀 𝒖𝒖 1 + 𝜀𝜀 2 𝒖𝒖 2 + 𝑂𝑂 (𝜀𝜀 3 ) . (2.6) The problem has axisymmetric solutions. 2.2.1 The first order solutions According to the homogenous boundary conditions of 𝑢𝑢 1 𝑟𝑟 and 𝑢𝑢 1 𝜃𝜃 , 𝑢𝑢 1 𝑟𝑟 = 𝑢𝑢 1 𝜃𝜃 = 0 is satisfied everywhere. The azimuthal velocity component 𝑢𝑢 1 𝜙𝜙 satisfies the set ∂ 𝑢𝑢 1 𝜙𝜙 ∂ 𝑡𝑡 = 1 | 𝑀𝑀 | 2 � 1 𝑟𝑟 2 ∂ ∂ 𝑟𝑟 �𝑟𝑟 2 ∂ 𝑢𝑢 1 𝜙𝜙 ∂ 𝑟𝑟 � + 1 𝑟𝑟 2 sin 𝜃𝜃 ∂ ∂ 𝜃𝜃 �sin 𝜃𝜃 ∂ 𝑢𝑢 1 𝜙𝜙 ∂ 𝜃𝜃 � − 𝑢𝑢 1 𝜙𝜙 𝑟𝑟 2 sin 2 𝜃𝜃 � , (2.7) 𝑟𝑟 = 𝑅𝑅 𝑖𝑖 𝜔𝜔 = 𝛾𝛾 𝑢𝑢 1 𝜙𝜙 = sin 𝑡𝑡 sin 𝜃𝜃 , 𝑟𝑟 = 1 𝑢𝑢 1 𝜙𝜙 = 0. (2.8) The solution is 𝑢𝑢 1 𝜙𝜙 = 𝑔𝑔 1 (𝑟𝑟 )𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. , 9 𝑔𝑔 1 ( 𝑟𝑟 ) = − 𝑖𝑖 2 𝛾𝛾 2 𝑟𝑟 2 ( cos 𝑘𝑘 + 𝑘𝑘 sin 𝑘𝑘 )( sin 𝑘𝑘 𝑟𝑟 − 𝑘𝑘 𝑟𝑟 cos 𝑘𝑘 𝑟𝑟 ) −(sin 𝑘𝑘 − 𝑘𝑘 cos 𝑘𝑘 )(cos 𝑘𝑘 𝑟𝑟 + 𝑘𝑘 𝑟𝑟 sin 𝑘𝑘 𝑟𝑟 ) ( cos 𝑘𝑘 + 𝑘𝑘 sin 𝑘𝑘 )( sin 𝑘𝑘 𝛾𝛾 − 𝑘𝑘 𝛾𝛾 cos 𝑘𝑘 𝛾𝛾 ) −(sin 𝑘𝑘 − 𝑘𝑘 cos 𝑘𝑘 )(cos 𝑘𝑘 𝛾𝛾 + 𝑘𝑘 𝛾𝛾 sin 𝑘𝑘 𝛾𝛾 ) , (2.9) where 𝑘𝑘 = | 𝑀𝑀 | 𝑒𝑒 − 𝑖𝑖 𝜋𝜋 4 , and c.c. denotes the complex conjugate. Using the following expressions 𝜏𝜏 𝑟𝑟 𝜃𝜃 = 𝜌𝜌 𝜈𝜈 �𝑟𝑟 𝜕𝜕 𝜕𝜕 𝑟𝑟 � 𝑢𝑢 𝜃𝜃 𝑟𝑟 � + 1 𝑟𝑟 𝜕𝜕 𝑢𝑢 𝑟𝑟 𝜕𝜕 𝜃𝜃 � , 𝜏𝜏 𝜙𝜙 𝜃𝜃 = 𝜌𝜌 𝜈𝜈 � sin 𝜃𝜃 𝑟𝑟 ∂ ∂ 𝜃𝜃 � 𝑢𝑢 𝜙𝜙 sin 𝜃𝜃 � + 1 𝑟𝑟 sin 𝜃𝜃 ∂ 𝑢𝑢 𝜃𝜃 ∂ 𝜙𝜙 � , 𝜏𝜏 𝑟𝑟 𝜙𝜙 = 𝜌𝜌 𝜈𝜈 �𝑟𝑟 𝜕𝜕 𝜕𝜕 𝑟𝑟 � 𝑢𝑢 𝜙𝜙 𝑟𝑟 � + 1 𝑟𝑟 sin 𝜃𝜃 ∂ 𝑢𝑢 𝑟𝑟 ∂ 𝜙𝜙 � , (2.10) the shear stress is given in terms of dimensionless quantities as: 𝜏𝜏 1 𝑟𝑟 𝜃𝜃 = 𝜏𝜏 1 𝜙𝜙 𝜃𝜃 = 0, 𝜏𝜏 1 𝑟𝑟 𝜙𝜙 = � 𝐴𝐴 �−3 sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 2 𝑟𝑟 3 + 3 cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 𝑟𝑟 2 + sin 𝑘𝑘 𝑟𝑟 𝑟𝑟 � + 𝐵𝐵 �3 cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 2 𝑟𝑟 3 + 3 sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 𝑟𝑟 2 − cos 𝑘𝑘 𝑟𝑟 𝑟𝑟 �� 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. , (2.11) where 𝐴𝐴 = − 𝑖𝑖 2 1 � s in 𝑘𝑘 𝛾𝛾 𝑘𝑘 2 𝛾𝛾 2 − cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 𝛾𝛾 � + s in 𝑘𝑘 − 𝑘𝑘 cos 𝑘𝑘 cos 𝑘𝑘 + 𝑘𝑘 s in 𝑘𝑘 � − cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 2 𝛾𝛾 2 − s in 𝑘𝑘 𝛾𝛾 𝑘𝑘 𝛾𝛾 � , 𝐵𝐵 = − 𝑖𝑖 2 1 cos 𝑘𝑘 + 𝑘𝑘 s in 𝑘𝑘 s in 𝑘𝑘 − 𝑘𝑘 cos 𝑘𝑘 � s in 𝑘𝑘 𝛾𝛾 𝑘𝑘 2 𝛾𝛾 2 − cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 𝛾𝛾 � + � − cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 2 𝛾𝛾 2 − sin 𝑘𝑘 𝛾𝛾 𝑘𝑘 𝛾𝛾 � . (2.12) 2.2.2 The second order solutions The second order solution is much more complex than the first order. Also, we are more interested in the steady streaming. So we separate the time-independent part from second- order time-dependent variables. 𝒖𝒖 2 = 𝒖𝒖 2 0 + � 𝒖𝒖 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. �, 𝑝𝑝 2 = 𝑝𝑝 2 0 + { 𝑝𝑝 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c.}, 𝒖𝒖 1 ∙ 𝛁𝛁 𝒖𝒖 1 = 𝓕𝓕 0 + { 𝓕𝓕 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c.}, (2.13) 10 Note that 𝒖𝒖 2 0 , 𝑝𝑝 2 0 and 𝓕𝓕 0 are steady variables. The steady component of order 𝜀𝜀 2 takes the forms ∇ 2 𝒖𝒖 2 0 − 𝛁𝛁 𝑝𝑝 2 0 = | 𝑀𝑀 | 2 𝓕𝓕 0 , 𝛁𝛁 ∙ 𝒖𝒖 2 0 = 0, (2.14) with the homogenous boundary conditions 𝑟𝑟 = 𝛾𝛾 𝑢𝑢 2 0 𝑟𝑟 = 𝑢𝑢 2 0 𝜃𝜃 = 𝑢𝑢 2 0 𝜙𝜙 = 0, 𝑟𝑟 = 1 𝑢𝑢 2 0 𝑟𝑟 = 𝑢𝑢 2 0 𝜃𝜃 = 𝑢𝑢 2 0 𝜙𝜙 = 0. (2.15) According to Quartapelle and Verri[84], the vector spherical harmonics can be used to expand the pressure and velocity fields, which are defined through the relationships 𝑷𝑷 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 ) = 𝒓𝒓 �( 𝜃𝜃 , 𝜙𝜙 ) 𝑌𝑌 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 ) (𝑛𝑛 ≥ 0), 𝑩𝑩 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 ) = 1 𝑠𝑠 𝑛𝑛 𝑟𝑟 𝛁𝛁 𝑌𝑌 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 ) (𝑛𝑛 > 0), 𝑪𝑪 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 ) = 1 𝑠𝑠 𝑛𝑛 𝛁𝛁 × [𝒓𝒓 ( 𝜃𝜃 , 𝜙𝜙 ) 𝑌𝑌 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 )] (𝑛𝑛 > 0), (2.16) where 𝒓𝒓 �( 𝜃𝜃 , 𝜙𝜙 ) is the radial unit vector, 𝑠𝑠 𝑛𝑛 = � 𝑛𝑛 (𝑛𝑛 + 1), and functions 𝑌𝑌 𝑛𝑛 𝑚𝑚 ( 𝜃𝜃 , 𝜙𝜙 ), 𝑛𝑛 ≥ 0,− 𝑛𝑛 ≤ 𝑚𝑚 ≤ 𝑛𝑛 are orthonormal spherical harmonics. The basis of vector harmonics has orthonomality over the unit sphere. For 𝑚𝑚 = 0, 𝑌𝑌 𝑛𝑛 0 ( 𝜃𝜃 , 𝜙𝜙 ) = � 2 𝑛𝑛 + 1 4 𝜋𝜋 𝑃𝑃 𝑛𝑛 (cos 𝜃𝜃 ), 𝑷𝑷 𝑛𝑛 0 ( 𝜃𝜃 , 𝜙𝜙 ) = � 2 𝑛𝑛 + 1 4 𝜋𝜋 𝑃𝑃 𝑛𝑛 (cos 𝜃𝜃 )𝒓𝒓 �, 𝑩𝑩 𝑛𝑛 0 ( 𝜃𝜃 , 𝜙𝜙 ) = 1 𝑠𝑠 𝑛𝑛 ∂ 𝑌𝑌 𝑛𝑛 0 ∂ 𝜃𝜃 𝜽𝜽 � , 𝑪𝑪 𝑛𝑛 0 ( 𝜃𝜃 , 𝜙𝜙 ) = − 1 𝑠𝑠 𝑛𝑛 ∂ 𝑌𝑌 𝑛𝑛 0 ∂ 𝜃𝜃 𝝓𝝓 � , (2.17) where 𝑃𝑃 𝑛𝑛 (cos 𝜃𝜃 ) denotes the nth Legendre Polynomial. 11 Upon the application of orthonormality on 𝓕𝓕 0 = − 2 𝑔𝑔 1 ( 𝑟𝑟 ) 𝑔𝑔 1 ( 𝑟𝑟 ) � � � � � � � � 𝑟𝑟 �sin 2 𝜃𝜃 𝒓𝒓 � + sin 𝜃𝜃 cos 𝜃𝜃 𝜽𝜽 � �, (2.18) the following is obtained 𝓕𝓕 0 = ℱ 𝑃𝑃 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝑃𝑃 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝐵𝐵 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 ( 𝜃𝜃 , 𝜙𝜙 ), (2.19) where ℱ 𝑃𝑃 0 = − 8 3 √ 𝜋𝜋 𝑔𝑔 1 ( 𝑟𝑟 ) 𝑔𝑔 1 ( 𝑟𝑟 ) � � � � � � � � 𝑟𝑟 , ℱ 𝑃𝑃 2 = 8 1 5 √5 𝜋𝜋 𝑔𝑔 1 ( 𝑟𝑟 ) 𝑔𝑔 1 ( 𝑟𝑟 ) � � � � � � � � 𝑟𝑟 , ℱ 𝐵𝐵 2 = 4 1 5 √30 𝜋𝜋 𝑔𝑔 1 ( 𝑟𝑟 ) 𝑔𝑔 1 ( 𝑟𝑟 ) � � � � � � � � 𝑟𝑟 . (2.20) It is reasonable to write the pressure 𝑝𝑝 2 0 and velocity 𝒖𝒖 2 0 = (𝑢𝑢 2 0 𝑟𝑟 , 𝑢𝑢 2 0 𝜃𝜃 , 𝑢𝑢 2 0 𝜙𝜙 ) in the form 𝒖𝒖 2 0 = 𝑢𝑢 2 0 𝑟𝑟 , 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 + 𝑢𝑢 2 0 𝑟𝑟 , 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 + 𝑢𝑢 2 0 𝜃𝜃 , 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 , (2.21) 𝑝𝑝 2 0 = 𝑝𝑝 2 0, 0 𝑌𝑌 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑝𝑝 2 0, 2 𝑌𝑌 2 0 ( 𝜃𝜃 , 𝜙𝜙 ). (2.22) The resulting differential equations to order 𝜀𝜀 2 become d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 0 − 2 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 − d d 𝑟𝑟 𝑝𝑝 2 0, 0 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 , 1 𝑟𝑟 2 d d 𝑟𝑟 � 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 � = 0, d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − 8 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 − d d 𝑟𝑟 𝑝𝑝 2 0, 2 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 2 , d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 − 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑝𝑝 2 0, 2 = | 𝑀𝑀 | 2 ℱ 𝐵𝐵 2 , d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 = 0. (2.23) For the homogenous boundary conditions, the solution is found to be 𝑢𝑢 2 0 𝑟𝑟 , 0 = 0, 12 𝑝𝑝 2 0, 0 = 𝑃𝑃 2 0 + ∫ −| 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 ( 𝑟𝑟 ′ )d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 , 𝑢𝑢 2 0 𝑟𝑟 , 2 = 𝑐𝑐 1 𝑟𝑟 + 𝑐𝑐 2 1 𝑟𝑟 2 + 𝑐𝑐 3 𝑟𝑟 3 + 𝑐𝑐 4 1 𝑟𝑟 4 + 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 2 + 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 3 + 𝐻𝐻 4 (𝑟𝑟 ) 1 𝑟𝑟 4 , 𝑢𝑢 2 0 𝜃𝜃 , 2 = 1 √ 6 �3 𝑐𝑐 1 𝑟𝑟 + 5 𝑐𝑐 3 𝑟𝑟 3 − 𝑐𝑐 4 2 𝑟𝑟 4 + 3 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 5 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 3 − 𝐻𝐻 4 ( 𝑟𝑟 ) 2 𝑟𝑟 4 � , (2.24) where 𝑐𝑐 1 𝐷𝐷 = 2 𝛾𝛾 6 (5 𝐴𝐴 1 − 𝐴𝐴 2 ) − 5 𝛾𝛾 4 (2 𝐴𝐴 3 + 𝐴𝐴 4 ) + 7 𝐴𝐴 4 𝛾𝛾 2 + 7 𝐴𝐴 2 𝛾𝛾 − 5 𝛾𝛾 (2 𝐴𝐴 1 + 𝐴𝐴 2 ) + 2 𝛾𝛾 3 (5 𝐴𝐴 3 − 𝐴𝐴 4 ) , 𝑐𝑐 2 𝐷𝐷 = 2 𝛾𝛾 4 (2 𝐴𝐴 3 + 𝐴𝐴 4 ) − 5 𝛾𝛾 3 (5 𝐴𝐴 1 − 𝐴𝐴 2 ) + 7 𝛾𝛾 (3 𝐴𝐴 1 − 𝐴𝐴 2 ) + 3 𝛾𝛾 (7 𝐴𝐴 3 − 𝐴𝐴 4 ) − 5 𝛾𝛾 3 (5 𝐴𝐴 3 − 𝐴𝐴 4 ) + 2 𝛾𝛾 4 (2 𝐴𝐴 1 + 𝐴𝐴 2 ) , 𝑐𝑐 3 𝐷𝐷 = − 2 𝛾𝛾 6 (3 𝐴𝐴 1 − 𝐴𝐴 2 ) + 3 𝛾𝛾 4 (2 𝐴𝐴 3 + 𝐴𝐴 4 ) − 5 𝐴𝐴 2 𝛾𝛾 3 − 5 𝐴𝐴 4 𝛾𝛾 2 + 3 𝛾𝛾 (2 𝐴𝐴 1 + 𝐴𝐴 2 ) − 2 𝛾𝛾 (3 𝐴𝐴 3 − 𝐴𝐴 4 ) , 𝑐𝑐 4 𝐷𝐷 = − 2 𝐴𝐴 4 𝛾𝛾 2 + 3 𝛾𝛾 (5 𝐴𝐴 1 − 𝐴𝐴 2 ) − 5 𝛾𝛾 (3 𝐴𝐴 1 − 𝐴𝐴 2 + 3 𝐴𝐴 3 − 𝐴𝐴 4 ) + 5 𝛾𝛾 3 (3 𝐴𝐴 3 − 𝐴𝐴 4 ) − 2 𝛾𝛾 4 𝐴𝐴 2 , (2.25) with 𝐷𝐷 = 4 𝛾𝛾 6 − 2 5 𝛾𝛾 3 + 4 2 𝛾𝛾 − 25 𝛾𝛾 + 4 𝛾𝛾 4 , 𝐴𝐴 1 = − 𝐻𝐻 1 (1) − 𝐻𝐻 2 (1) − 𝐻𝐻 3 (1) − 𝐻𝐻 4 (1) , 𝐴𝐴 2 = −3 𝐻𝐻 1 (1) − 5 𝐻𝐻 3 (1) + 2 𝐻𝐻 4 (1) , 𝐴𝐴 3 = − 𝛾𝛾 𝐻𝐻 1 ( 𝛾𝛾 ) − 1 𝛾𝛾 2 𝐻𝐻 2 ( 𝛾𝛾 ) − 𝛾𝛾 3 𝐻𝐻 3 ( 𝛾𝛾 ) − 1 𝛾𝛾 4 𝐻𝐻 4 ( 𝛾𝛾 ) , 𝐴𝐴 4 = −3 𝛾𝛾 𝐻𝐻 1 ( 𝛾𝛾 ) − 5 𝛾𝛾 3 𝐻𝐻 3 ( 𝛾𝛾 ) + 2 𝛾𝛾 4 𝐻𝐻 4 ( 𝛾𝛾 ) , 𝐻𝐻 1 = − | 𝑀𝑀 | 2 1 0 �[ 𝑟𝑟 ′ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 2 ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ )d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 2 = | 𝑀𝑀 | 2 1 0 �[𝑟𝑟 ′ 4 ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 5 ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 3 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 3 = 3| 𝑀𝑀 | 2 7 0 � ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ � 𝛾𝛾 𝑟𝑟 , 𝐻𝐻 4 = − 3| 𝑀𝑀 | 2 7 0 �[𝑟𝑟 ′ 6 ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 7 ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 5 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � . (2.26) 13 As a result, the velocity has the following expression 𝒖𝒖 2 0 = 1 4 � 5 𝜋𝜋 � 𝑢𝑢 2 0 𝑟𝑟 , 2 (3 cos 2 𝜃𝜃 − 1), − √6 𝑢𝑢 2 0 𝜃𝜃 , 2 cos 𝜃𝜃 sin 𝜃𝜃 , 0 � . (2.27) The stream function takes the form 𝜓𝜓 2 0 = 1 4 � 5 𝜋𝜋 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 sin 2 𝜃𝜃 cos 𝜃𝜃 . (2.28) The second order shear stress induced by steady streaming is 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 = � − 1 4 � 5 𝜋𝜋 �6 𝑐𝑐 1 + 6 𝑐𝑐 2 1 𝑟𝑟 3 + 16 𝑐𝑐 3 𝑟𝑟 2 + 16 𝑐𝑐 4 1 𝑟𝑟 5 + 6 𝐻𝐻 1 ( 𝑟𝑟 ) + 6 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 3 + 16 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 2 + 16 𝐻𝐻 4 ( 𝑟𝑟 ) 1 𝑟𝑟 5 + 3 𝐻𝐻 1 ′ ( 𝑟𝑟 ) 𝑟𝑟 + 5 𝐻𝐻 3 ′ ( 𝑟𝑟 ) 𝑟𝑟 3 − 2 𝐻𝐻 4 ′ (𝑟𝑟 ) 1 𝑟𝑟 4 �� cos 𝜃𝜃 sin 𝜃𝜃 . (2.29) According to Eqn. (2.29), 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 ( 𝑟𝑟 , 𝜃𝜃 ) of same radius reaches its peak value at 𝜃𝜃 = 𝜋𝜋 4 , and equals to 0 on the equatorial plane ( 𝜃𝜃 = 𝜋𝜋 2 ) and the oscillation axis ( 𝜃𝜃 = 0). The value of 𝜀𝜀 2 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 makes little contribution to shear stress compared to 𝜀𝜀 𝜏𝜏 1 𝑟𝑟 𝜃𝜃 under same Womersley number | 𝑀𝑀 |, due to small values of 𝜀𝜀 . 2.3 Theoretical results and discussion The shear stress profiles in the torsional oscillation case under different | 𝑀𝑀 | values are shown in Figure 2.2. The maximum shear stress appears on the inner boundary, and increases as | 𝑀𝑀 | goes up. While on the outer boundary, shear stress decreases with increasing | 𝑀𝑀 |. As | 𝑀𝑀 | rises to 20, shear stress close to inner boundary has a dramatic decrease and vanishes around 𝑟𝑟 = 0.7. The shear stress to first order has maximum value on equatorial plane, according to Eqn. (2.11). Also, shear stress is proportional to sin 𝑡𝑡 , same as oscillation velocity; so the shear 14 stress is maximum as oscillation velocity reaches its maximum value, when 𝑡𝑡 = �𝑛𝑛 + 1 2 � 𝜋𝜋 and 𝑛𝑛 = 0,1. The steady component of the flow (streaming) arises from the nonlinear terms in the Navier- Stokes equation. The steady streaming described by Eqn. (2.28) shows that the character of the streaming depends on the Womersley number | 𝑀𝑀 | = � 𝜔𝜔 𝐷𝐷 2 /𝜈𝜈 , and therefore on the frequency. The streamlines in the torsional oscillation case under different | 𝑀𝑀 | values have been plotted in Figure 2.3. The arrows present the steady component of second order velocity 𝒖𝒖 2 0 . Increasing | 𝑀𝑀 | doesn’t change the steady streaming flow pattern. Under both low and high | 𝑀𝑀 |, the region is filled with a standing vortex, which is directed counterclockwise. The relation between streaming intensity and Womersley number | 𝑀𝑀 | is compared in Figure 2.4. The absolute values of the maximum streaming velocity | 𝑢𝑢 2 0 | 𝑚𝑚 under different polar angle 𝜃𝜃 are provided to present the streaming intensity. Using Taylor expansion, when | 𝑀𝑀 | ≪ 1, 𝑔𝑔 1 ( 𝑟𝑟 ) in Eqn. (2.9) can be written as, 𝑔𝑔 1 ( 𝑟𝑟 ) = − 𝑖𝑖 2 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � �𝑟𝑟 − 1 𝑟𝑟 2 � + 1 2 0 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � | 𝑀𝑀 | 2 �𝑟𝑟 3 − 5 𝑟𝑟 + 5 − 1 𝑟𝑟 2 � + 𝑂𝑂 (| 𝑀𝑀 | 4 ) . (2.30) To obtain the steady streaming, expressions for the terms in Eqs. (2.20), (2.25) and (2.26) are in the form as follows, ℱ 𝑃𝑃 2 = 2 1 5 √5 𝜋𝜋 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 �𝑟𝑟 − 2 𝑟𝑟 2 − 1 𝑟𝑟 5 � + 2 1 5 0 0 √5 𝜋𝜋 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 | 𝑀𝑀 | 4 �𝑟𝑟 5 − 10 𝑟𝑟 3 + 10 𝑟𝑟 2 + 25 𝑟𝑟 − 52 + 2 5 𝑟𝑟 + 1 0 𝑟𝑟 2 − 1 0 𝑟𝑟 3 + 1 𝑟𝑟 5 � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝐻𝐻 1 = √ 5 𝜋𝜋 5 0 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 | 𝑀𝑀 | 2 �4 � 1 𝑟𝑟 − 1 𝛾𝛾 � + � 1 𝑟𝑟 4 − 1 𝛾𝛾 4 �� + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝐻𝐻 2 = √ 5 𝜋𝜋 2 5 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 | 𝑀𝑀 | 2 �( 𝑟𝑟 2 − 𝛾𝛾 2 ) − 2 � 1 𝑟𝑟 − 1 𝛾𝛾 �� + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 15 𝐻𝐻 3 = − √ 5 𝜋𝜋 1 7 5 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 | 𝑀𝑀 | 2 �2 � 1 𝑟𝑟 3 − 1 𝛾𝛾 3 � + � 1 𝑟𝑟 6 − 1 𝛾𝛾 6 �� + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝐻𝐻 4 = − 3 √ 5 𝜋𝜋 3 5 0 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 | 𝑀𝑀 | 2 [( 𝑟𝑟 4 − 𝛾𝛾 4 ) − 4( 𝑟𝑟 − 𝛾𝛾 )] + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝑐𝑐 1 𝐷𝐷 = − | 𝑀𝑀 | 2 1 0 � 2 1 5 √5 𝜋𝜋 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 �− 6 𝛾𝛾 1 0 + 5 1 𝛾𝛾 7 − 4 5 𝛾𝛾 6 − 2 1 2 𝛾𝛾 5 + 7 5 2 𝛾𝛾 3 − 1 4 7 𝛾𝛾 2 + 3 1 5 2 𝛾𝛾 + 54 − 2 2 5 2 𝛾𝛾 + 21 𝛾𝛾 3 �� + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝑐𝑐 2 𝐷𝐷 = − | 𝑀𝑀 | 2 1 0 � 2 1 5 √5 𝜋𝜋 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 � 9 𝛾𝛾 7 − 1 0 2 𝛾𝛾 4 + 2 2 5 2 𝛾𝛾 3 + 4 2 𝛾𝛾 2 + 1 2 3 2 𝛾𝛾 + 75 − 1 4 7 2 𝛾𝛾 − 2 7 2 𝛾𝛾 3 + 45 𝛾𝛾 4 − 12 𝛾𝛾 6 �� + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝑐𝑐 3 𝐷𝐷 = − | 𝑀𝑀 | 2 1 0 � 2 1 5 √5 𝜋𝜋 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 � 1 2 7 𝛾𝛾 1 2 − 5 1 7 𝛾𝛾 9 − 9 2 𝛾𝛾 7 − 1 2 3 7 𝛾𝛾 6 − 6 7 5 1 4 𝛾𝛾 5 + 1 2 6 𝛾𝛾 4 − 2 2 5 2 𝛾𝛾 3 − 2 4 3 7 𝛾𝛾 2 + 1 3 5 2 𝛾𝛾 − 8 1 7 𝛾𝛾 � � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝑐𝑐 4 𝐷𝐷 = − | 𝑀𝑀 | 2 1 0 � 2 1 5 √5 𝜋𝜋 � 𝛾𝛾 2 𝛾𝛾 3 − 1 � 2 �− 3 3 7 𝛾𝛾 5 + 4 0 8 7 𝛾𝛾 2 − 1 3 5 2 𝛾𝛾 − 27 − 6 4 5 1 4 𝛾𝛾 − 4 5 0 7 𝛾𝛾 2 + 1 8 9 2 𝛾𝛾 3 − 45 𝛾𝛾 4 − 8 1 1 4 𝛾𝛾 5 + 1 8 7 𝛾𝛾 8 �� + 𝑂𝑂 (| 𝑀𝑀 | 6 ) . (2.31) Then, take ratio of boundary radii γ = 0.5 for instance. 𝑢𝑢 2 0 𝑟𝑟 , 2 = 2 √ 5 𝜋𝜋 1 2 5 1 9 5 | 𝑀𝑀 | 2 1 𝑟𝑟 4 (1 − 𝑟𝑟 ) 2 (1 − 2 𝑟𝑟 ) 2 (121 + 215 𝑟𝑟 + 237 𝑟𝑟 2 + 79 𝑟𝑟 3 ) + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝑢𝑢 2 0 𝜃𝜃 , 2 = √ 3 0 𝜋𝜋 3 7 5 5 8 5 | 𝑀𝑀 | 2 1 𝑟𝑟 4 (1 − 𝑟𝑟 )(1 − 2 𝑟𝑟 )( −242 − 215 𝑟𝑟 − 161 𝑟𝑟 2 − 53 𝑟𝑟 3 + 1185 𝑟𝑟 4 + 790 𝑟𝑟 5 ) + 𝑂𝑂 (| 𝑀𝑀 | 6 ) . (2.32) Therefore, with the analytical calculation, the steady streaming velocity and stream function are found to be 𝑢𝑢 2 0 = | 𝑀𝑀 | 2 5 0 0 7 8 � 1 𝑟𝑟 4 (1 − 𝑟𝑟 ) 2 (1 − 2 𝑟𝑟 ) 2 (121 + 215 𝑟𝑟 + 237 𝑟𝑟 2 + 79 𝑟𝑟 3 )(3 cos 2 𝜃𝜃 − 1), − 1 𝑟𝑟 4 (1 − 𝑟𝑟 )(1 − 2 𝑟𝑟 )( −242 − 215 𝑟𝑟 − 161 𝑟𝑟 2 − 53 𝑟𝑟 3 + 1185 𝑟𝑟 4 + 790 𝑟𝑟 5 ) cos 𝜃𝜃 sin 𝜃𝜃 , 0 � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 16 𝜓𝜓 2 0 = | 𝑀𝑀 | 2 5 0 0 7 8 𝑟𝑟 2 (1 − 𝑟𝑟 ) 2 (1 − 2 𝑟𝑟 ) 2 (121 + 215 𝑟𝑟 + 237 𝑟𝑟 2 + 79 𝑟𝑟 3 ) sin 2 𝜃𝜃 cos 𝜃𝜃 + 𝑂𝑂 (| 𝑀𝑀 | 6 ) . (2.33) In the limit of small Womersley number | 𝑀𝑀 |, the intensity of the steady streaming is proportional to | 𝑀𝑀 | 2 for torsional oscillation case in our paper. Note this is different from the case of a rotating sphere completely filled with fluid, in which intensity of steady streaming is proportional to | 𝑀𝑀 | 6 for small | 𝑀𝑀 | values. 17 Figure 2.2 First order shear stress profiles [Eqn. (2.11)] on the equatorial plane over one time period for the torsional oscillation with 𝛾𝛾 =0.5. The dashed curve shows the time average of absolute shear stress values over one period. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=5, (c) | 𝑀𝑀 |=20. 18 Figure 2.3 The streaming flow pattern for torsional oscillation with 𝛾𝛾 =0.5. (a)| 𝑀𝑀 |=1, (b)| 𝑀𝑀 |=20. 19 Figure 2.4 Streaming intensity as a function of Womersley number | 𝑀𝑀 | at various polar angle 𝜃𝜃 for the torsional oscillation with 𝛾𝛾 =0.5. 20 CHAPTER 3 TRANSVERSE OSCILLATION OF HEMISPHERE 3.1 Introduction In this chapter we consider flow in a hemispherical bowl with a rounded rod undergoing transverse oscillation. The proposed experiment system is shown in Figure 3.1. In Section 3.2, the model is introduced. The details of deriving and solving the governing equations to get first order solutions and steady streaming are given. In Section 3.3, the theoretical results are discussed. Figure 3.1 Schematic of the proposed experiment. The circular rod can be vibrated laterally. 21 3.2 Model and theoretical development A hemisphere of radius 𝑅𝑅 𝑖𝑖 moving along the centerline, taken to be 𝑧𝑧 -axis, at angular frequency 𝜔𝜔 and velocity amplitude 𝑈𝑈 0 is considered. This hemisphere is placed in a hemispherical container. The two hemispheres are concentric, and the container radius 𝐷𝐷 is assumed to be larger than that of the oscillating hemisphere. The equations of motion are as follows: ∂ 𝒖𝒖 ∂ 𝑡𝑡 + 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 1 𝜌𝜌 𝛁𝛁 𝑝𝑝 + ν ∇ 2 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0, (3.1) with boundary conditions 𝑟𝑟 = 𝑅𝑅 𝑖𝑖 𝑢𝑢 𝑟𝑟 = 𝑈𝑈 0 𝑠𝑠𝑠𝑠 𝑛𝑛 ( 𝜔𝜔 𝑡𝑡 ) 𝑐𝑐𝑐𝑐 𝑠𝑠𝜃𝜃 , 𝑢𝑢 𝜃𝜃 = − 𝑈𝑈 0 𝑠𝑠𝑠𝑠 𝑛𝑛 ( 𝜔𝜔 𝑡𝑡 ) 𝑠𝑠𝑠𝑠 𝑛𝑛 𝜃𝜃 , 𝑢𝑢 𝜙𝜙 = 0, 𝑟𝑟 = 𝐷𝐷 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 𝑢𝑢 𝜙𝜙 = 0. (3.2) With the same scaling adopted as torsional oscillation in Chapter 2: 𝒖𝒖 ∗ = 𝒖𝒖 𝜔𝜔𝜔𝜔 , 𝑡𝑡 ∗ = 𝜔𝜔 𝑡𝑡 , 𝑟𝑟 ∗ = 𝑟𝑟 𝜔𝜔 , 𝑝𝑝 ∗ = 𝑝𝑝 𝜌𝜌 𝜈𝜈 𝜔𝜔 , 𝛁𝛁 ∗ = 𝐷𝐷 𝛁𝛁 , (3.3) same dimensionless governing equation is obtained, | 𝑀𝑀 | 2 ∂ 𝒖𝒖 ∂ t + | 𝑀𝑀 | 2 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 𝛁𝛁 𝑝𝑝 + ∇ 2 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0, (3.4) but with different boundary conditions 𝑟𝑟 = 𝛾𝛾 𝑢𝑢 𝑟𝑟 = 𝜀𝜀 sin 𝑡𝑡 cos𝜃𝜃 , 𝑢𝑢 𝜃𝜃 = − 𝜀𝜀 sin 𝑡𝑡 sin 𝜃𝜃 , 𝑢𝑢 𝜙𝜙 = 0, 𝑟𝑟 = 1 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 𝑢𝑢 𝜙𝜙 = 0. (3.5) 22 3.2.1 The first order solutions Applying the perturbation method as Eqn. (2.6) and then, to order 𝜀𝜀 , the flow can be described by the following equation: | 𝑀𝑀 | 2 ∂ 𝑢𝑢 1 ∂ 𝑡𝑡 = − 𝛁𝛁 𝑝𝑝 1 + ∇ 2 𝑢𝑢 1 , 𝛁𝛁 ∙ 𝒖𝒖 = 0 (3.6) with the boundary conditions 𝑟𝑟 = 𝛾𝛾 𝑢𝑢 1 𝑟𝑟 = sin 𝑡𝑡 cos𝜃𝜃 , 𝑢𝑢 1 𝜃𝜃 = −sin 𝑡𝑡 sin 𝜃𝜃 , 𝑢𝑢 1 𝜙𝜙 = 0, 𝑟𝑟 = 1 𝑢𝑢 1 𝑟𝑟 = 𝑢𝑢 1 𝜃𝜃 = 𝑢𝑢 1 𝜙𝜙 = 0. (3.7) For this axisymmetric flow, 𝑢𝑢 1 𝜙𝜙 = 0 and the Stokes steam function can be introduced as 𝑢𝑢 1 = 𝛁𝛁 × � 𝜓𝜓 1 𝑟𝑟 sin 𝜃𝜃 𝝓𝝓 � � . (3.8) Eliminating the pressure term in first order equation, it is obtained that �𝐿𝐿 − 1 − | 𝑀𝑀 | 2 ∂ ∂ 𝑡𝑡 � 𝐿𝐿 − 1 ( 𝜓𝜓 1 ) = 0, (3.9) where 𝐿𝐿 − 1 = ∂ 2 ∂ 𝑟𝑟 2 + sin 𝜃𝜃 𝑟𝑟 2 ∂ ∂ 𝜃𝜃 � 1 sin 𝜃𝜃 ∂ ∂ 𝜃𝜃 � denotes the Stokes operator. The solution is assumed to be an eigenfunction expansion in Gegenbauer polynomials as 𝜓𝜓 1 ( 𝑟𝑟 , 𝜃𝜃 ) = ∑ 𝜓𝜓 1 𝑛𝑛 (𝑟𝑟 , 𝑡𝑡 )𝐶𝐶 𝑛𝑛 + 1 − 1 2 (cos 𝜃𝜃 ) ∞ 𝑛𝑛 = 0 . (3.10) To satisfy the boundary conditions Eqn. (3.7), only the term 𝑛𝑛 = 1 is needed. Therefore, the solution is 𝜓𝜓 1 ( 𝑟𝑟 , 𝜃𝜃 ) = ℎ 1 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 𝐶𝐶 2 − 1 2 (cos 𝜃𝜃 ) + 𝑐𝑐 . 𝑐𝑐 . = ℎ 1 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 1 2 sin 2 𝜃𝜃 + c.c. , 𝑢𝑢 1 𝑟𝑟 = 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 cos 𝜃𝜃 + c.c. , 𝑢𝑢 1 𝜃𝜃 = 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. , 23 ℎ 1 ( 𝑟𝑟 ) = 𝐴𝐴 ∗ �− sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 4 𝑟𝑟 + cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 3 � + 𝐵𝐵 ∗ � cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 4 𝑟𝑟 + sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 3 � + 𝐶𝐶 ∗ 𝑟𝑟 2 + 𝐷𝐷 ∗ 1 𝑟𝑟 , 𝑓𝑓 1 ( 𝑟𝑟 ) = 𝐴𝐴 ∗ �− sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 4 𝑟𝑟 3 + cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 3 𝑟𝑟 2 � + 𝐵𝐵 ∗ � cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 4 𝑟𝑟 3 + sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 3 𝑟𝑟 2 � + 𝐶𝐶 ∗ + 𝐷𝐷 ∗ 1 𝑟𝑟 3 , 𝑠𝑠 1 ( 𝑟𝑟 ) = 1 2 � 𝐴𝐴 ∗ �− sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 4 𝑟𝑟 3 + cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 3 𝑟𝑟 2 + sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 2 𝑟𝑟 � + 𝐵𝐵 ∗ � cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 4 𝑟𝑟 3 + sin 𝑘𝑘 𝑟𝑟 𝑘𝑘 3 𝑟𝑟 2 − cos 𝑘𝑘 𝑟𝑟 𝑘𝑘 2 𝑟𝑟 � − 2 𝐶𝐶 ∗ + 𝐷𝐷 ∗ 1 𝑟𝑟 3 � , (3.11) where 𝐴𝐴 ∗ 𝐹𝐹 = − 3 𝑖𝑖 2 (2 𝑏𝑏 1 + 𝑏𝑏 2 − 2 𝑏𝑏 3 𝑑𝑑 − 𝑏𝑏 4 𝑑𝑑 ) , 𝐵𝐵 ∗ 𝐹𝐹 = 3 𝑖𝑖 2 (2 𝐷𝐷 1 + 𝐷𝐷 2 − 2 𝐷𝐷 3 𝑑𝑑 − 𝐷𝐷 4 𝑑𝑑 ) , 𝐶𝐶 ∗ 𝐹𝐹 = 𝑖𝑖 2 [( 𝐷𝐷 3 − 𝐷𝐷 4 )(2 𝑏𝑏 1 + 𝑏𝑏 2 ) − ( 𝑏𝑏 3 − 𝑏𝑏 4 )(2 𝐷𝐷 1 + 𝐷𝐷 2 ) − 3 𝑑𝑑 ( 𝐷𝐷 3 𝑏𝑏 4 − 𝐷𝐷 4 𝑏𝑏 3 )] , 𝐷𝐷 ∗ 𝐹𝐹 = − 𝑖𝑖 2 [(2 𝐷𝐷 1 + 𝐷𝐷 2 )(2 𝑏𝑏 3 + 𝑏𝑏 4 ) − (2 𝑏𝑏 1 + 𝑏𝑏 2 )(2 𝐷𝐷 3 + 𝐷𝐷 4 )] , 𝐹𝐹 = ( 𝑏𝑏 1 − 𝑏𝑏 2 − 𝑏𝑏 3 + 𝑏𝑏 4 )(2 𝐷𝐷 3 𝑑𝑑 + 𝐷𝐷 4 𝑑𝑑 − 3 𝐷𝐷 2 − 2 𝐷𝐷 3 + 2 𝐷𝐷 4 ) − ( 𝐷𝐷 1 − 𝐷𝐷 2 − 𝐷𝐷 3 + 𝐷𝐷 4 )(2 𝑏𝑏 3 𝑑𝑑 + 𝑏𝑏 4 𝑑𝑑 − 3 𝑏𝑏 2 − 2 𝑏𝑏 3 + 2 𝑏𝑏 4 ) , (3.12) and 𝐷𝐷 1 = − sin 𝑘𝑘 𝛾𝛾 𝑘𝑘 4 𝛾𝛾 3 + cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 3 𝛾𝛾 2 , 𝐷𝐷 2 = − sin 𝑘𝑘 𝛾𝛾 𝑘𝑘 4 𝛾𝛾 3 + cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 3 𝛾𝛾 2 + sin 𝑘𝑘 𝛾𝛾 𝑘𝑘 2 𝛾𝛾 , 𝐷𝐷 3 = − sin 𝑘𝑘 𝑘𝑘 4 + cos 𝑘𝑘 𝑘𝑘 3 , 𝐷𝐷 4 = − sin 𝑘𝑘 𝑘𝑘 4 + cos 𝑘𝑘 𝑘𝑘 3 + sin 𝑘𝑘 𝑘𝑘 2 , 𝑏𝑏 1 = cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 4 𝛾𝛾 3 + sin 𝑘𝑘 𝛾𝛾 𝑘𝑘 3 𝛾𝛾 2 , 𝑏𝑏 2 = cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 4 𝛾𝛾 3 + sin 𝑘𝑘 𝛾𝛾 𝑘𝑘 3 𝛾𝛾 2 − cos 𝑘𝑘 𝛾𝛾 𝑘𝑘 2 𝛾𝛾 , 𝑏𝑏 3 = cos 𝑘𝑘 𝑘𝑘 4 + sin 𝑘𝑘 𝑘𝑘 3 , 𝑏𝑏 4 = cos 𝑘𝑘 𝑘𝑘 4 + sin 𝑘𝑘 𝑘𝑘 3 − cos 𝑘𝑘 𝑘𝑘 2 , 𝑑𝑑 = 1 𝛾𝛾 3 . (3.13) The dimensionless shear stresses are found to be 𝜏𝜏 1𝜙𝜙 𝜃𝜃 = 𝜏𝜏 1 𝑟𝑟 𝜙𝜙 = 0, 24 𝜏𝜏 1 𝑟𝑟 𝜃𝜃 = { 𝐴𝐴 ∗ �sin 𝑘𝑘 𝑟𝑟 � 3 𝑘𝑘 4 𝑟𝑟 4 − 3 2 𝑘𝑘 2 𝑟𝑟 2 � + cos 𝑘𝑘 𝑟𝑟 �− 3 𝑘𝑘 3 𝑟𝑟 3 + 1 2 𝑘𝑘 𝑟𝑟 �� + 𝐵𝐵 ∗ [sin 𝑘𝑘 𝑟𝑟 �− 3 𝑘𝑘 3 𝑟𝑟 3 + 1 2 𝑘𝑘 𝑟𝑟 � + cos 𝑘𝑘 𝑟𝑟 �− 3 𝑘𝑘 4 𝑟𝑟 4 + 3 2 𝑘𝑘 2 𝑟𝑟 2 �] − 𝐷𝐷 ∗ 3 𝑟𝑟 4 } 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. . (3.14) 3.2.2 The second order solutions To study steady streaming, the terms are decomposed into time-independent and time- dependent parts, as for torsional oscillation. Only the time-independent part is required. 𝒖𝒖 2 = 𝒖𝒖 2 0 + � 𝒖𝒖 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 𝑝𝑝 2 = 𝑝𝑝 2 0 + { 𝑝𝑝 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c.} , 𝒖𝒖 1 ∙ 𝛁𝛁 𝒖𝒖 1 = 𝓕𝓕 0 + { 𝓕𝓕 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c.} , (3.15) where 𝒖𝒖 2 0 , 𝑝𝑝 2 0 and ℱ 0 are steady variables. The steady component of order 𝜀𝜀 2 takes the forms ∇ 2 𝒖𝒖 2 0 − 𝛁𝛁 𝑝𝑝 2 0 = | 𝑀𝑀 | 2 𝓕𝓕 0 , 𝛁𝛁 ∙ 𝒖𝒖 2 0 = 0, (3.16) with homogenous boundary conditions 𝑟𝑟 = 𝛾𝛾 𝑢𝑢 2 0 𝑟𝑟 = 𝑢𝑢 2 0 𝜃𝜃 = 𝑢𝑢 2 0 𝜙𝜙 = 0, 𝑟𝑟 = 1 𝑢𝑢 2 0 𝑟𝑟 = 𝑢𝑢 2 0 𝜃𝜃 = 𝑢𝑢 2 0 𝜙𝜙 = 0. (3.17) The vector spherical harmonics are still used to expand the pressure and velocity fields. Same procedures are used as in torsional oscillation to obtain steady part of nonlinear term, 𝓕𝓕 0 = ℱ 𝑃𝑃 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝑃𝑃 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝐵𝐵 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) , (3.18) where 𝓕𝓕 0 = � � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � 𝑓𝑓 1 ′ ( 𝑟𝑟 ) � cos 2 𝜃𝜃 − 1 𝑟𝑟 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � � sin 2 𝜃𝜃 � 𝒓𝒓 � + � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ′ ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 1 𝑟𝑟 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � � � sin 𝜃𝜃 cos 𝜃𝜃 𝜽𝜽 � , ℱ 𝑃𝑃 0 = 2 3 √ 𝜋𝜋 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � 𝑓𝑓 1 ′ ( 𝑟𝑟 ) − 2 𝑟𝑟 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � � � , 25 ℱ 𝑃𝑃 2 = 4 1 5 √5 𝜋𝜋 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � 𝑓𝑓 1 ′ ( 𝑟𝑟 ) + 1 𝑟𝑟 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � � � , ℱ 𝐵𝐵 2 = − 2 1 5 √30 𝜋𝜋 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ′ ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 1 𝑟𝑟 � 𝑓𝑓 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑓𝑓 1 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 1 ( 𝑟𝑟 ) 𝑠𝑠 1 ( 𝑟𝑟 ) � � � � � � � � � . (3.19) Then steady parts of velocities and pressure can be expanded as 𝒖𝒖 2 0 = 𝑢𝑢 2 0 𝑟𝑟 , 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 + 𝑢𝑢 2 0 𝑟𝑟 , 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 + 𝑢𝑢 2 0 𝜃𝜃 , 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 , 𝑝𝑝 2 0 = 𝑝𝑝 2 0, 0 𝑌𝑌 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑝𝑝 2 0, 2 𝑌𝑌 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) . (3.20) The resulting ordinary differential equations to order 𝜀𝜀 2 become d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 0 − 2 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 − d d 𝑟𝑟 𝑝𝑝 2 0, 0 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 , 1 𝑟𝑟 2 d d 𝑟𝑟 � 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 � = 0, d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − 8 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 − d d 𝑟𝑟 𝑝𝑝 2 0, 2 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 2 , d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 − 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑝𝑝 2 0, 2 = | 𝑀𝑀 | 2 ℱ 𝐵𝐵 2 , d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 = 0, (3.21) with the homogenous boundary conditions. The solution is 𝑢𝑢 2 0 𝑟𝑟 , 0 = 0, 𝑝𝑝 2 0, 0 = 𝑃𝑃 2 0 + ∫ −| 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 ( 𝑟𝑟 ′ )d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 , 𝑢𝑢 2 0 𝑟𝑟 , 2 = 𝑐𝑐 1 𝑟𝑟 + 𝑐𝑐 2 1 𝑟𝑟 2 + 𝑐𝑐 3 𝑟𝑟 3 + 𝑐𝑐 4 1 𝑟𝑟 4 + 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 2 + 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 3 + 𝐻𝐻 4 (𝑟𝑟 ) 1 𝑟𝑟 4 , 𝑢𝑢 2 0 𝜃𝜃 , 2 = 1 √ 6 [3 𝑐𝑐 1 𝑟𝑟 + 5 𝑐𝑐 3 𝑟𝑟 3 − 2 𝑐𝑐 4 1 𝑟𝑟 4 + 3 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 5 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 3 − 2 𝐻𝐻 4 ( 𝑟𝑟 ) 1 𝑟𝑟 4 ] , (3.22) where 𝑐𝑐 1 𝐷𝐷 = 2 𝛾𝛾 6 (5 𝐴𝐴 1 − 𝐴𝐴 2 ) − 5 𝛾𝛾 4 (2 𝐴𝐴 3 + 𝐴𝐴 4 ) + 7 𝐴𝐴 4 𝛾𝛾 2 + 7 𝐴𝐴 2 𝛾𝛾 − 5 𝛾𝛾 (2 𝐴𝐴 1 + 𝐴𝐴 2 ) + 2 𝛾𝛾 3 (5 𝐴𝐴 3 − 𝐴𝐴 4 ) , 26 𝑐𝑐 2 𝐷𝐷 = 2 𝛾𝛾 4 (2 𝐴𝐴 3 + 𝐴𝐴 4 ) − 5 𝛾𝛾 3 (5 𝐴𝐴 1 − 𝐴𝐴 2 ) + 7 𝛾𝛾 (3 𝐴𝐴 1 − 𝐴𝐴 2 ) + 3 𝛾𝛾 (7 𝐴𝐴 3 − 𝐴𝐴 4 ) − 5 𝛾𝛾 3 (5 𝐴𝐴 3 − 𝐴𝐴 4 ) + 2 𝛾𝛾 4 (2 𝐴𝐴 1 + 𝐴𝐴 2 ) , 𝑐𝑐 3 𝐷𝐷 = − 2 𝛾𝛾 6 (3 𝐴𝐴 1 − 𝐴𝐴 2 ) + 3 𝛾𝛾 4 (2 𝐴𝐴 3 + 𝐴𝐴 4 ) − 5 𝐴𝐴 2 𝛾𝛾 3 − 5 𝐴𝐴 4 𝛾𝛾 2 + 3 𝛾𝛾 (2 𝐴𝐴 1 + 𝐴𝐴 2 ) − 2 𝛾𝛾 (3 𝐴𝐴 3 − 𝐴𝐴 4 ) , 𝑐𝑐 4 𝐷𝐷 = − 2 𝐴𝐴 4 𝛾𝛾 2 + 3 𝛾𝛾 (5 𝐴𝐴 1 − 𝐴𝐴 2 ) − 5 𝛾𝛾 (3 𝐴𝐴 1 − 𝐴𝐴 2 + 3 𝐴𝐴 3 − 𝐴𝐴 4 ) + 5 𝛾𝛾 3 (3 𝐴𝐴 3 − 𝐴𝐴 4 ) − 2 𝛾𝛾 4 𝐴𝐴 2 , (3.23) with 𝐷𝐷 = 4 𝛾𝛾 6 − 2 5 𝛾𝛾 3 + 4 2 𝛾𝛾 − 25 𝛾𝛾 + 4 𝛾𝛾 4 , 𝐴𝐴 1 = − 𝐻𝐻 1 (1) − 𝐻𝐻 2 (1) − 𝐻𝐻 3 (1) − 𝐻𝐻 4 (1) , 𝐴𝐴 2 = −3 𝐻𝐻 1 (1) − 5 𝐻𝐻 3 (1) + 2 𝐻𝐻 4 (1) , 𝐴𝐴 3 = − 𝛾𝛾 𝐻𝐻 1 ( 𝛾𝛾 ) − 1 𝛾𝛾 2 𝐻𝐻 2 ( 𝛾𝛾 ) − 𝛾𝛾 3 𝐻𝐻 3 ( 𝛾𝛾 ) − 1 𝛾𝛾 4 𝐻𝐻 4 ( 𝛾𝛾 ) , 𝐴𝐴 4 = −3 𝛾𝛾 𝐻𝐻 1 ( 𝛾𝛾 ) − 5 𝛾𝛾 3 𝐻𝐻 3 ( 𝛾𝛾 ) + 2 𝛾𝛾 4 𝐻𝐻 4 ( 𝛾𝛾 ) , 𝐻𝐻 1 = − | 𝑀𝑀 | 2 5 � 1 √ 6 [𝑟𝑟 ′ ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ )d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 2 = | 𝑀𝑀 | 2 5 � 1 √ 6 [𝑟𝑟 ′ 4 ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 3 √ 6 ∫ ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 3 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 3 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 3 = 3| 𝑀𝑀 | 2 3 5 � 1 √ 6 � ℱ 𝐵𝐵 2 (𝑟𝑟 ′ ) 𝑟𝑟 ′ � 𝛾𝛾 𝑟𝑟 + 2 √ 6 ∫ ℱ 𝐵𝐵 2 (𝑟𝑟 ′ ) 𝑟𝑟 ′ 2 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 2 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 4 = − 3| 𝑀𝑀 | 2 3 5 � 1 √ 6 [𝑟𝑟 ′ 6 ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 5 √ 6 ∫ ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 5 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 5 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � . (3.24) As a result, the velocity has the following expression 𝒖𝒖 2 0 = 1 4 � 5 𝜋𝜋 � 𝑢𝑢 2 0 𝑟𝑟 , 2 (3 cos 2 𝜃𝜃 − 1), − √6 𝑢𝑢 2 0 𝜃𝜃 , 2 cos 𝜃𝜃 sin 𝜃𝜃 , 0 � . (3.25) The stream function takes the form 𝜓𝜓 2 0 = 1 4 � 5 𝜋𝜋 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 sin 2 𝜃𝜃 cos 𝜃𝜃 . (3.26) 27 The second order shear stress induced by steady streaming is 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 = � − 1 4 � 5 𝜋𝜋 �6 𝑐𝑐 1 + 6 𝑐𝑐 2 1 𝑟𝑟 3 + 16 𝑐𝑐 3 𝑟𝑟 2 + 16 𝑐𝑐 4 1 𝑟𝑟 5 + 6 𝐻𝐻 1 ( 𝑟𝑟 ) + 6 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 3 + 16 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 2 + 16 𝐻𝐻 4 ( 𝑟𝑟 ) 1 𝑟𝑟 5 + 3 𝐻𝐻 1 ′ ( 𝑟𝑟 ) 𝑟𝑟 + 5 𝐻𝐻 3 ′ ( 𝑟𝑟 ) 𝑟𝑟 3 − 2 𝐻𝐻 4 ′ (𝑟𝑟 ) 1 𝑟𝑟 4 �� cos 𝜃𝜃 sin 𝜃𝜃 . (3.27) Similar as torsional oscillation, 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 ( 𝑟𝑟 , 𝜃𝜃 ) is still a function of cos 𝜃𝜃 sin 𝜃𝜃 . When 𝜃𝜃 = 𝜋𝜋 4 , value of 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 ( 𝑟𝑟 , 𝜃𝜃 ) for same radius is maximum, while it equals to 0 on the equatorial plane ( 𝜃𝜃 = 𝜋𝜋 2 ) and the oscillation axis ( 𝜃𝜃 = 0). Even when 𝜃𝜃 = 𝜋𝜋 4 , the contribution of 𝜀𝜀 2 𝜏𝜏 2 0 𝑟𝑟 𝜃𝜃 to shear stress is quite small, compared to 𝜀𝜀 𝜏𝜏 1 𝑟𝑟 𝜃𝜃 under same Womersley number | 𝑀𝑀 |. Note the form of solution for transverse oscillation is the same as torsional oscillation, except the equations of ℱ 𝑃𝑃 0 , ℱ 𝑃𝑃 2 , ℱ 𝐵𝐵 2 , 𝐻𝐻 1 ( 𝑟𝑟 ), 𝐻𝐻 2 ( 𝑟𝑟 ), 𝐻𝐻 3 ( 𝑟𝑟 ) and 𝐻𝐻 4 ( 𝑟𝑟 ). 3.3 Theoretical results and discussion Shear stress profiles in the transverse oscillation case have a different trend, compared to torsional oscillation (see Fig. 3.2). The minimum of shear stress is located between two boundaries, while the maximum located on the inner boundary, as well as the time average of absolute shear stress values. Increasing | 𝑀𝑀 | results in growth of shear stress for the inner region, but shear stress on the outer wall remains constant. For all | 𝑀𝑀 |,in the range 1~20, the shear stress has a sharp decline with increasing 𝑟𝑟 near the inner wall region, and drops to zero around a value of 𝑟𝑟 = 𝑟𝑟 𝑖𝑖 . This location of zero stress moves from 0.7 to 0.85 as | 𝑀𝑀 | rises from 1 to 20. It suggests that as the increase of oscillation frequency, presented by | 𝑀𝑀 |, oscillation of 28 inner hemisphere has a further influence to the outer region. Also, compared to torsional oscillation, transverse oscillation has a deeper influence. Similar to the torsional oscillations case, maximum value of first order shear stress can be found on equatorial plane. Also, shear stress is maximum as oscillation velocity reaches its maximum value, when 𝑡𝑡 = �𝑛𝑛 + 1 2 � 𝜋𝜋 and 𝑛𝑛 = 0,1. Figure 3.3 shows the steady streaming for the transverse oscillation. The arrows present the steady component of second order velocity 𝒖𝒖 2 0 . Similar as torsional oscillation, there is only one standing vortex in the region, and the steady streaming flow pattern doesn’t change with increase of Womersley number | 𝑀𝑀 |. When | 𝑀𝑀 | is as high as 20, it can be seen that the center of vortex moves to the inner boundary. The difference from torsional oscillation is the direction of the vortex is clockwise. Figure 3.4 shows the streaming intensity, presented by the absolute values of the maximum streaming velocity | 𝑢𝑢 2 0 | 𝑚𝑚 under different polar angle 𝜃𝜃 , versus Womersley number | 𝑀𝑀 |. Similar as torsional oscillation in Figure 2.4, | 𝑢𝑢 2 0 | 𝑚𝑚 rises with increase in | 𝑀𝑀 |. But the value difference illustrates that, transverse oscillation induces intenser streaming than torsional oscillation. Using Taylor expansion, when | 𝑀𝑀 | ≪ 1, 𝑓𝑓 1 ( 𝑟𝑟 ) and 𝑠𝑠 1 ( 𝑟𝑟 ) in Eqn. (3.11) can be written as, 𝑓𝑓 1 ( 𝑟𝑟 ) = 6 0 𝑖𝑖 𝛾𝛾 ( 1 − 𝛾𝛾 )[ 4( 𝛾𝛾 5 − 1) + 5 𝛾𝛾 ( 1 − 𝛾𝛾 2 )( 1 + 𝛾𝛾 )] � 1 4 0 (1 − 𝛾𝛾 2 ) 𝑟𝑟 2 + �− 3 4 0 + 5 4 8 𝛾𝛾 2 − 1 8 𝛾𝛾 4 + 1 3 0 𝛾𝛾 5 � + 1 2 0 (1 − 𝛾𝛾 5 ) 1 𝑟𝑟 − 1 6 0 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 3 � + 𝑂𝑂 (| 𝑀𝑀 | 2 ) , 𝑠𝑠 1 ( 𝑟𝑟 ) = 6 0 𝑖𝑖 𝛾𝛾 ( 1 − 𝛾𝛾 )[ 4( 𝛾𝛾 5 − 1) + 5 𝛾𝛾 ( 1 − 𝛾𝛾 2 )( 1 + 𝛾𝛾 )] �− 1 2 0 (1 − 𝛾𝛾 2 ) 𝑟𝑟 2 + � 3 4 0 − 5 4 8 + 1 8 𝛾𝛾 4 − 1 3 0 𝛾𝛾 5 � − 1 4 0 (1 − 𝛾𝛾 5 ) 1 𝑟𝑟 − 1 1 2 0 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 3 � + 𝑂𝑂 (| 𝑀𝑀 | 2 ) . (3.28) The terms in Eqn. (3.24) are expressed as follows, 29 𝐻𝐻 1 = − 3 √ 5 𝜋𝜋 2 5 𝛾𝛾 2 ( 1 − 𝛾𝛾 ) 2 [ 4( 𝛾𝛾 5 − 1) + 5 𝛾𝛾 ( 1 − 𝛾𝛾 2 )( 1 + 𝛾𝛾 )] 2 | 𝑀𝑀 | 2 � 𝛾𝛾 2 (1 − 𝛾𝛾 3 )(1 − 𝛾𝛾 5 ) � 1 𝑟𝑟 4 − 1 𝛾𝛾 4 � − 6(1 − 𝛾𝛾 5 ) 2 � 1 𝑟𝑟 2 − 1 𝛾𝛾 2 � − ( −18 + 25 𝛾𝛾 2 − 30 𝛾𝛾 4 + 8 𝛾𝛾 5 )(1 − 𝛾𝛾 5 ) � 1 𝑟𝑟 − 1 𝛾𝛾 � + 6(1 − 𝛾𝛾 2 )(1 − 𝛾𝛾 5 )( 𝑟𝑟 − 𝛾𝛾 ) − 2(1 − 𝛾𝛾 2 ) 2 ( 𝑟𝑟 4 − 𝛾𝛾 4 ) � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝐻𝐻 2 = 3 √ 5 𝜋𝜋 2 5 𝛾𝛾 2 ( 1 − 𝛾𝛾 ) 2 [ 4( 𝛾𝛾 5 − 1) + 5 𝛾𝛾 ( 1 − 𝛾𝛾 2 )( 1 + 𝛾𝛾 )] 2 | 𝑀𝑀 | 2 �4 𝛾𝛾 2 (1 − 𝛾𝛾 3 )(1 − 𝛾𝛾 5 ) � 1 𝑟𝑟 − 1 𝛾𝛾 � + 12(1 − 𝛾𝛾 5 ) 2 ( 𝑟𝑟 − 𝛾𝛾 ) + 1 2 ( −18 + 25 𝛾𝛾 2 − 30 𝛾𝛾 4 + 8 𝛾𝛾 5 )(1 − 𝛾𝛾 5 )( 𝑟𝑟 2 − 𝛾𝛾 2 ) + 3 2 (1 − 𝛾𝛾 2 )(1 − 𝛾𝛾 5 )( 𝑟𝑟 4 − 𝛾𝛾 4 ) − 8 7 (1 − 𝛾𝛾 2 ) 2 ( 𝑟𝑟 7 − 𝛾𝛾 7 ) � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝐻𝐻 3 = 3 √ 5 𝜋𝜋 1 7 5 𝛾𝛾 2 ( 1 − 𝛾𝛾 ) 2 [ 4( 𝛾𝛾 5 − 1) + 5 𝛾𝛾 ( 1 − 𝛾𝛾 2 )( 1 + 𝛾𝛾 )] 2 | 𝑀𝑀 | 2 �2 𝛾𝛾 2 (1 − 𝛾𝛾 3 )(1 − 𝛾𝛾 5 ) � 1 𝑟𝑟 6 − 1 𝛾𝛾 6 � − 9(1 − 𝛾𝛾 5 ) 2 � 1 𝑟𝑟 4 − 1 𝛾𝛾 4 � − ( −18 + 25 𝛾𝛾 2 − 30 𝛾𝛾 4 + 8 𝛾𝛾 5 )(1 − 𝛾𝛾 5 ) � 1 𝑟𝑟 3 − 1 𝛾𝛾 3 � − 18(1 − 𝛾𝛾 2 )(1 − 𝛾𝛾 5 ) � 1 𝑟𝑟 − 1 𝛾𝛾 � − 12(1 − 𝛾𝛾 2 ) 2 ( 𝑟𝑟 2 − 𝛾𝛾 2 ) � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝐻𝐻 4 = 3 √ 5 𝜋𝜋 1 7 5 𝛾𝛾 2 ( 1 − 𝛾𝛾 ) 2 [ 4( 𝛾𝛾 5 − 1) + 5 𝛾𝛾 ( 1 − 𝛾𝛾 2 )( 1 + 𝛾𝛾 )] 2 | 𝑀𝑀 | 2 �12 𝛾𝛾 2 (1 − 𝛾𝛾 3 )(1 − 𝛾𝛾 5 )( 𝑟𝑟 − 𝛾𝛾 ) − 12(1 − 𝛾𝛾 5 ) 2 ( 𝑟𝑟 3 − 𝛾𝛾 3 ) − 3 4 ( −18 + 25 𝛾𝛾 2 − 30 𝛾𝛾 4 + 8 𝛾𝛾 5 )(1 − 𝛾𝛾 5 )( 𝑟𝑟 4 − 𝛾𝛾 4 ) − 3(1 − 𝛾𝛾 2 )(1 − 𝛾𝛾 5 )( 𝑟𝑟 6 − 𝛾𝛾 6 ) + 8 3 (1 − 𝛾𝛾 2 ) 2 ( 𝑟𝑟 9 − 𝛾𝛾 9 ) � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) . (3.29) Again, ratio of boundary radii 𝛾𝛾 = 1 2 is taken as example, 𝑢𝑢 2 0 𝑟𝑟 , 2 = √ 5 𝜋𝜋 2 0 6 7 5 0 6 0 | 𝑀𝑀 | 2 1 𝑟𝑟 4 (1 − 𝑟𝑟 ) 2 (1 − 2 𝑟𝑟 ) 2 ( −19463 + 659431 𝑟𝑟 − 230925 𝑟𝑟 2 + 120782 𝑟𝑟 3 − 147168 𝑟𝑟 4 − 49056 𝑟𝑟 5 ) + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝑢𝑢 2 0 𝜃𝜃 , 2 = √ 3 0 𝜋𝜋 1 2 4 0 5 0 3 6 0 | 𝑀𝑀 | 2 1 𝑟𝑟 4 (1 − 𝑟𝑟 )(1 − 2 𝑟𝑟 )(38926 − 659431 𝑟𝑟 − 2056145 𝑟𝑟 2 + 5462918 𝑟𝑟 3 − 3228774 𝑟𝑟 4 + 2826668 𝑟𝑟 5 − 1030176 𝑟𝑟 6 − 686784 𝑟𝑟 7 ) + 𝑂𝑂 (| 𝑀𝑀 | 6 ) . (3.30) Therefore, with the analytical calculation, the steady streaming velocity and stream function are found to be 30 𝑢𝑢 2 0 = | 𝑀𝑀 | 2 1 6 5 4 0 0 4 8 � 1 𝑟𝑟 4 (1 − 𝑟𝑟 ) 2 (1 − 2 𝑟𝑟 ) 2 ( −19463 + 659431 𝑟𝑟 − 230925 𝑟𝑟 2 + 120782 𝑟𝑟 3 − 147168 𝑟𝑟 4 − 49056 𝑟𝑟 5 )(3 cos 2 𝜃𝜃 − 1), − 1 𝑟𝑟 4 (1 − 𝑟𝑟 )(1 − 2 𝑟𝑟 )(38926 − 659431 𝑟𝑟 − 2056145 𝑟𝑟 2 + 5462918 𝑟𝑟 3 − 3228774 𝑟𝑟 4 + 2826668 𝑟𝑟 5 − 1030176 𝑟𝑟 6 − 686784 𝑟𝑟 7 ) cos 𝜃𝜃 sin 𝜃𝜃 , 0 � + 𝑂𝑂 (| 𝑀𝑀 | 6 ) , 𝜓𝜓 2 0 = | 𝑀𝑀 | 2 1 6 5 4 0 0 4 8 𝑟𝑟 2 {(1 − 𝑟𝑟 ) 2 (1 − 2 𝑟𝑟 ) 2 ( −19463 + 659431 𝑟𝑟 − 230925 𝑟𝑟 2 + 120782 𝑟𝑟 3 − 147168 𝑟𝑟 4 − 49056 𝑟𝑟 5 )} sin 2 𝜃𝜃 cos 𝜃𝜃 + 𝑂𝑂 (| 𝑀𝑀 | 6 ) . (3.31) Same as torsional oscillation in Chapter 2, the intensity of the steady streaming is still proportional to | 𝑀𝑀 | 2 in the small | 𝑀𝑀 | limit. 31 Figure 3.2 First order shear stress profiles [Eqn. (3.14)] on the equatorial plane over one time period for the transverse oscillation with 𝛾𝛾 =0.5. The dashed curve shows the time average of absolute shear stress values over one period. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=5, (c) | 𝑀𝑀 |=20. 32 Figure 3.3 The streaming flow pattern for the transverse oscillation with 𝛾𝛾 =0.5. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=20. 33 Figure 3.4 Streaming intensity as a function of Womersley number | 𝑀𝑀 | at various polar angle 𝜃𝜃 for the transverse oscillation with 𝛾𝛾 =0.5. 34 CHAPTER 4 COMBINED OSCILLATIONS OF SPHERE 4.1 Introduction The study in this chapter, while motivated by the protein-shear work in Chapter 2 and 3, is a more fundamental development concerning streaming flows in a spherical annular region. In this investigation, the two different types of oscillations are combined with same angular frequency 𝜔𝜔 , but a phase lag 𝛽𝛽 ′ and amplitude difference. Due to strong shear stresses resulting from the interaction of two oscillations as compared to single type of oscillation, such a case is of particular fundamental interest. It should also be noted that with the same angular frequency 𝜔𝜔 , the effect of the interaction of two oscillations is greatest, and thus, the magnitude of steady streaming induced is strongest, according to Kelly[62]. In this regard, there is a strong interest in examining the effect of superposed torsional and transverse oscillations in generating the steady streaming with interesting phenomenon. The schematic is shown in Figure 4.1. In Section 4.2, limiting forms of the solution to the governing equations are considered for low- frequency (| 𝑀𝑀 | 2 ≪ 1) and high-frequency (| 𝑀𝑀 | 2 ≫ 1), as well as the case of unrestricted | 𝑀𝑀 | values. In Section 4.3, the theoretical results are discussed. 35 Figure 4.1 Schematic of the combined oscillations case. The inner sphere can oscillate both rotationally and laterally. 4.2 Model and theoretical development There is a viscous fluid between two concentric spheres, the outer one with radius a and the inner one R i <a. The outer spherical container is held fixed, while the inner sphere is oscillating both torsionally and transversely along z-axis with phase lag 𝛽𝛽 ′ and having amplitude difference. This problem is three-dimensional axisymmetric, and we formulate it in the spherical coordinates. The dimensional Navier-Stokes and continuity equations are 𝜕𝜕 𝒖𝒖 𝜕𝜕 𝑡𝑡 + 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 1 𝜌𝜌 𝛁𝛁 𝑝𝑝 + ν ∇ 2 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0. (4.1) The boundary conditions are 36 𝑟𝑟 = 𝑅𝑅 𝑖𝑖 𝑢𝑢 𝑟𝑟 = 𝑈𝑈 0 sin 𝜔𝜔 𝑡𝑡 cos 𝜃𝜃 , 𝑢𝑢 𝜃𝜃 = − 𝑈𝑈 0 sin 𝜔𝜔 𝑡𝑡 sin 𝜃𝜃 , 𝑢𝑢 𝜙𝜙 = 𝛼𝛼 𝑈𝑈 0 sin[ 𝜔𝜔 ( 𝑡𝑡 + 𝛽𝛽 ′ )] sin 𝜃𝜃 , 𝑟𝑟 = 𝐷𝐷 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 𝑢𝑢 𝜙𝜙 = 0. (4.2) The stream function, 𝜓𝜓 , and angular circulation, Ω, related to the velocity components are introduced as, 𝑢𝑢 𝑟𝑟 = − 1 𝑟𝑟 2 𝜕𝜕 𝜓𝜓 𝜕𝜕𝜕𝜕 , 𝑢𝑢 𝜃𝜃 = − � 1 − 𝜕𝜕 2 � − 1 2 𝑟𝑟 𝜕𝜕 𝜓𝜓 𝜕𝜕 𝑟𝑟 , 𝑢𝑢 𝜙𝜙 = Ω 𝑟𝑟 ( 1 − 𝜕𝜕 2 ) 1 2 , (4.3) where 𝜇𝜇 = 𝑐𝑐𝑐𝑐 𝑠𝑠 𝜃𝜃 . The following dimensionless scaling is adopted, 𝒖𝒖 ∗ = 𝒖𝒖 𝑈𝑈 0 , 𝜓𝜓 ∗ = 𝜓𝜓 𝑈𝑈 0 𝜔𝜔 2 , Ω ∗ = Ω 𝑈𝑈 0 𝜔𝜔 , 𝑡𝑡 ∗ = 𝜔𝜔 𝑡𝑡 , 𝑟𝑟 ∗ = 𝑟𝑟 𝜔𝜔 , 𝑝𝑝 ∗ = 𝑝𝑝 𝜌𝜌 𝜈𝜈 𝑈𝑈 0 / 𝜔𝜔 , 𝛁𝛁 ∗ = 𝐷𝐷 𝛁𝛁 . (4.4) The dimensionless governing equations are 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜓𝜓 ) + 𝜀𝜀 𝑟𝑟 2 � 𝜕𝜕 � 𝜓𝜓 , 𝐷𝐷 2 𝜓𝜓 � 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) + 2 𝐷𝐷 2 𝜓𝜓 𝐿𝐿 𝜓𝜓 + 2 Ω 𝐿𝐿 Ω � = 1 | 𝑀𝑀 | 2 𝐷𝐷 4 𝜓𝜓 , 𝜕𝜕 Ω 𝜕𝜕 𝑡𝑡 + 𝜀𝜀 𝑟𝑟 2 � 𝜕𝜕 ( 𝜓𝜓 , Ω) 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) � = 1 | 𝑀𝑀 | 2 𝐷𝐷 2 Ω , (4.5) with boundary conditions − 1 𝑟𝑟 2 𝜕𝜕 𝜓𝜓 𝜕𝜕𝜕𝜕 = 𝜇𝜇 sin 𝑡𝑡 , − � 1 − 𝜕𝜕 2 � − 1 2 𝑟𝑟 𝜕𝜕 𝜓𝜓 𝜕𝜕 𝑟𝑟 = − sin 𝑡𝑡 (1 − 𝜇𝜇 2 ) 1 2 , and Ω 𝑟𝑟 ( 1 − 𝜕𝜕 2 ) 1 2 = 𝛼𝛼 sin( 𝑡𝑡 + 𝛽𝛽 ) (1 − 𝜇𝜇 2 ) 1 2 on 𝑟𝑟 = 𝛾𝛾 , 𝜓𝜓 = 𝜕𝜕 𝜓𝜓 𝜕𝜕 𝑟𝑟 = 0, Ω = 0 on 𝑟𝑟 = 1. (4.6) where 𝜇𝜇 = cos 𝜃𝜃 , 𝛽𝛽 = 𝜔𝜔 𝛽𝛽 ′ , 𝜀𝜀 = 𝑈𝑈 1 𝜔𝜔𝜔𝜔 , | 𝑀𝑀 | 2 = 𝜔𝜔 𝜔𝜔 2 𝜈𝜈 , 𝛾𝛾 = 𝑅𝑅 𝑖𝑖 𝜔𝜔 , 𝐷𝐷 2 = 𝜕𝜕 2 𝜕𝜕 𝑟𝑟 2 + 1 − 𝜕𝜕 2 𝑟𝑟 2 𝜕𝜕 2 𝜕𝜕 𝜕𝜕 2 and 𝐿𝐿 = 𝜕𝜕 1 − 𝜕𝜕 2 𝜕𝜕 𝜕𝜕 𝑟𝑟 + 1 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 . (4.7) 37 4.2.1 The leading order solutions We start out with the following perturbation expansion, 𝜓𝜓 = 𝜓𝜓 0 + 𝜀𝜀 𝜓𝜓 1 + H.O.T. , Ω = Ω 0 + 𝜀𝜀 Ω 1 + H.O.T. . (4.8) where H.O.T. means higher order terms. Upon substitution into the momentum equation (4.5), the leading order solutions emerge as 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜓𝜓 0 ) = 1 | 𝑀𝑀 | 2 𝐷𝐷 4 𝜓𝜓 0 , 𝜕𝜕 Ω 0 𝜕𝜕 𝑡𝑡 = 1 | 𝑀𝑀 | 2 𝐷𝐷 2 Ω 0 , (4.9) with the boundary conditions − 1 𝑟𝑟 2 𝜕𝜕 𝜓𝜓 0 𝜕𝜕𝜕𝜕 = 𝜇𝜇 sin 𝑡𝑡 , − � 1 − 𝜕𝜕 2 � − 1 2 𝑟𝑟 𝜕𝜕 𝜓𝜓 0 𝜕𝜕 𝑟𝑟 = − sin 𝑡𝑡 (1 − 𝜇𝜇 2 ) 1 2 , and Ω 0 𝑟𝑟 ( 1 − 𝜕𝜕 2 ) 1 2 = 𝛼𝛼 sin( 𝑡𝑡 + 𝛽𝛽 ) (1 − 𝜇𝜇 2 ) 1 2 on 𝑟𝑟 = 𝛾𝛾 , 𝜓𝜓 0 = 𝜕𝜕 𝜓𝜓 0 𝜕𝜕 𝑟𝑟 = Ω 0 = 0 on 𝑟𝑟 = 1. (4.10) The leading order solutions are 𝜓𝜓 0 = �𝐴𝐴 0 𝑟𝑟 2 + 𝐵𝐵 0 𝑟𝑟 + 𝐶𝐶 0 � 1 𝑀𝑀𝑟𝑟 − 1 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐷𝐷 0 � 1 𝑀𝑀𝑟𝑟 + 1 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = �𝐴𝐴 0 ∗ � 1 𝑀𝑀𝑟𝑟 − 1 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐵𝐵 0 ∗ � 1 𝑀𝑀𝑟𝑟 + 1 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , (4.11) and 𝑢𝑢 0 𝑟𝑟 = 2 �𝐴𝐴 0 + 𝐵𝐵 0 𝑟𝑟 3 + 𝐶𝐶 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 − 1 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐷𝐷 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 + 1 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � 𝜇𝜇 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝑢𝑢 0 𝜃𝜃 = �−2 𝐴𝐴 0 + 𝐵𝐵 0 𝑟𝑟 3 + 𝐶𝐶 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 − 1 + 𝑀𝑀 𝑟𝑟 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐷𝐷 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 + 1 + 𝑀𝑀 𝑟𝑟 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝑢𝑢 0 𝜙𝜙 = � 𝐴𝐴 0 ∗ 𝑟𝑟 � 1 𝑀𝑀𝑟𝑟 − 1 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐵𝐵 0 ∗ 𝑟𝑟 � 1 𝑀𝑀𝑟𝑟 + 1 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , (4.12) 38 where 𝐴𝐴 0 = − 1 4 𝑖𝑖 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) � 3 + 3𝑀𝑀 𝛾𝛾 + 𝑀𝑀 2 𝛾𝛾 2 � − 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) � 3 − 3𝑀𝑀 𝛾𝛾 + 𝑀𝑀 2 𝛾𝛾 2 � − 6 𝑀𝑀 1 2 𝑀𝑀 + 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) � 3 � 1 𝛾𝛾 − 1 � − 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � − 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) � 3 � 1 𝛾𝛾 − 1 � + 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � , 𝐵𝐵 0 = 1 4 𝑖𝑖 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) � 3 𝑀𝑀 + 3 𝛾𝛾 + 𝑀𝑀 𝛾𝛾 2 � � 3 𝑀𝑀 − 3 + 𝑀𝑀 � − 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) � 3 𝑀𝑀 − 3 𝛾𝛾 + 𝑀𝑀 𝛾𝛾 2 � � 3 𝑀𝑀 + 3 + 𝑀𝑀 � 1 2 𝑀𝑀 + 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) � 3 � 1 𝛾𝛾 − 1 � − 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � − 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) � 3 � 1 𝛾𝛾 − 1 � + 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � , 𝐶𝐶 0 = − 1 4 𝑖𝑖 𝑒𝑒 − 𝑀𝑀 𝛾𝛾 � 9 𝑀𝑀 + 9 𝛾𝛾 + 3 𝑀𝑀 𝛾𝛾 2 � − 𝑒𝑒 − 𝑀𝑀 � 9 𝑀𝑀 + 9 + 3 𝑀𝑀 � 1 2 𝑀𝑀 + 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) � 3 � 1 𝛾𝛾 − 1 � − 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � − 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) � 3 � 1 𝛾𝛾 − 1 � + 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � , 𝐷𝐷 0 = 1 4 𝑖𝑖 𝑒𝑒 𝑀𝑀 𝛾𝛾 � 9 𝑀𝑀 − 9 𝛾𝛾 + 3 𝑀𝑀 𝛾𝛾 2 � − 𝑒𝑒 𝑀𝑀 � 9 𝑀𝑀 − 9 + 3 𝑀𝑀 � 1 2 𝑀𝑀 + 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) � 3 � 1 𝛾𝛾 − 1 � − 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � − 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) � 3 � 1 𝛾𝛾 − 1 � + 3 𝑀𝑀 � 1 𝛾𝛾 + 𝛾𝛾 � + 𝑀𝑀 2 � 1 𝛾𝛾 − 𝛾𝛾 2 � � , 𝐴𝐴 0 ∗ = − 1 2 𝑖𝑖 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑒𝑒 − 𝑀𝑀 � 1 𝑀𝑀 + 1 � � 1 𝑀𝑀 − 1 � � 1 𝑀𝑀 𝛾𝛾 2 + 1 𝛾𝛾 � 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) − � 1 𝑀𝑀 + 1 � � 1 𝑀𝑀 𝛾𝛾 2 − 1 𝛾𝛾 � 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) , 𝐵𝐵 0 ∗ = 1 2 𝑖𝑖 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝑒𝑒 𝑀𝑀 � 1 𝑀𝑀 − 1 � � 1 𝑀𝑀 − 1 � � 1 𝑀𝑀 𝛾𝛾 2 + 1 𝛾𝛾 � 𝑒𝑒 𝑀𝑀 ( 1 − 𝛾𝛾 ) − � 1 𝑀𝑀 + 1 � � 1 𝑀𝑀 𝛾𝛾 2 − 1 𝛾𝛾 � 𝑒𝑒 𝑀𝑀 ( 𝛾𝛾 − 1) , 𝑀𝑀 2 = 𝑠𝑠 | 𝑀𝑀 | 2 , (4.13) and c.c. denotes the complex conjugate. The dimensionless shear stresses are 𝜏𝜏 0 𝑟𝑟 𝜃𝜃 = �− 6 𝐵𝐵 0 𝑟𝑟 4 + 𝐶𝐶 0 �− 6 𝑀𝑀 𝑟𝑟 4 + 6 𝑟𝑟 3 − 3 𝑀𝑀 𝑟𝑟 2 + 𝑀𝑀 2 𝑟𝑟 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐷𝐷 0 �− 6 𝑀𝑀 𝑟𝑟 4 − 6 𝑟𝑟 3 − 3 𝑀𝑀 𝑟𝑟 2 − 𝑀𝑀 2 𝑟𝑟 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝜏𝜏 0𝜙𝜙 𝜃𝜃 = 0, 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 = �𝐴𝐴 0 ∗ �− 3 𝑀𝑀 𝑟𝑟 3 + 3 𝑟𝑟 2 − 𝑀𝑀 𝑟𝑟 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐵𝐵 0 ∗ �− 3 𝑀𝑀 𝑟𝑟 3 − 3 𝑟𝑟 2 − 𝑀𝑀 𝑟𝑟 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. . (4.14) 4.2.2 The solutions to order 𝜺𝜺 First, the leading order velocity components are written in the form, 𝑢𝑢 0 𝑟𝑟 = 𝑓𝑓 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 cos 𝜃𝜃 + c.c. , 39 𝑢𝑢 0 𝜃𝜃 = 𝑠𝑠 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. , 𝑢𝑢 0 𝜙𝜙 = 𝑔𝑔 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. , (4.15) where 𝑓𝑓 ( 𝑟𝑟 ) = 2 �𝐴𝐴 0 + 𝐵𝐵 0 𝑟𝑟 3 + 𝐶𝐶 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 − 1 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐷𝐷 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 + 1 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 � , 𝑠𝑠 ( 𝑟𝑟 ) = −2 𝐴𝐴 0 + 𝐵𝐵 0 𝑟𝑟 3 + 𝐶𝐶 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 − 1 + 𝑀𝑀 𝑟𝑟 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐷𝐷 0 𝑟𝑟 2 � 1 𝑀𝑀𝑟𝑟 + 1 + 𝑀𝑀 𝑟𝑟 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 , 𝑔𝑔 ( 𝑟𝑟 ) = 𝐴𝐴 0 ∗ 𝑟𝑟 � 1 𝑀𝑀𝑟𝑟 − 1 � 𝑒𝑒 𝑀𝑀𝑟𝑟 + 𝐵𝐵 0 ∗ 𝑟𝑟 � 1 𝑀𝑀𝑟𝑟 + 1 � 𝑒𝑒 − 𝑀𝑀𝑟𝑟 . (4.16) Then the variables to order 𝜀𝜀 are divided into time-independent and time-dependent parts as 𝒖𝒖 1 = 𝒖𝒖 1 0 + � 𝒖𝒖 1 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 𝑝𝑝 1 = 𝑝𝑝 1 0 + � 𝑝𝑝 1 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 𝒖𝒖 1 ⋅ 𝛁𝛁 𝒖𝒖 1 = 𝓕𝓕 0 + � 𝓕𝓕 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � . (4.17) The steady part of first order solutions satisfies ∇ 2 𝒖𝒖 1 0 − 𝛁𝛁 𝑝𝑝 1 0 = | 𝑀𝑀 | 2 𝓕𝓕 0 , 𝛁𝛁 ⋅ 𝒖𝒖 1 0 = 0, (4.18) with homogeneous boundary conditions 𝑟𝑟 = 𝛾𝛾 𝑢𝑢 1 0 = 0, 𝑟𝑟 = 1 𝑢𝑢 1 0 = 0. (4.19) The vector spherical harmonics are used to expand the steady part of nonlinear term, 𝓕𝓕 0 = ℱ 𝑝𝑝 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝑝𝑝 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝐵𝐵 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝐶𝐶2 ( 𝑟𝑟 ) 𝑪𝑪 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) , (4.20) where 𝓕𝓕 0 = � � 𝑓𝑓 ( 𝑟𝑟 ) 𝑓𝑓 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � 𝑓𝑓 ′ ( 𝑟𝑟 ) � cos 2 𝜃𝜃 − 𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 𝑠𝑠 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 sin 2 𝜃𝜃 � 𝒓𝒓 � + �𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 ′ ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 𝑠𝑠 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � − 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 � sin 𝜃𝜃 cos 𝜃𝜃 𝜽𝜽 � + �𝑓𝑓 ( 𝑟𝑟 ) 𝑔𝑔 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑔𝑔 ′ ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � + 𝑔𝑔 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � + 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 � sin 𝜃𝜃 cos 𝜃𝜃 𝝓𝝓 � , 40 ℱ 𝑃𝑃 0 = 2 3 √ 𝜋𝜋 �𝑓𝑓 ( 𝑟𝑟 ) 𝑓𝑓 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � 𝑓𝑓 ′ ( 𝑟𝑟 ) − 2 𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 𝑠𝑠 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 � , ℱ 𝑃𝑃 2 = 4 1 5 √5 𝜋𝜋 �𝑓𝑓 ( 𝑟𝑟 ) 𝑓𝑓 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � 𝑓𝑓 ′ ( 𝑟𝑟 ) + 𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 𝑠𝑠 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 � , ℱ 𝐵𝐵 2 = − 2 1 5 √30 𝜋𝜋 �𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 ′ ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � + 𝑠𝑠 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � − 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 � , ℱ 𝐶𝐶2 = 2 1 5 √30 𝜋𝜋 �𝑓𝑓 ( 𝑟𝑟 ) 𝑔𝑔 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑔𝑔 ′ ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 𝑓𝑓 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � + 𝑔𝑔 ( 𝑟𝑟 ) 𝑓𝑓 ( 𝑟𝑟 ) � � � � � � + 2 𝑠𝑠 ( 𝑟𝑟 ) 𝑔𝑔 ( 𝑟𝑟 ) � � � � � � + 2 𝑔𝑔 ( 𝑟𝑟 ) 𝑠𝑠 ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 � . (4.21) Next, we expand the steady parts of velocity and pressure in forms similar to the nonlinear term above, i.e., 𝒖𝒖 1 0 = 𝑢𝑢 1 0 𝑟𝑟 , 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑢𝑢 1 0 𝑟𝑟 , 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑢𝑢 1 0 𝜃𝜃 , 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑢𝑢 1 0 𝜙𝜙 , 2 ( 𝑟𝑟 ) 𝑪𝑪 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) , 𝑝𝑝 1 0 = 𝑝𝑝 1 0, 0 𝑌𝑌 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑝𝑝 1 0, 2 𝑌𝑌 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) . (4.22) The resulting differential equations to order 𝜀𝜀 become d 2 d 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 0 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 1 0 𝑟𝑟 , 0 − 2 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 0 − d d 𝑟𝑟 𝑝𝑝 1 0, 0 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 , 1 𝑟𝑟 2 d d 𝑟𝑟 � 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 0 � = 0, d 2 d 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 1 0 𝑟𝑟 , 2 − 8 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 1 0 𝜃𝜃 , 2 − d d 𝑟𝑟 𝑝𝑝 1 0, 2 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 2 , d 2 d 𝑟𝑟 2 𝑢𝑢 1 0 𝜃𝜃 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 1 0 𝜃𝜃 , 2 − 6 𝑟𝑟 2 𝑢𝑢 1 0 𝜃𝜃 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑝𝑝 1 0, 2 = | 𝑀𝑀 | 2 ℱ 𝐵𝐵 2 , d d 𝑟𝑟 𝑢𝑢 1 0 𝑟𝑟 , 2 + 2 𝑟𝑟 𝑢𝑢 1 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑢𝑢 1 0 𝜃𝜃 , 2 = 0, d 2 d 𝑟𝑟 2 𝑢𝑢 1 0 𝜙𝜙 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 1 0 𝜙𝜙 , 2 − 6 𝑟𝑟 2 𝑢𝑢 1 0 𝜙𝜙 , 2 = | 𝑀𝑀 | 2 ℱ 𝐶𝐶2 , (4.23) with homogeneous boundary conditions. The solutions are 𝑢𝑢 1 0 𝑟𝑟 , 0 = 0, 𝑝𝑝 1 0, 0 = 𝑃𝑃 1 0 + ∫ −| 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 ( 𝑟𝑟 ′ )d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 , 𝑢𝑢 1 0 𝑟𝑟 , 2 = 𝑐𝑐 1 𝑟𝑟 + 𝑐𝑐 2 1 𝑟𝑟 2 + 𝑐𝑐 3 𝑟𝑟 3 + 𝑐𝑐 4 1 𝑟𝑟 4 + 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 2 + 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 3 + 𝐻𝐻 4 (𝑟𝑟 ) 1 𝑟𝑟 4 , 41 𝑢𝑢 1 0 𝜃𝜃 , 2 = 1 √ 6 �3 𝑐𝑐 1 𝑟𝑟 + 5 𝑐𝑐 3 𝑟𝑟 3 − 𝑐𝑐 4 2 𝑟𝑟 4 + 3 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 5 𝐻𝐻 3 ( 𝑟𝑟 ) 𝑟𝑟 3 − 𝐻𝐻 4 ( 𝑟𝑟 ) 2 𝑟𝑟 4 � , 𝑢𝑢 1 0 𝜙𝜙 , 2 = 𝑐𝑐 5 1 𝑟𝑟 3 + 𝑐𝑐 6 𝑟𝑟 2 + 𝐻𝐻 5 ( 𝑟𝑟 ) 1 𝑟𝑟 3 + 𝐻𝐻 6 ( 𝑟𝑟 ) 𝑟𝑟 2 , (4.24) where 𝑐𝑐 1 𝐷𝐷 ∗ = 2 𝛾𝛾 6 (5 𝐴𝐴 1 1 − 𝐴𝐴 1 2 ) − 5 𝛾𝛾 4 (2 𝐴𝐴 1 3 + 𝐴𝐴 1 4 ) + 7 𝛾𝛾 2 𝐴𝐴 1 4 + 7 𝛾𝛾 𝐴𝐴 1 2 − 5 𝛾𝛾 (2 𝐴𝐴 1 1 + 𝐴𝐴 1 2 ) + 2 𝛾𝛾 3 (5 𝐴𝐴 1 3 − 𝐴𝐴 1 4 ) , 𝑐𝑐 2 𝐷𝐷 ∗ = 2 𝛾𝛾 4 (2 𝐴𝐴 1 3 + 𝐴𝐴 1 4 ) − 5 𝛾𝛾 3 (5 𝐴𝐴 1 1 − 𝐴𝐴 1 2 ) + 7 𝛾𝛾 (3 𝐴𝐴 1 1 − 𝐴𝐴 1 2 ) + 3 𝛾𝛾 (7 𝐴𝐴 1 3 − 𝐴𝐴 1 4 ) − 5 𝛾𝛾 3 (5 𝐴𝐴 1 3 − 𝐴𝐴 1 4 ) + 2 𝛾𝛾 4 (2 𝐴𝐴 1 1 + 𝐴𝐴 1 2 ) , 𝑐𝑐 3 𝐷𝐷 ∗ = − 2 𝛾𝛾 6 (3 𝐴𝐴 1 1 − 𝐴𝐴 1 2 ) + 3 𝛾𝛾 4 (2 𝐴𝐴 1 3 + 𝐴𝐴 1 4 ) − 5 𝛾𝛾 3 𝐴𝐴 1 2 − 5 𝛾𝛾 2 𝐴𝐴 1 4 + 3 𝛾𝛾 (2 𝐴𝐴 1 1 + 𝐴𝐴 1 2 ) − 2 𝛾𝛾 (3 𝐴𝐴 1 3 − 𝐴𝐴 1 4 ) , 𝑐𝑐 4 𝐷𝐷 ∗ = − 2 𝛾𝛾 2 𝐴𝐴 1 4 + 3 𝛾𝛾 (5 𝐴𝐴 1 1 − 𝐴𝐴 1 2 ) − 5 𝛾𝛾 (3 𝐴𝐴 1 1 − 𝐴𝐴 1 2 + 3 𝐴𝐴 1 3 − 𝐴𝐴 1 4 ) + 5 𝛾𝛾 3 (3 𝐴𝐴 1 3 − 𝐴𝐴 1 4 ) − 2 𝛾𝛾 4 𝐴𝐴 1 2 , 𝑐𝑐 5 = 1 1 − 𝛾𝛾 5 [ − 𝐻𝐻 5 ( 𝛾𝛾 ) − 𝐻𝐻 6 ( 𝛾𝛾 ) 𝛾𝛾 5 + 𝐻𝐻 5 (1) 𝛾𝛾 5 + 𝐻𝐻 6 (1) 𝛾𝛾 5 ] , 𝑐𝑐 6 = 1 1 − 𝛾𝛾 5 [ 𝐻𝐻 5 ( 𝛾𝛾 ) + 𝐻𝐻 6 ( 𝛾𝛾 ) 𝛾𝛾 5 − 𝐻𝐻 5 (1) − 𝐻𝐻 6 (1)] , (4.25) with 𝐷𝐷 ∗ = 4 𝛾𝛾 6 − 2 5 𝛾𝛾 3 + 4 2 𝛾𝛾 − 25 𝛾𝛾 + 4 𝛾𝛾 4 , 𝐴𝐴 1 1 = − 𝐻𝐻 1 (1) − 𝐻𝐻 2 (1) − 𝐻𝐻 3 (1) − 𝐻𝐻 4 (1) , 𝐴𝐴 1 2 = −3 𝐻𝐻 1 (1) − 5 𝐻𝐻 3 (1) + 2 𝐻𝐻 4 (1) , 𝐴𝐴 1 3 = − 𝛾𝛾 𝐻𝐻 1 ( 𝛾𝛾 ) − 1 𝛾𝛾 2 𝐻𝐻 2 ( 𝛾𝛾 ) − 𝛾𝛾 3 𝐻𝐻 3 ( 𝛾𝛾 ) − 1 𝛾𝛾 4 𝐻𝐻 4 ( 𝛾𝛾 ) , 𝐴𝐴 1 4 = −3 𝛾𝛾 𝐻𝐻 1 ( 𝛾𝛾 ) − 5 𝛾𝛾 3 𝐻𝐻 3 ( 𝛾𝛾 ) + 2 𝛾𝛾 4 𝐻𝐻 4 ( 𝛾𝛾 ) , 𝐻𝐻 1 = − | 𝑀𝑀 | 2 5 � 1 √ 6 [ 𝑟𝑟 ′ ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ )d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 42 𝐻𝐻 2 = | 𝑀𝑀 | 2 5 � 1 √ 6 [𝑟𝑟 ′ 4 ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 3 √ 6 ∫ ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 3 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 3 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 3 = 3| 𝑀𝑀 | 2 3 5 � 1 √ 6 � ℱ 𝐵𝐵 2 (𝑟𝑟 ′ ) 𝑟𝑟 ′ � 𝛾𝛾 𝑟𝑟 + 2 √ 6 ∫ ℱ 𝐵𝐵 2 (𝑟𝑟 ′ ) 𝑟𝑟 ′ 2 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 − ∫ ℱ 𝑃𝑃 2 (𝑟𝑟 ′ ) 𝑟𝑟 ′ 2 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 4 = − 3| 𝑀𝑀 | 2 3 5 � 1 √ 6 [𝑟𝑟 ′ 6 ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ )] 𝛾𝛾 𝑟𝑟 − 5 √ 6 ∫ ℱ 𝐵𝐵 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 5 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 − ∫ ℱ 𝑃𝑃 2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 5 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 � , 𝐻𝐻 5 = − | 𝑀𝑀 | 2 5 ∫ ℱ 𝐶𝐶2 ( 𝑟𝑟 ′ ) 𝑟𝑟 ′ 4 d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 , 𝐻𝐻 6 = | 𝑀𝑀 | 2 5 ∫ ℱ 𝐶𝐶 2 � 𝑟𝑟 ′ � 𝑟𝑟 ′ d𝑟𝑟 ′ 𝑟𝑟 𝛾𝛾 . (4.26) The velocity can now be expressed as 𝒖𝒖 1 0 = 1 4 � 5 𝜋𝜋 � 𝑢𝑢 1 0 𝑟𝑟 , 2 (3 cos 2 𝜃𝜃 − 1), − √6 𝑢𝑢 1 0 𝜃𝜃 , 2 cos 𝜃𝜃 sin 𝜃𝜃 , √6 𝑢𝑢 1 0 𝜙𝜙 , 2 cos 𝜃𝜃 sin 𝜃𝜃 � . (4.27) The stream function and angular circulation take the form 𝜓𝜓 1 0 = 1 4 � 5 𝜋𝜋 𝑟𝑟 2 𝑢𝑢 1 0 𝑟𝑟 , 2 sin 2 𝜃𝜃 cos 𝜃𝜃 , Ω 1 0 = 1 4 � 3 0 𝜋𝜋 𝑟𝑟 𝑢𝑢 1 0 𝜙𝜙 , 2 sin 2 𝜃𝜃 cos 𝜃𝜃 , (4.28) While these results are for unrestricted | 𝑀𝑀 |, they are mathematically complex. A better insight is achieved with some asymptotic analysis. Therefore, the problem is also discussed separately in low-frequency (| 𝑀𝑀 | 2 ≪ 1) and high-frequency (| 𝑀𝑀 | 2 ≫ 1) limits. 4.2.3 The low-frequency limit, 𝜺𝜺 ≪ 𝟏𝟏 ≪ | 𝑴𝑴 | − 𝟐𝟐 For low frequency, the viscous diffusion thickness is larger than the characteristic dimension of the body, which means, vorticity diffuses over a much wider region. We now seek a perturbation solution of governing equations in the form 𝜓𝜓 = 𝜓𝜓 0 + 𝑅𝑅 𝜓𝜓 1 + 𝑅𝑅 2 𝜓𝜓 2 + 𝑂𝑂 ( 𝑅𝑅 3 ) , Ω = Ω 0 + 𝑅𝑅 Ω 1 + 𝑅𝑅 2 Ω 2 + 𝑂𝑂 ( 𝑅𝑅 3 ) , (4.29) 43 where 𝑅𝑅 is the oscillatory Reynolds number defined as 𝑅𝑅 = 𝜀𝜀 | 𝑀𝑀 | 2 = 𝑈𝑈 0 𝜔𝜔 𝜈𝜈 ≪ 1, (4.30) Then, in the limit of | 𝑀𝑀 | → 0, Eqn. (4.11) and Eqn. (4.14) can be simplified to 𝜓𝜓 0 = 3 2 𝑖𝑖 𝛾𝛾 ( 1 − 𝛾𝛾 ) 4 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) �− 1 3 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 + (1 − 𝛾𝛾 5 ) 𝑟𝑟 − 1 6 (9 − 5 𝛾𝛾 2 − 4 𝛾𝛾 5 ) 𝑟𝑟 2 + 1 2 (1 − 𝛾𝛾 2 ) 𝑟𝑟 4 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. + 𝑂𝑂 (| 𝑀𝑀 | 2 ) , Ω 0 = 1 2 𝑖𝑖 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝛾𝛾 2 1 − 𝛾𝛾 3 � 1 𝑟𝑟 − 𝑟𝑟 2 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. + 𝑂𝑂 (| 𝑀𝑀 | 2 ) , (4.31) and 𝜏𝜏 0 𝑟𝑟 𝜃𝜃 = 3 2 𝑖𝑖 𝛾𝛾 ( 1 − 𝛾𝛾 ) 4 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) �2 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 4 − 3(1 − 𝛾𝛾 2 ) 𝑟𝑟 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝜏𝜏 0𝜙𝜙 𝜃𝜃 = 0, 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 = − 3 2 𝑖𝑖 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝛾𝛾 2 1 − 𝛾𝛾 3 1 𝑟𝑟 3 (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. . (4.32) Next, solutions of 𝑂𝑂 ( 𝑅𝑅 ) satisfy 𝐷𝐷 4 𝜓𝜓 1 = 1 𝑟𝑟 2 � 𝜕𝜕 � 𝜓𝜓 0 , 𝐷𝐷 2 𝜓𝜓 0 � 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) + 2 𝐷𝐷 2 𝜓𝜓 0 𝐿𝐿 𝜓𝜓 0 + 2 Ω 0 𝐿𝐿 Ω 0 � + 𝑂𝑂 (| 𝑀𝑀 |) , 𝐷𝐷 2 Ω 1 = 1 𝑟𝑟 2 𝜕𝜕 ( 𝜓𝜓 0 , Ω 0 ) 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) + 𝑂𝑂 (| 𝑀𝑀 |) . (4.33) As with velocity and pressure fields, we can write 𝜓𝜓 1 = 𝜓𝜓 1 ( 𝑠𝑠 ) + 𝜓𝜓 1 ( 𝑢𝑢 ) = { 𝜓𝜓 1 0 ( 𝑟𝑟 , 𝜇𝜇 ) + 𝜓𝜓 1 2 ( 𝑟𝑟 , 𝜇𝜇 ) 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 } + c.c. , Ω 1 = Ω 1 ( 𝑠𝑠 ) + Ω 1 ( 𝑢𝑢 ) = { Ω 1 0 ( 𝑟𝑟 , 𝜇𝜇 ) + Ω 1 2 ( 𝑟𝑟 , 𝜇𝜇 ) 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 } + c.c. , (4.34) where the superscripts (𝑠𝑠 ) and (𝑢𝑢 ) refer to steady and unsteady parts, respectively, with boundary conditions − 1 𝑟𝑟 2 𝜕𝜕 𝜓𝜓 1 𝑛𝑛 𝜕𝜕𝜕𝜕 = 0, − � 1 − 𝜕𝜕 2 � − 1 2 𝑟𝑟 𝜕𝜕 𝜓𝜓 1 𝑛𝑛 𝜕𝜕 𝑟𝑟 = 0 and Ω 1 𝑛𝑛 𝑟𝑟 ( 1 − 𝜕𝜕 2 ) 1 2 = 0 on = 𝑅𝑅 𝑖𝑖 𝜔𝜔 = 𝛾𝛾 , 44 𝜓𝜓 1 𝑛𝑛 = ∂ 𝜓𝜓 1 𝑛𝑛 ∂ 𝑟𝑟 = 0, Ω 1 𝑛𝑛 = 0 on 𝑟𝑟 = 1 (𝑛𝑛 = 0,2). (4.35) The solutions for steady streaming are 𝜓𝜓 1 ( 𝑠𝑠 ) = � �𝐷𝐷 3 1 𝑟𝑟 2 + 𝐷𝐷 4 𝑟𝑟 3 + 𝐷𝐷 5 𝑟𝑟 5 + 𝐷𝐷 6 � + 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 �2 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 + 6(1 − 𝛾𝛾 5 ) 𝑟𝑟 − (9 − 5 𝛾𝛾 2 − 4 𝛾𝛾 5 ) 𝑟𝑟 2 − 3(1 − 𝛾𝛾 2 ) 𝑟𝑟 4 � + 𝛼𝛼 2 1 6 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 � 1 𝑟𝑟 + 𝑟𝑟 2 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , Ω 1 ( 𝑠𝑠 ) = � �𝐷𝐷 1 1 𝑟𝑟 2 + 𝐷𝐷 2 𝑟𝑟 3 � + 3 1 1 2 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝛾𝛾 3 ( 1 − 𝛾𝛾 ) 4 ( 1 − 𝛾𝛾 3 )( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) �28(1 − 𝛾𝛾 5 ) 1 𝑟𝑟 − 14(3 − 𝛾𝛾 2 − 2 𝛾𝛾 5 ) + 7(7 − 5 𝛾𝛾 2 − 2 𝛾𝛾 5 ) 𝑟𝑟 2 + 4(1 − 𝛾𝛾 2 ) 𝑟𝑟 5 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. . (4.36) Thus, velocities for steady streaming are 𝑢𝑢 1 𝑟𝑟 ( 𝑠𝑠 ) = � �𝐷𝐷 3 1 𝑟𝑟 4 + 𝐷𝐷 4 𝑟𝑟 + 𝐷𝐷 5 𝑟𝑟 3 + 𝐷𝐷 6 1 𝑟𝑟 2 � + 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 �2 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 3 + 6(1 − 𝛾𝛾 5 ) 1 𝑟𝑟 − (9 − 5 𝛾𝛾 2 − 4 𝛾𝛾 5 ) − 3(1 − 𝛾𝛾 2 ) 𝑟𝑟 2 � + 𝛼𝛼 2 1 6 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 � 1 𝑟𝑟 3 + 1 �� (3 𝜇𝜇 2 − 1) + c.c. , 𝑢𝑢 1 𝜃𝜃 ( 𝑠𝑠 ) = � �2 𝐷𝐷 3 1 𝑟𝑟 4 − 3 𝐷𝐷 4 𝑟𝑟 − 5 𝐷𝐷 5 𝑟𝑟 3 � + 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 �2 𝛾𝛾 2 (1 − 𝛾𝛾 3 ) 1 𝑟𝑟 3 − 6(1 − 𝛾𝛾 5 ) 1 𝑟𝑟 + 2(9 − 5 𝛾𝛾 2 − 4 𝛾𝛾 5 ) + 12(1 − 𝛾𝛾 2 ) 𝑟𝑟 2 � + 𝛼𝛼 2 1 6 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 � 1 𝑟𝑟 3 − 2 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) 1 2 + c.c. , 𝑢𝑢 1 𝜙𝜙 ( 𝑠𝑠 ) = � �𝐷𝐷 1 1 𝑟𝑟 3 + 𝐷𝐷 2 𝑟𝑟 2 � + 3 1 1 2 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝛾𝛾 3 ( 1 − 𝛾𝛾 ) 4 ( 1 − 𝛾𝛾 3 )( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) �28(1 − 𝛾𝛾 5 ) 1 𝑟𝑟 2 − 14(3 − 𝛾𝛾 2 − 2 𝛾𝛾 5 ) 1 𝑟𝑟 + 7(7 − 5 𝛾𝛾 2 − 2 𝛾𝛾 5 ) 𝑟𝑟 + 4(1 − 𝛾𝛾 2 ) 𝑟𝑟 4 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) 1 2 + c.c. , (4.37) where 𝐷𝐷 1 = − 3 1 1 2 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝛾𝛾 5 ( 1 − 𝛾𝛾 ) 4 ( 1 − 𝛾𝛾 3 )( 1 − 𝛾𝛾 5 )( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) � 2 8 𝛾𝛾 − 42 + 63 𝛾𝛾 2 − 39 𝛾𝛾 3 − 63 𝛾𝛾 4 + 57 𝛾𝛾 5 − 18 𝛾𝛾 7 + 14 𝛾𝛾 8 � , 45 𝐷𝐷 2 = − 3 1 1 2 𝛼𝛼 𝑒𝑒 𝑖𝑖 𝑖𝑖 𝛾𝛾 5 ( 1 − 𝛾𝛾 ) 4 ( 1 − 𝛾𝛾 3 )( 1 − 𝛾𝛾 5 )( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) � 3 9 𝛾𝛾 2 − 2 8 𝛾𝛾 + 17 − 63 𝛾𝛾 2 − 14 𝛾𝛾 3 + 63 𝛾𝛾 4 − 32 𝛾𝛾 5 + 18 𝛾𝛾 7 � , 𝐷𝐷 3 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 [ 𝑇𝑇 1 ( −15 𝛾𝛾 4 + 15 𝛾𝛾 2 ) + 𝑇𝑇 2 ( −2 𝛾𝛾 7 + 5 𝛾𝛾 4 − 3 𝛾𝛾 2 ) + 𝑇𝑇 3 (15 𝛾𝛾 4 − 15 𝛾𝛾 2 ) + 𝑇𝑇 4 ( −3 𝛾𝛾 5 + 5 𝛾𝛾 3 − 2)] , 𝐷𝐷 4 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 �𝑇𝑇 1 �−10 𝛾𝛾 4 + 10 1 𝛾𝛾 3 � + 𝑇𝑇 2 �7 𝛾𝛾 2 − 5 𝛾𝛾 4 − 2 𝛾𝛾 3 � + 𝑇𝑇 3 �10 𝛾𝛾 4 − 10 1 𝛾𝛾 3 � + 𝑇𝑇 4 �−2 𝛾𝛾 5 − 5 1 𝛾𝛾 2 + 7 �� , 𝐷𝐷 5 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 �𝑇𝑇 1 �6 𝛾𝛾 2 − 6 1 𝛾𝛾 3 � + 𝑇𝑇 2 �3 𝛾𝛾 2 − 5 + 2 𝛾𝛾 3 � + 𝑇𝑇 3 �−6 𝛾𝛾 2 + 6 1 𝛾𝛾 3 � + 𝑇𝑇 4 �2 𝛾𝛾 3 − 5 + 3 𝛾𝛾 2 �� , 𝐷𝐷 6 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 �𝑇𝑇 1 (4 𝛾𝛾 7 + 21 𝛾𝛾 2 − 25) + 𝑇𝑇 2 (2 𝛾𝛾 7 − 7 𝛾𝛾 2 + 5) + 𝑇𝑇 3 �−25 𝛾𝛾 4 + 21 𝛾𝛾 2 + 4 1 𝛾𝛾 3 � + 𝑇𝑇 4 �5 𝛾𝛾 5 − 7 𝛾𝛾 3 + 2 𝛾𝛾 2 �� , (4.38) and 𝑇𝑇 1 = − 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 ( −6 + 10 𝛾𝛾 2 − 4 𝛾𝛾 5 ) − 1 8 𝛼𝛼 2 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 , 𝑇𝑇 2 = − 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 ( −24 + 20 𝛾𝛾 2 + 4 𝛾𝛾 5 ) − 1 1 6 𝛼𝛼 2 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 , 𝑇𝑇 3 = − 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 (8 𝛾𝛾 − 9 𝛾𝛾 2 − 3 𝛾𝛾 6 + 4 𝛾𝛾 7 ) − 1 1 6 𝛼𝛼 2 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 � 1 𝛾𝛾 + 𝛾𝛾 2 � , 𝑇𝑇 4 = − 3 1 6 𝛾𝛾 2 � 1 − 𝛾𝛾 5 � ( 1 − 𝛾𝛾 ) 8 ( 4 + 7 𝛾𝛾 + 4 𝛾𝛾 2 ) 2 (4 − 18 𝛾𝛾 + 6 𝛾𝛾 5 + 8 𝛾𝛾 6 ) − 1 1 6 𝛼𝛼 2 � 𝛾𝛾 2 1 − 𝛾𝛾 3 � 2 �− 1 𝛾𝛾 2 + 2 𝛾𝛾 � . (4.39) 46 4.2.4 The high-frequency limit, 𝜺𝜺 ≪ 𝟏𝟏 ≪ | 𝑴𝑴 | 𝟐𝟐 In this limit, the viscous diffusion thickness is very small. The vorticity is confined to narrow Stokes layers on the two boundaries. The space between the two spheres can be divided into three parts, the inner region around the inner sphere, the inner region around the container, and the outer region which is the bulk of the annular region. A schematic of the different regions is depicted in Figure 4.2. Figure 4.2 A schematic of the streaming flow regions at high frequency. 47 4.2.4.1 The oscillating field First we consider the inner region, which is the boundary layer around the inner sphere. The inner variables with the Stokes layers are scaled as follows: 𝜂𝜂 = ( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 and Ψ = 𝜓𝜓 | 𝑀𝑀 | √ 2 , (4.40) As a result, we have the governing equations become 𝜕𝜕 𝜕𝜕 𝑡𝑡 � 𝜕𝜕 2 Ψ 𝜕𝜕 𝜂𝜂 2 � + 𝜀𝜀 𝛾𝛾 2 � 𝜕𝜕 � Ψ, 𝜕𝜕 2 Ψ 𝜕𝜕 𝜂𝜂 2 � 𝜕𝜕 ( 𝜂𝜂 , 𝜕𝜕 ) + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) 𝜕𝜕 Ψ 𝜕𝜕 𝜂𝜂 𝜕𝜕 2 Ψ 𝜕𝜕 𝜂𝜂 2 + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) Ω 𝜕𝜕 Ω 𝜕𝜕 𝜂𝜂 � = 1 2 𝜕𝜕 4 Ψ 𝜕𝜕 𝜂𝜂 4 + H.O.T. , 𝜕𝜕 Ω 𝜕𝜕 𝑡𝑡 + 𝜀𝜀 𝛾𝛾 2 � 𝜕𝜕 ( Ψ, Ω) 𝜕𝜕 ( 𝜂𝜂 , 𝜕𝜕 ) � = 1 2 𝜕𝜕 2 Ω 𝜕𝜕 𝜂𝜂 2 + H.O.T. , (4.41) with boundary conditions 𝜂𝜂 = 0, 𝜕𝜕 Ψ 𝜕𝜕𝜕𝜕 = − 𝛾𝛾 2 | 𝑀𝑀 | √ 2 𝜇𝜇 sin 𝑡𝑡 , 𝜕𝜕 Ψ 𝜕𝜕 𝜂𝜂 = 𝛾𝛾 (1 − 𝜇𝜇 2 ) sin 𝑡𝑡 and Ω = 𝛾𝛾𝛼𝛼 (1 − 𝜇𝜇 2 ) sin( 𝑡𝑡 + 𝛽𝛽 ) . (4.42) In this case, Ψ and Ω can be expressed as Ψ = Ψ 0 + 𝜀𝜀 Ψ 1 + 𝑂𝑂 ( 𝜀𝜀 2 ) , Ω = Ω 0 + 𝜀𝜀 Ω 1 + 𝑂𝑂 ( 𝜀𝜀 2 ) . (4.43) We have for the leading order, 𝜕𝜕 𝜕𝜕 𝑡𝑡 � 𝜕𝜕 2 Ψ 0 𝜕𝜕 𝜂𝜂 2 � = 1 2 𝜕𝜕 4 Ψ 0 𝜕𝜕 𝜂𝜂 4 , 𝜕𝜕 Ω 0 𝜕𝜕 𝑡𝑡 = 1 2 𝜕𝜕 2 Ω 0 𝜕𝜕 𝜂𝜂 2 . (4.44) These equations have solutions Ψ 0 = � 𝐷𝐷 0 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 + � 𝛾𝛾 2 𝑖𝑖 + (1 + 𝑠𝑠 ) 𝐷𝐷 0 � 𝜂𝜂 + | 𝑀𝑀 | 4 √ 2 𝑖𝑖 𝛾𝛾 2 − 𝐷𝐷 0 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 𝛼𝛼 𝛾𝛾 2 𝑖𝑖 (1 − 𝜇𝜇 2 ) 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 𝑒𝑒 𝑖𝑖 ( 𝑡𝑡 + 𝑖𝑖 ) + c.c. . (4.45) Next, we have two equations for the outer region, 48 𝐷𝐷 2 𝜓𝜓 0 = 0, 𝜕𝜕 Ω 0 𝜕𝜕 𝑡𝑡 = 0. (4.46) These equations have solutions 𝜓𝜓 0 = �𝐷𝐷 0 ∗ 𝑟𝑟 2 + 𝑏𝑏 0 ∗ 1 𝑟𝑟 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 𝐹𝐹 ( 𝑟𝑟 ) . (4.47) Finally, in the boundary layer on the container, with boundary layer variables introduced as, 𝜂𝜂 ̂ = (1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 and Ψ � = 𝜓𝜓 | 𝑀𝑀 | √ 2 , (4.48) the governing equations become 𝜕𝜕 𝜕𝜕 𝑡𝑡 � 𝜕𝜕 2 Ψ � 𝜕𝜕 𝜂𝜂 � 2 � − 𝜀𝜀 � 𝜕𝜕 � Ψ � , 𝜕𝜕 2 Ψ � 𝜕𝜕 𝜂𝜂 2 � 𝜕𝜕 ( 𝜂𝜂 � , 𝜕𝜕 ) + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) 𝜕𝜕 Ψ � 𝜕𝜕 𝜂𝜂 � 𝜕𝜕 2 Ψ � 𝜕𝜕 𝜂𝜂 � 2 + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) Ω 𝜕𝜕 Ω 𝜕𝜕 𝜂𝜂 � � = 1 2 𝜕𝜕 4 Ψ � 𝜕𝜕 𝜂𝜂 � 4 + H.O.T. , 𝜕𝜕 Ω 𝜕𝜕 𝑡𝑡 − 𝜀𝜀 � 𝜕𝜕 ( Ψ � , Ω) 𝜕𝜕 ( 𝜂𝜂 � , 𝜕𝜕 ) � = 1 2 𝜕𝜕 2 Ω 𝜕𝜕 𝜂𝜂 � 2 + H.O.T. , (4.49) with boundary conditions 𝜂𝜂 ̂ = 0, Ψ � = 𝜕𝜕 Ψ � 𝜕𝜕 𝜂𝜂 � = Ω = 0. (4.50) After the perturbation method Ψ � = Ψ � 0 + 𝜀𝜀 Ψ � 1 + 𝑂𝑂 ( 𝜀𝜀 2 ), Ω = Ω 0 + 𝜀𝜀 Ω 1 + 𝑂𝑂 ( 𝜀𝜀 2 ), (4.51) into the governing equation (4.49), equations to order 𝑂𝑂 (1) in the boundary layer on the container are 𝜕𝜕 𝜕𝜕 𝑡𝑡 � 𝜕𝜕 2 Ψ � 0 𝜕𝜕 𝜂𝜂 � 2 � = 1 2 𝜕𝜕 4 Ψ � 0 𝜕𝜕 𝜂𝜂 � 4 , 𝜕𝜕 Ω 0 𝜕𝜕 𝑡𝑡 = 1 2 𝜕𝜕 2 Ω 0 𝜕𝜕 𝜂𝜂 � 2 . (4.52) Solutions are 49 Ψ � 0 = 𝑑𝑑 0 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + (1 + 𝑠𝑠 ) 𝜂𝜂 ̂ − 1 �(1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 0 . (4.53) The coefficients in the solutions of each region are yet undetermined and can be obtained later by matching with the solutions in other region next to them, according to the requirement, Ψ 0 | 𝜂𝜂 →∞ = | 𝑀𝑀 | √ 2 𝜓𝜓 0 | 𝑟𝑟 →𝛾𝛾 , Ψ � 0 | 𝜂𝜂 � →∞ = | 𝑀𝑀 | √ 2 𝜓𝜓 0 | 𝑟𝑟 →1 , (4.54) The left sides of the matching requirement are Ψ 0 | 𝜂𝜂 →∞ = � � 𝛾𝛾 2 𝑖𝑖 + (1 + 𝑠𝑠 ) 𝐷𝐷 0 � 𝜂𝜂 + | 𝑀𝑀 | 4 √ 2 𝑖𝑖 𝛾𝛾 2 − 𝐷𝐷 0 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ψ � 0 | 𝜂𝜂 � →∞ = 𝑑𝑑 0 [(1 + 𝑠𝑠 ) 𝜂𝜂 ̂ − 1](1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. . (4.55) Then we take Taylor expansion of 𝜓𝜓 0 about 𝑟𝑟 = 𝛾𝛾 and replace 𝑟𝑟 − 𝛾𝛾 with √ 2 | 𝑀𝑀 | 𝜂𝜂 to get 𝜓𝜓 0 ~ �𝐷𝐷 0 ∗ �𝛾𝛾 2 + 2 𝛾𝛾 √ 2 | 𝑀𝑀 | 𝜂𝜂 � + 𝑏𝑏 0 ∗ � 1 𝛾𝛾 − 1 𝛾𝛾 2 √ 2 | 𝑀𝑀 | 𝜂𝜂 �� (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. = ��𝐷𝐷 0 ∗ 𝛾𝛾 2 + 𝑏𝑏 0 ∗ 1 𝛾𝛾 � + �𝐷𝐷 0 ∗ 2 𝛾𝛾 − 𝑏𝑏 0 ∗ 1 𝛾𝛾 2 � √ 2 | 𝑀𝑀 | 𝜂𝜂 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. . (4.56) Similarly, Taylor expansion of 𝜓𝜓 0 about 𝑟𝑟 = 1 and replacement of 1 − 𝑟𝑟 with √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ leads to 𝜓𝜓 0 ~ �𝐷𝐷 0 ∗ �1 − 2 √ 2 | 𝑀𝑀 | 𝜂𝜂 � + 𝑏𝑏 0 ∗ �1 + √ 2 | 𝑀𝑀 | 𝜂𝜂 �� (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. = �( 𝐷𝐷 0 ∗ + 𝑏𝑏 0 ∗ ) + ( −2 𝐷𝐷 0 ∗ + 𝑏𝑏 0 ∗ ) √ 2 | 𝑀𝑀 | 𝜂𝜂 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. . (4.57) As | 𝑀𝑀 | 2 ≫ 1, the coefficients are obtained, 𝐷𝐷 0 ∗ = − 1 4 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � , 𝑏𝑏 0 ∗ = 1 4 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � , 50 𝐷𝐷 0 = 3( 1 + 𝑖𝑖 ) 8 𝛾𝛾 1 − 𝛾𝛾 3 , 𝑑𝑑 0 = − 3( 1 + 𝑖𝑖 ) 8 𝛾𝛾 3 1 − 𝛾𝛾 3 . (4.58) To sum up, the matching yields leading order solutions as follows: For outer region, 𝜓𝜓 0 = − 1 4 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � �𝑟𝑟 2 − 1 𝑟𝑟 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 0 . (4.59) For the inner region around the inner sphere, Ψ 0 = � 3( 1 + 𝑖𝑖 ) 8 𝛾𝛾 1 − 𝛾𝛾 3 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) − 1 � + | 𝑀𝑀 | 4 √ 2 𝑠𝑠 � 2 𝛾𝛾 � 1 + 𝛾𝛾 3 � 1 − 𝛾𝛾 3 ( 𝑟𝑟 − 𝛾𝛾 ) − 𝛾𝛾 2 �� (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 𝛼𝛼 𝛾𝛾 2 𝑖𝑖 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 ( 𝑡𝑡 + 𝑖𝑖 ) + c.c. . (4.60) For the boundary layer around the container, Ψ � 0 = − 3( 1 + 𝑖𝑖 ) 8 𝛾𝛾 3 1 − 𝛾𝛾 3 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 1 − 𝑟𝑟 ) + (1 + 𝑠𝑠 ) | 𝑀𝑀 | √ 2 (1 − 𝑟𝑟 ) − 1 � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 0 . (4.61) The composite solution set is 𝜓𝜓 0 = � √ 2 | 𝑀𝑀 | � 3( 1 + 𝑖𝑖 ) 8 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) − 3( 1 + 𝑖𝑖 ) 8 𝛾𝛾 3 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 1 − 𝑟𝑟 ) � − 1 4 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � �𝑟𝑟 2 − 1 𝑟𝑟 �� (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 0 = 𝛼𝛼 𝛾𝛾 2 𝑖𝑖 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 ( 𝑡𝑡 + 𝑖𝑖 ) + c.c. . (4.62) Velocities are 51 𝑢𝑢 0 𝑟𝑟 = � √ 2 | 𝑀𝑀 | 3( 1 + 𝑖𝑖 ) 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 1 𝑟𝑟 2 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) − √ 2 | 𝑀𝑀 | 3( 1 + 𝑖𝑖 ) 4 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � 1 𝑟𝑟 2 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 1 − 𝑟𝑟 ) − 1 2 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � �1 − 1 𝑟𝑟 3 �� 𝜇𝜇 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝑢𝑢 0 𝜃𝜃 = � 3 𝑖𝑖 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 1 𝑟𝑟 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) + 3 𝑖𝑖 4 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � 1 𝑟𝑟 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 1 − 𝑟𝑟 ) + 1 4 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � �2 + 1 𝑟𝑟 3 �� (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝑢𝑢 0 𝜙𝜙 = 1 2 𝑖𝑖 𝛼𝛼𝛾𝛾 𝑒𝑒 𝑖𝑖 𝑖𝑖 1 𝑟𝑟 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. . (4.63) Shear stresses are 𝜏𝜏 0 𝑟𝑟 𝜃𝜃 = � 3 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � � �− 2 𝑖𝑖 𝑟𝑟 2 − √ 2 | 𝑀𝑀 | (1 + 𝑠𝑠 ) 1 𝑟𝑟 3 + | 𝑀𝑀 | √ 2 (1 − 𝑠𝑠 ) 1 𝑟𝑟 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) � + 3 4 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � � �− 2 𝑖𝑖 𝑟𝑟 2 + √ 2 | 𝑀𝑀 | (1 + 𝑠𝑠 ) 1 𝑟𝑟 3 − | 𝑀𝑀 | √ 2 (1 − 𝑠𝑠 ) 1 𝑟𝑟 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 1 − 𝑟𝑟 ) � − 3 2 𝑖𝑖 � 𝛾𝛾 3 1 − 𝛾𝛾 3 � 1 𝑟𝑟 4 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 = 𝛼𝛼 𝛾𝛾 𝑒𝑒 𝑖𝑖𝑖𝑖 2 𝑖𝑖 �− 2 𝑟𝑟 2 − | 𝑀𝑀 | √ 2 (1 + 𝑠𝑠 ) 1 𝑟𝑟 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝜏𝜏 0𝜙𝜙 𝜃𝜃 = 0. (4.64) 4.2.4.2 Steady streaming To order 𝜀𝜀 , the streaming is still studied separately in three regions, and then solutions are matched to determine coefficients in those solutions. In the boundary layer around the inner sphere, Ψ 1 and Ω 1 satisfy 𝜕𝜕 𝜕𝜕 𝑡𝑡 � 𝜕𝜕 2 Ψ 1 𝜕𝜕 𝜂𝜂 2 � − 1 2 𝜕𝜕 4 Ψ 1 𝜕𝜕 𝜂𝜂 4 = − 1 𝛾𝛾 2 � 𝜕𝜕 � Ψ 0 , 𝜕𝜕 2 Ψ 0 𝜕𝜕 𝜂𝜂 2 � 𝜕𝜕 ( 𝜂𝜂 , 𝜕𝜕 ) + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) 𝜕𝜕 Ψ 0 𝜕𝜕 𝜂𝜂 𝜕𝜕 2 Ψ 0 𝜕𝜕 𝜂𝜂 2 + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) Ω 0 𝜕𝜕 Ω 0 𝜕𝜕 𝜂𝜂 � , 𝜕𝜕 Ω 1 𝜕𝜕 𝑡𝑡 − 1 2 𝜕𝜕 2 Ω 1 𝜕𝜕 𝜂𝜂 2 = − 1 𝛾𝛾 2 𝜕𝜕 ( Ψ 0 , Ω 0 ) 𝜕𝜕 ( 𝜂𝜂 , 𝜕𝜕 ) , (4.65) with boundary conditions 𝜂𝜂 = 0, 𝜕𝜕 Ψ 1 𝜕𝜕𝜕𝜕 = 𝜕𝜕 Ψ 1 𝜕𝜕 𝜂𝜂 = Ω 1 = 0. (4.66) 52 The solutions of Ψ 1 and Ω 1 are sought in the form, Ψ 1 = Ψ 1 ( 𝑠𝑠 ) + Ψ 1 ( 𝑢𝑢 ) = � Ψ 1 0 ( 𝜂𝜂 ) + Ψ 1 2 ( 𝜂𝜂 ) 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , Ω 1 = Ω 1 ( 𝑠𝑠 ) + Ω 1 ( 𝑢𝑢 ) = � Ω 1 0 ( 𝜂𝜂 ) + Ω 1 2 ( 𝜂𝜂 ) 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. . (4.67) Solutions are Ψ 1 0 = 2 𝛾𝛾 2 � − 3 3 2 �2 𝑠𝑠 | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 32(1 + 𝑠𝑠 ) 𝛾𝛾 − 3(5 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 + 1 + 𝑖𝑖 3 2 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � 𝑒𝑒 − 2 𝜂𝜂 − 3 𝑖𝑖 8 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� 𝛾𝛾 1 − 𝛾𝛾 3 𝜂𝜂 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + 𝐴𝐴 1 𝜂𝜂 3 + 𝐵𝐵 1 𝜂𝜂 2 + 𝐶𝐶 1 𝜂𝜂 + 𝐷𝐷 1 , Ω 1 0 = 𝛼𝛼 𝑒𝑒 𝑖𝑖𝑖𝑖 2 𝛾𝛾 � − 3( 1 − 𝑖𝑖 ) 1 6 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 − 2 𝜂𝜂 − 1 − 𝑖𝑖 4 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� 𝜂𝜂 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 + 1 8 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + 𝐴𝐴 1 ∗ 𝜂𝜂 + 𝐵𝐵 1 ∗ , (4.68) where 𝐵𝐵 1 ∗ = − 2 𝛼𝛼 𝑒𝑒 𝑖𝑖𝑖𝑖 𝛾𝛾 � − 3( 1 − 𝑖𝑖 ) 1 6 𝛾𝛾 1 − 𝛾𝛾 3 + 1 8 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 �� , 𝐶𝐶 1 = − 2 𝛾𝛾 2 � − 1 + 𝑖𝑖 1 6 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � + 3 1 6 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 � , 𝐷𝐷 1 = − 2 𝛾𝛾 2 � 1 + 𝑖𝑖 3 2 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � − 3 3 2 �2 𝑠𝑠 | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 8(1 + 𝑠𝑠 ) 𝛾𝛾 − 3(5 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 � . (4.69) 𝐴𝐴 1 , 𝐵𝐵 1 and 𝐴𝐴 1 ∗ need to be determined by matching with the out flow. In the boundary layer on the container, we have 𝜕𝜕 𝜕𝜕 𝑡𝑡 � 𝜕𝜕 2 Ψ � 1 𝜕𝜕 𝜂𝜂 � 2 � − 1 2 𝜕𝜕 4 Ψ � 1 𝜕𝜕 𝜂𝜂 4 = 𝜕𝜕 � Ψ � 0 , 𝜕𝜕 2 Ψ � 0 𝜕𝜕 𝜂𝜂 2 � 𝜕𝜕 ( 𝜂𝜂 � , 𝜕𝜕 ) + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) 𝜕𝜕 Ψ � 0 𝜕𝜕 𝜂𝜂 � 𝜕𝜕 2 Ψ � 0 𝜕𝜕 𝜂𝜂 � 2 + 2 𝜕𝜕 ( 1 − 𝜕𝜕 2 ) Ω 0 𝜕𝜕 Ω 0 𝜕𝜕 𝜂𝜂 � , 𝜕𝜕 Ω 1 𝜕𝜕 𝑡𝑡 − 1 2 𝜕𝜕 2 Ω 1 𝜕𝜕 𝜂𝜂 � 2 = 𝜕𝜕 ( Ψ � 0 , Ω 0 ) 𝜕𝜕 ( 𝜂𝜂 � , 𝜕𝜕 ) , (4.70) with boundary conditions 𝜂𝜂 ̂ = 0, 𝜕𝜕 Ψ � 1 𝜕𝜕𝜕𝜕 = 𝜕𝜕 Ψ � 1 𝜕𝜕 𝜂𝜂 � = Ω 1 = 0. (4.71) 53 Similarly, we write Ψ � 1 and Ω 1 as Ψ � 1 = Ψ � 1 ( 𝑠𝑠 ) + Ψ � 1 ( 𝑢𝑢 ) = � Ψ � 1 0 ( 𝜂𝜂 ̂) + Ψ � 1 2 ( 𝜂𝜂 ̂) 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , Ω 1 = Ω 1 ( 𝑠𝑠 ) + Ω 1 ( 𝑢𝑢 ) = � Ω 1 0 ( 𝜂𝜂 ̂) + Ω 1 2 ( 𝜂𝜂 ̂) 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. . (4.72) Solutions are Ψ � 1 0 = − 9 6 4 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 �(1 + 𝑠𝑠 ) 𝑒𝑒 − 2 𝜂𝜂 � + 4(5 + 3 𝑠𝑠 ) 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + 8 𝑠𝑠 𝜂𝜂 ̂ 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � � + 𝐴𝐴 2 𝜂𝜂 ̂ 3 + 𝐵𝐵 2 𝜂𝜂 ̂ 2 + 𝐶𝐶 2 𝜂𝜂 ̂ + 𝐷𝐷 2 , Ω 1 0 = 𝐴𝐴 2 ∗ 𝜂𝜂 ̂, (4.73) where 𝐶𝐶 2 = − 9( 5 + 1 3 𝑖𝑖 ) 3 2 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 , 𝐷𝐷 2 = 9( 2 1 + 1 3 𝑖𝑖 ) 6 4 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 . (4.74) 𝐴𝐴 2 , 𝐵𝐵 2 and 𝐴𝐴 2 ∗ need to be determined by boundary conditions and matching with the outer flow. Finally in the outer region, the steady flow is characterized by the streaming Reynolds number, which is defined as 𝑅𝑅 𝑠𝑠 = 𝑈𝑈 0 2 𝜔𝜔𝜈𝜈 = 𝜀𝜀 2 | 𝑀𝑀 | 2 = � Displacement amplitude of torsional oscillation Stokes-layer thickness � 2 . (4.75) This is not an independent parameter, because it is determined by 𝜀𝜀 and | 𝑀𝑀 | together. Here, only 𝑅𝑅 𝑠𝑠 ≪ 1 is considered, in which case, the governing equation is the Stokes equation. Accordingly, in the outer region, we write 𝜓𝜓 = 𝜓𝜓 0 + 𝜀𝜀𝜀𝜀 + 𝑂𝑂 ( 𝜀𝜀 2 ) , Ω = Ω 0 + 𝜀𝜀𝜀𝜀 + 𝑂𝑂 ( 𝜀𝜀 2 ) , (4.76) where 𝜓𝜓 0 and Ω 0 are given before. Since from Eqn. (4.46), we know 54 𝐷𝐷 2 𝜓𝜓 0 = 0 and Ω 0 = 0, and 𝜀𝜀 and 𝜀𝜀 satisfy 𝑅𝑅 𝑠𝑠 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜀𝜀 ) + 𝜀𝜀 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝜓𝜓 0 + 𝜀𝜀𝜀𝜀 ) 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝐷𝐷 2 𝜀𝜀 ) − 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝜓𝜓 0 + 𝜀𝜀𝜀𝜀 ) 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝐷𝐷 2 𝜀𝜀 ) + 2 𝐷𝐷 2 𝜀𝜀𝐿𝐿 ( 𝜓𝜓 0 + 𝜀𝜀𝜀𝜀 ) + 2 𝜀𝜀𝜀𝜀𝐿𝐿 𝜀𝜀 � = 𝜀𝜀 2 𝐷𝐷 4 𝜀𝜀 , 𝑅𝑅 𝑠𝑠 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑡𝑡 + 𝜀𝜀 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝜓𝜓 0 + 𝜀𝜀𝜀𝜀 ) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 − 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝜓𝜓 0 + 𝜀𝜀𝜀𝜀 ) 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑟𝑟 � = 𝜀𝜀 2 𝐷𝐷 2 𝜀𝜀 , (4.77) where the operator 𝐿𝐿 is defined in Eqn. (4.7). We now expand 𝜀𝜀 ( 𝑟𝑟 , 𝜇𝜇 , 𝑡𝑡 , 𝜀𝜀 ) and 𝜀𝜀 ( 𝑟𝑟 , 𝜇𝜇 , 𝑡𝑡 , 𝜀𝜀 ) in the form 𝜀𝜀 = 𝜀𝜀 1 + 𝜀𝜀 𝜀𝜀 2 + ⋯ , 𝜀𝜀 = 𝜀𝜀 1 + 𝜀𝜀 𝜀𝜀 2 + ⋯ . (4.78) The terms of 𝑂𝑂 (1) require that 𝜀𝜀 1 and 𝜀𝜀 1 satisfy 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜀𝜀 1 ) = 0, 𝜕𝜕 𝜕𝜕 1 𝜕𝜕 𝑡𝑡 = 0, (4.79) the solutions of which are 𝜀𝜀 1 = 𝐹𝐹 1 0 ( 𝑟𝑟 , 𝜇𝜇 ) + 𝐹𝐹 1 2 ( 𝑟𝑟 , 𝜇𝜇 ) 𝑇𝑇 � 1 ( 𝑡𝑡 ) + c.c. , 𝜀𝜀 1 = 𝐺𝐺 1 0 ( 𝑟𝑟 , 𝜇𝜇 ) + c.c. , (4.80) with 𝐷𝐷 2 𝐹𝐹 1 2 ( 𝑟𝑟 , 𝜇𝜇 ) = 0. (4.81) The terms of 𝑂𝑂 ( 𝜀𝜀 ) give the equation for 𝜀𝜀 2 and 𝜀𝜀 2 as 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜀𝜀 2 ) + 1 𝑟𝑟 2 � 𝜕𝜕 𝜓𝜓 0 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝐷𝐷 2 𝜀𝜀 1 ) − 𝜕𝜕 𝜓𝜓 0 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝐷𝐷 2 𝜀𝜀 1 ) + 2 𝐷𝐷 2 𝜀𝜀 1 𝐿𝐿 𝜓𝜓 0 � = 0, 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝜀𝜀 2 ) + 1 𝑟𝑟 2 � 𝜕𝜕 𝜓𝜓 0 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕 1 𝜕𝜕𝜕𝜕 − 𝜕𝜕 𝜓𝜓 0 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 1 𝜕𝜕 𝑟𝑟 � = 0, (4.82) 55 for which, once again, 𝜀𝜀 2 = 𝐹𝐹 2 0 ( 𝑟𝑟 , 𝜇𝜇 ) + 𝐹𝐹 2 2 ( 𝑟𝑟 , 𝜇𝜇 ) 𝑇𝑇 � 2 ( 𝑡𝑡 ) + c.c. , 𝜀𝜀 2 = 𝐺𝐺 2 0 ( 𝑟𝑟 , 𝜇𝜇 ) + 𝐺𝐺 2 2 ( 𝑟𝑟 , 𝜇𝜇 ) 𝑇𝑇 � 2 ∗ ( 𝑡𝑡 ) + c.c. . (4.83) Since in Eqn. (4.82), 𝐷𝐷 2 𝜀𝜀 1 is independent of 𝑡𝑡 from Eqn. (4.79), and 𝜓𝜓 0 ∝ sin 𝑡𝑡 from Eqn. (4.59), we can write 𝐷𝐷 2 𝜀𝜀 2 in the form 𝐷𝐷 2 𝜀𝜀 2 ∝ cos 𝑡𝑡 . (4.84) Similarly, for 𝜀𝜀 2 , we have 𝜀𝜀 2 ∝ cos 𝑡𝑡 . (4.85) Then, the equations for 𝜀𝜀 3 and 𝜀𝜀 3 , obtained from the 𝑂𝑂 ( 𝜀𝜀 2 ) terms are 𝑅𝑅 𝑠𝑠 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜀𝜀 3 ) + 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝜓𝜓 0 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝐷𝐷 2 𝜀𝜀 2 ) − 𝜕𝜕 𝜓𝜓 0 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝐷𝐷 2 𝜀𝜀 2 ) + 2 𝐿𝐿 𝜓𝜓 0 𝐷𝐷 2 𝜀𝜀 2 � + 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝜉𝜉 1 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝐷𝐷 2 𝜀𝜀 1 ) − 𝜕𝜕 𝜉𝜉 1 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝐷𝐷 2 𝜀𝜀 1 ) + 2 𝐿𝐿 𝜀𝜀 1 𝐷𝐷 2 𝜀𝜀 1 + 2 𝜀𝜀 1 𝐿𝐿 𝜀𝜀 1 � = 𝐷𝐷 4 𝜀𝜀 1 , 𝑅𝑅 𝑠𝑠 𝜕𝜕 𝜕𝜕 3 𝜕𝜕 𝑡𝑡 + 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝜓𝜓 0 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕 2 𝜕𝜕𝜕𝜕 − 𝜕𝜕 𝜓𝜓 0 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 2 𝜕𝜕 𝑟𝑟 � + 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝜉𝜉 1 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕 1 𝜕𝜕𝜕𝜕 − 𝜕𝜕 𝜉𝜉 1 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 1 𝜕𝜕 𝑟𝑟 � = 𝐷𝐷 2 𝜀𝜀 1 . (4.86) We can conclude 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜀𝜀 3 ) contains no steady part. Otherwise, 𝜀𝜀 3 would contain terms of 𝑂𝑂 ( 𝑡𝑡 ), which would lead to unbounded growth in time, according to Rosenblat[85]. It is similar for 𝜕𝜕 𝜕𝜕 3 𝜕𝜕 𝑡𝑡 . Because the second terms of Eqn. (4.86) are in proportion to sin 𝑡𝑡 cos 𝑡𝑡 , they have no steady part. We may equate the time-independent parts of 𝑂𝑂 ( 𝜀𝜀 2 ) equations, to get for the steady part 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝐹𝐹 1 0 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 ( 𝐷𝐷 2 𝐹𝐹 1 0 ) − 𝜕𝜕 𝐹𝐹 1 0 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝑟𝑟 ( 𝐷𝐷 2 𝐹𝐹 1 0 ) + 2 𝐿𝐿 𝐹𝐹 1 0 𝐷𝐷 2 𝐹𝐹 1 0 + 2 𝐺𝐺 1 0 𝐿𝐿 𝐺𝐺 1 0 � = 𝐷𝐷 4 𝐹𝐹 1 0 , 𝑅𝑅 𝑠𝑠 𝑟𝑟 2 � 𝜕𝜕 𝐹𝐹 1 0 𝜕𝜕 𝑟𝑟 𝜕𝜕 𝐺𝐺 1 0 𝜕𝜕𝜕𝜕 − 𝜕𝜕 𝐹𝐹 1 0 𝜕𝜕𝜕𝜕 𝜕𝜕 𝐺𝐺 1 0 𝜕𝜕 𝑟𝑟 � = 𝐷𝐷 2 𝐺𝐺 1 0 . (4.87) Now consider streaming Reynolds number 𝑅𝑅 𝑠𝑠 ≪ 1 in Eqn. (4.87) resulting in Stokes-like steady flow, i.e. , 56 𝐷𝐷 4 𝐹𝐹 1 0 = 0, 𝐷𝐷 2 𝐺𝐺 1 0 = 0. (4.88) The appropriate form of solutions should be 𝐹𝐹 1 0 ( 𝑟𝑟 , 𝜇𝜇 ) = � 𝐴𝐴 3 𝑟𝑟 5 + 𝐵𝐵 3 𝑟𝑟 3 + 𝐶𝐶 3 + 𝐷𝐷 3 𝑟𝑟 2 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) , 𝐺𝐺 1 0 ( 𝑟𝑟 , 𝜇𝜇 ) = �𝐴𝐴 3 ∗ 𝑟𝑟 3 + 𝐵𝐵 3 ∗ 𝑟𝑟 2 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) . (4.89) Then, Ψ 1 0 | 𝜂𝜂 →∞ = 𝐴𝐴 1 𝜂𝜂 3 + 𝐵𝐵 1 𝜂𝜂 2 + 𝐶𝐶 1 𝜂𝜂 + 𝐷𝐷 1 , Ψ � 1 0 | 𝜂𝜂 � →∞ = 𝐴𝐴 2 𝜂𝜂 ̂ 3 + 𝐵𝐵 2 𝜂𝜂 ̂ 2 + 𝐶𝐶 2 𝜂𝜂 ̂ + 𝐷𝐷 2 , (4.90) where 𝐴𝐴 1 , 𝐵𝐵 1 , 𝐴𝐴 2 and 𝐵𝐵 2 are unknown. Now we take Taylor expansion of 𝐹𝐹 1 0 ( 𝑟𝑟 ) about 𝑟𝑟 = 𝛾𝛾 and replace 𝑟𝑟 − 𝛾𝛾 with √ 2 | 𝑀𝑀 | 𝜂𝜂 , 𝐹𝐹 1 0 ( 𝑟𝑟 )~ �𝐴𝐴 3 𝛾𝛾 5 + 𝐵𝐵 3 𝛾𝛾 3 + 𝐶𝐶 3 + 𝐷𝐷 3 𝛾𝛾 2 � + �5 𝐴𝐴 3 𝛾𝛾 4 + 3 𝐵𝐵 3 𝛾𝛾 2 − 2 𝐷𝐷 3 𝛾𝛾 3 � √ 2 | 𝑀𝑀 | 𝜂𝜂 + �10 𝐴𝐴 3 𝛾𝛾 3 + 3 𝐵𝐵 3 𝛾𝛾 + 3 𝐷𝐷 3 𝛾𝛾 4 � � √ 2 | 𝑀𝑀 | 𝜂𝜂 � 2 + �10 𝐴𝐴 3 𝛾𝛾 2 + 𝐵𝐵 3 − 4 𝐷𝐷 3 𝛾𝛾 5 � � √ 2 | 𝑀𝑀 | 𝜂𝜂 � 3 + ⋯ . (4.91) Taylor expansion of 𝐹𝐹 1 0 ( 𝑟𝑟 ) about 𝑟𝑟 = 1 and replace 1 − 𝑟𝑟 with √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ , 𝐹𝐹 1 0 ( 𝑟𝑟 )~( 𝐴𝐴 3 + 𝐵𝐵 3 + 𝐶𝐶 3 + 𝐷𝐷 3 ) + (5 𝐴𝐴 3 + 3 𝐵𝐵 3 − 2 𝐷𝐷 3 ) �− √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ � +(10 𝐴𝐴 3 + 3 𝐵𝐵 3 + 3 𝐷𝐷 3 ) �− √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ � 2 + (10 𝐴𝐴 3 + 𝐵𝐵 3 − 4 𝐷𝐷 3 ) �− √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ � 3 + ⋯ . (4.92) Then, according to the matching requirement similar to Eqn. (4.54), these coefficients are obtained 𝐴𝐴 3 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �2 𝛾𝛾 3 − 5 + 3 𝛾𝛾 2 � + 𝐷𝐷 1 �− 6 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 + 6 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 � + 𝐶𝐶 2 �−3 𝛾𝛾 2 + 5 − 2 𝛾𝛾 3 � + 𝐷𝐷 2 � 6 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 − 6 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 �� , 57 𝐵𝐵 3 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �−2 𝛾𝛾 5 + 7 − 5 𝛾𝛾 2 � + 𝐷𝐷 1 � 1 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 − 1 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 � + 𝐶𝐶 2 �5 𝛾𝛾 4 − 7 𝛾𝛾 2 + 2 𝛾𝛾 3 � + 𝐷𝐷 2 �− 1 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 + 1 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 �� , 𝐶𝐶 3 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �5 𝛾𝛾 5 − 7 𝛾𝛾 3 + 2 𝛾𝛾 2 � + 𝐷𝐷 1 �− 2 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 + 2 1 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 + 4 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 � + 𝐶𝐶 2 ( −2 𝛾𝛾 7 + 7 𝛾𝛾 2 − 5) + 𝐷𝐷 2 � 4 √ 2 | 𝑀𝑀 | 𝛾𝛾 7 + 2 1 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 − 2 5 √ 2 | 𝑀𝑀 | �� , 𝐷𝐷 3 = 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 ( −3 𝛾𝛾 5 + 5 𝛾𝛾 3 − 2) + 𝐷𝐷 1 � 1 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 − 1 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 � + 𝐶𝐶 2 (2 𝛾𝛾 7 − 5 𝛾𝛾 4 + 3 𝛾𝛾 2 ) + 𝐷𝐷 2 � − 1 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 + 1 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 �� , (4.93) and 𝐴𝐴 1 = 2 | 𝑀𝑀 | 2 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �18 𝛾𝛾 5 − 50 𝛾𝛾 2 + 49 − 2 5 𝛾𝛾 2 + 8 𝛾𝛾 5 � + 𝐷𝐷 1 �− 5 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 + 5 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 � + 𝐶𝐶 2 �−25 𝛾𝛾 4 + 35 𝛾𝛾 2 − 1 0 𝛾𝛾 3 � + 𝐷𝐷 2 � 5 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 − 5 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 �� , 𝐵𝐵 1 = √ 2 | 𝑀𝑀 | 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �14 𝛾𝛾 6 − 50 𝛾𝛾 3 + 42 𝛾𝛾 − 6 𝛾𝛾 4 � + 𝐷𝐷 1 �− 3 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 5 + 1 0 5 √ 2 | 𝑀𝑀 | − 7 5 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 2 � + 𝐶𝐶 2 �−15 𝛾𝛾 5 + 35 𝛾𝛾 3 − 35 + 1 5 𝛾𝛾 2 � + 𝐷𝐷 2 � 3 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 5 − 1 0 5 √ 2 | 𝑀𝑀 | + 7 5 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 2 �� , 𝐴𝐴 2 = 2 | 𝑀𝑀 | 2 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �−10 𝛾𝛾 5 + 35 − 2 5 𝛾𝛾 2 � + 𝐷𝐷 1 � 5 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 − 5 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 � + 𝐶𝐶 2 �8 𝛾𝛾 7 − 25 𝛾𝛾 4 + 49 𝛾𝛾 2 − 50 + 1 8 𝛾𝛾 3 � + 𝐷𝐷 2 �− 5 0 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 + 5 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 �� , 𝐵𝐵 2 = √ 2 | 𝑀𝑀 | 1 4 𝛾𝛾 7 − 2 5 𝛾𝛾 4 + 4 2 𝛾𝛾 2 − 2 5 + 4 𝛾𝛾 3 � 𝐶𝐶 1 �−15 𝛾𝛾 5 + 35 𝛾𝛾 3 − 35 + 1 5 𝛾𝛾 2 � + 𝐷𝐷 1 � 7 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 − 1 0 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 + 3 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 � + 𝐶𝐶 2 �6 𝛾𝛾 7 − 42 𝛾𝛾 2 + 50 − 1 4 𝛾𝛾 3 � + 𝐷𝐷 2 �− 7 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 4 + 1 0 5 √ 2 | 𝑀𝑀 | 𝛾𝛾 2 − 3 0 √ 2 | 𝑀𝑀 | 1 𝛾𝛾 3 �� . (4.94) 58 Similarly, the angular circulation Ω 1 0 for steady streaming in the two inner regions has far-field values, Ω 1 0 | 𝜂𝜂 →∞ = 𝐴𝐴 1 ∗ 𝜂𝜂 + 𝐵𝐵 1 ∗ , Ω 1 0 | 𝜂𝜂 � →∞ = 𝐴𝐴 2 ∗ 𝜂𝜂 ̂, (4.95) where 𝐴𝐴 1 ∗ and 𝐴𝐴 2 ∗ are unknown. Then, Taylor expansion of 𝐺𝐺 1 0 ( 𝑟𝑟 ) about 𝑟𝑟 = 𝛾𝛾 and replacement of 𝑟𝑟 − 𝛾𝛾 with √ 2 | 𝑀𝑀 | 𝜂𝜂 gives 𝐺𝐺 1 0 ( 𝑟𝑟 )~ �𝐴𝐴 3 ∗ 𝛾𝛾 3 + 𝐵𝐵 3 ∗ 𝛾𝛾 2 � + �3 𝐴𝐴 3 ∗ 𝛾𝛾 2 − 2 𝐵𝐵 3 ∗ 𝛾𝛾 3 � √ 2 | 𝑀𝑀 | 𝜂𝜂 + ⋯ . (4.96) Similarly, Taylor expansion of 𝐺𝐺 1 0 ( 𝑟𝑟 ) about 𝑟𝑟 = 1 and replacement of 1 − 𝑟𝑟 with √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ leads to 𝐺𝐺 1 0 ( 𝑟𝑟 )~( 𝐴𝐴 3 ∗ + 𝐵𝐵 3 ∗ ) + (3 𝐴𝐴 3 ∗ − 2 𝐵𝐵 3 ∗ ) �− √ 2 | 𝑀𝑀 | 𝜂𝜂 ̂ � + ⋯ . (4.97) Therefore, 𝐴𝐴 3 ∗ = 1 𝛾𝛾 3 − 1 𝛾𝛾 2 𝐵𝐵 1 ∗ , 𝐵𝐵 3 ∗ = − 1 𝛾𝛾 3 − 1 𝛾𝛾 2 𝐵𝐵 1 ∗ , 𝐴𝐴 1 ∗ = √ 2 | 𝑀𝑀 | � 3 𝛾𝛾 2 + 2 𝛾𝛾 3 𝛾𝛾 3 − 1 𝛾𝛾 2 � 𝐵𝐵 1 ∗ , 𝐴𝐴 2 ∗ = − √ 2 | 𝑀𝑀 | � 5 𝛾𝛾 3 − 1 𝛾𝛾 2 � 𝐵𝐵 1 ∗ . (4.98) In summary, steady streaming in the outer region is given by 𝜀𝜀 1 ( 𝑠𝑠 ) = �𝐴𝐴 3 𝑟𝑟 5 + 𝐵𝐵 3 𝑟𝑟 3 + 𝐶𝐶 3 + 𝐷𝐷 3 𝑟𝑟 2 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , ζ 1 ( 𝑠𝑠 ) = �𝐴𝐴 3 ∗ 𝑟𝑟 3 + 𝐵𝐵 3 ∗ 𝑟𝑟 2 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. . (4.99) In the inner region around the inner sphere, steady streaming has solutions 59 Ψ 1 ( 𝑠𝑠 ) = � 2 𝛾𝛾 2 � − 3 3 2 �2 𝑠𝑠 | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 8(1 + 𝑠𝑠 ) 𝛾𝛾 − 3(5 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 + 1 + 𝑖𝑖 3 2 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � 𝑒𝑒 − 2 𝜂𝜂 − 3 𝑖𝑖 8 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� 𝛾𝛾 1 − 𝛾𝛾 3 𝜂𝜂 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + 𝐴𝐴 1 𝜂𝜂 3 + 𝐵𝐵 1 𝜂𝜂 2 + 𝐶𝐶 1 𝜂𝜂 + 𝐷𝐷 1 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , Ω 1 ( 𝑠𝑠 ) = � 2 𝛼𝛼 𝑒𝑒 𝑖𝑖𝑖𝑖 𝛾𝛾 � − 3( 1 − 𝑖𝑖 ) 1 6 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 − 2 𝜂𝜂 − 1 − 𝑖𝑖 4 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� 𝜂𝜂 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 + 1 8 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6 𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + 𝐴𝐴 1 ∗ 𝜂𝜂 + 𝐵𝐵 1 ∗ � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. . (4.100) In the inner region around container outer shell, solutions for steady streaming are Ψ � 1 ( 𝑠𝑠 ) = � − 9 6 4 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 �(1 + 𝑠𝑠 ) 𝑒𝑒 − 2 𝜂𝜂 � + 4(5 + 3 𝑠𝑠 ) 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � + 8 𝑠𝑠 𝜂𝜂 ̂ 𝑒𝑒 −( 1 + 𝑖𝑖 ) 𝜂𝜂 � � + 𝐴𝐴 2 𝜂𝜂 ̂ 3 + 𝐵𝐵 2 𝜂𝜂 ̂ 2 + 𝐶𝐶 2 𝜂𝜂 ̂ + 𝐷𝐷 2 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , Ω 1 ( 𝑠𝑠 ) = 𝐴𝐴 2 ∗ 𝜂𝜂 ̂ 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. . (4.101) After matching, all the coefficients are determined, and the composite solutions can be written as, 𝜓𝜓 1 (𝑠𝑠 ) = � √ 2 | 𝑀𝑀 | 2 𝛾𝛾 2 � − 3 3 2 �2 𝑠𝑠 | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 8(1 + 𝑠𝑠 ) 𝛾𝛾 − 3(5 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 + 1 + 𝑖𝑖 3 2 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � 𝑒𝑒 − 2( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 − 3 𝑖𝑖 8 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� 𝛾𝛾 1 − 𝛾𝛾 3 ( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 � − 9 √ 2 6 4| 𝑀𝑀 | 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 �(1 + 𝑠𝑠 ) 𝑒𝑒 − 2( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 + 4(5 + 3 𝑠𝑠 ) 𝑒𝑒 −( 1 + 𝑖𝑖 )( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 + 8 𝑠𝑠 (1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 𝑒𝑒 −( 1 + 𝑖𝑖 )( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 � + �𝐴𝐴 3 𝑟𝑟 5 + 𝐵𝐵 3 𝑟𝑟 3 + 𝐶𝐶 3 + 𝐷𝐷 3 𝑟𝑟 2 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , Ω 1 (𝑠𝑠 ) = � 2 𝛼𝛼 𝑒𝑒 𝑖𝑖𝑖𝑖 𝛾𝛾 � − 3( 1 − 𝑖𝑖 ) 1 6 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 − 2( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 − 1 − 𝑖𝑖 4 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� ( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 + 1 8 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6 𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 � + 𝐴𝐴 3 ∗ 𝑟𝑟 3 + 𝐵𝐵 3 ∗ 𝑟𝑟 2 � 𝜇𝜇 (1 − 𝜇𝜇 2 ) + c.c. , (4.102) with velocities 60 𝑢𝑢 1 𝑟𝑟 ( 𝑠𝑠 ) = � √ 2 | 𝑀𝑀 | 2 𝛾𝛾 2 1 𝑟𝑟 2 � − 3 3 2 �2 𝑠𝑠 | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 8(1 + 𝑠𝑠 ) 𝛾𝛾 − 3(5 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 + 1 + 𝑖𝑖 3 2 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � 𝑒𝑒 − 2( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 − 3 𝑖𝑖 8 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� 𝛾𝛾 1 − 𝛾𝛾 3 ( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 � − 9 √ 2 6 4| 𝑀𝑀 | 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 1 𝑟𝑟 2 �(1 + 𝑠𝑠 ) 𝑒𝑒 − 2( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 + 4(5 + 3 𝑠𝑠 )𝑒𝑒 −( 1 + 𝑖𝑖 )( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 + 8 𝑠𝑠 (1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 𝑒𝑒 −( 1 + 𝑖𝑖 )( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 � + �𝐴𝐴 3 𝑟𝑟 3 + 𝐵𝐵 3 𝑟𝑟 + 𝐶𝐶 3 𝑟𝑟 2 + 𝐷𝐷 3 𝑟𝑟 4 �� (3 𝜇𝜇 2 − 1) + c.c. , 𝑢𝑢 1 𝜃𝜃 ( 𝑠𝑠 ) = � 2 𝛾𝛾 2 1 𝑟𝑟 � − 3 1 6 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6 𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 + 1 + 𝑖𝑖 1 6 � 9 4 � 𝛾𝛾 1 − 𝛾𝛾 3 � 2 − ( 𝛼𝛼𝛾𝛾 ) 2 � 𝑒𝑒 − 2( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 + 3( 1 − 𝑖𝑖 ) 8 �𝛾𝛾 − 3 2 ( 𝛾𝛾 1 − 𝛾𝛾 3 ) � 𝛾𝛾 1 − 𝛾𝛾 3 | 𝑀𝑀 | √ 2 ( 𝑟𝑟 − 𝛾𝛾 ) 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 � + 9 3 2 𝛾𝛾 6 ( 1 − 𝛾𝛾 3 ) 2 1 𝑟𝑟 �(1 + 𝑠𝑠 ) 𝑒𝑒 − 2( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 + 4(1 + 3 𝑠𝑠 ) 𝑒𝑒 −( 1 + 𝑖𝑖 )( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 − 4(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 (1 − 𝑟𝑟 ) 𝑒𝑒 −( 1 + 𝑖𝑖 )( 1 − 𝑟𝑟 ) | 𝑀𝑀 | √ 2 � + �−5 𝐴𝐴 3 𝑟𝑟 3 − 3 𝐵𝐵 3 𝑟𝑟 + 2 𝐷𝐷 3 𝑟𝑟 4 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) 1 2 + c.c. , 𝑢𝑢 1 𝜙𝜙 ( 𝑠𝑠 ) = � 2 𝛼𝛼 𝑒𝑒 𝑖𝑖𝑖𝑖 𝛾𝛾 1 𝑟𝑟 � − 3( 1 − 𝑖𝑖 ) 1 6 𝛾𝛾 1 − 𝛾𝛾 3 𝑒𝑒 − 2( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 − 1 − 𝑖𝑖 4 �𝛾𝛾 − 3 2 � 𝛾𝛾 1 − 𝛾𝛾 3 �� ( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 + 1 8 �−(1 − 𝑠𝑠 ) | 𝑀𝑀 | √ 2 𝛾𝛾 2 + 6 𝑠𝑠𝛾𝛾 − 3(1 + 3 𝑠𝑠 ) 𝛾𝛾 1 − 𝛾𝛾 3 � 𝑒𝑒 −( 1 + 𝑖𝑖 )( 𝑟𝑟 − 𝛾𝛾 ) | 𝑀𝑀 | √ 2 � + �𝐴𝐴 3 ∗ 𝑟𝑟 2 + 𝐵𝐵 3 ∗ 𝑟𝑟 3 �� 𝜇𝜇 (1 − 𝜇𝜇 2 ) 1 2 + c.c. . (4.103) 4.3 Theoretical results and discussion In Figure 4.3, the leading order shear stress profiles under different values of the Womersley number | 𝑀𝑀 | are investigated. According to Eqn. (4.14), transverse oscillations result in leading order shear stress 𝜏𝜏 0 𝑟𝑟 𝜃𝜃 , while torsional oscillations result in 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 . So the graphs for leading order shear stress components for combination oscillation are the same as those for single transverse and torsional oscillations (Figure 2.2 and Figure 3.2). Also, for both shear stresses, 61 the maximum values can be found on the inner boundary. These maximum values rise with increasing Womersley number | 𝑀𝑀 |. As | 𝑀𝑀 | increases to 20, torsional oscillations can only lead to shear stress 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 close to the inner boundary, while value of 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 is zero for the outside half part of the region. On the contrary, influence of transverse oscillation on shear stress 𝜏𝜏 0 𝑟𝑟 𝜃𝜃 close to the outer boundary remains the same as | 𝑀𝑀 | rises. Especially, the shear stress profiles for low frequency ( | 𝑀𝑀 | ≪ 1), shown in Figure 4.3(a), are similar as those for | 𝑀𝑀 | = 1 in Figure 4.3(b). At low frequency, a reduction of the Womersley number | 𝑀𝑀 | does not cause significant decrease in shear stress values. In Chapter 2 and 3, it is showed that the second order steady streaming, induced by a single transverse or torsional oscillations, is only on the 𝑟𝑟 - 𝜃𝜃 plane. For transverse oscillation only, the streaming vortex is clockwise, and counter-clockwise for torsional oscillation only. However, for combined transverse and torsional oscillations, the azimuthal velocity of steady streaming is non-zero. Steady streaming patterns on the 𝑟𝑟 - 𝜃𝜃 plane at low frequency for different values of the amplitude ratio 𝛼𝛼 are plotted in Figure 4.4,( for | 𝑀𝑀 | = 10 in Figure 4.5, and for high frequency in Figure 4.6). The vortices of streaming for only transverse oscillations ( 𝛼𝛼 = 0), shown in Figure 4.4(a), 4.5a) and 4.6(a), are the same as those in Chapter 3. As 𝛼𝛼 is increased, the effects of torsional oscillations enhance. We take low frequency for example. As the ratio of amplitude 𝛼𝛼 goes up from 1 to 3, the vortex changes direction from clockwise to counter- clockwise. This transient takes place at a value of 𝛼𝛼 between 2 and 3. On the other hand, for | 𝑀𝑀 | = 10, torsional oscillation play a dominant role until 𝛼𝛼 reaches 5 (Figure 4.5(c)), and 8 for high frequency (Figure 4.6(c)). Thus, at high frequency, effects of transverse oscillations are more obvious compared to those of torsional oscillation. 62 Corresponding to | 𝑀𝑀 | = 10, the transition for flow reversed (Figure 4.5(b) - (c)), is approximately at 𝛼𝛼 = 4. The flow field around this transition is detailed in Figure 4.7(a) - (c). Some interesting flow patterns have been predicted. With the increase of 𝛼𝛼 , a second vortex with opposite circulation is formed close to the inner boundary around 𝛼𝛼 = 3.9 (Figure 4.7(a)). It then grows bigger with increasing 𝛼𝛼 and eventually squeezes out the existing clockwise directed vortex (Figure 4.7(b) and (c)). The velocity profiles of streaming for different frequency are compared in Figure 4.8. When compare with low frequency, higher frequency oscillations can induce more intense streaming. The steady part of first order azimuthal velocity 𝑢𝑢 1 𝜙𝜙 (𝑠𝑠 ) is proportional to oscillation amplitude ratio 𝛼𝛼 according to Eqns. (4.27), (4.37) and (4.103). So increasing 𝛼𝛼 does not change the direction of azimuthal velocity, but only the value. The phase difference 𝛽𝛽 between the transverse and torsional oscillations does not have much influence on the leading order solutions, according to Eqns. (4.11) and (4.14). For the leading order, the radial and zenithal components of flow are induced by transverse oscillation, while azimuthal components by torsional oscillation. Since the azimuthal component is function of 𝑒𝑒 𝑖𝑖 ( 𝑡𝑡 + 𝑖𝑖 ) , the time average of the shear stress 𝜏𝜏 0 𝑟𝑟 𝜙𝜙 values over one time period has no significant change with 𝛽𝛽 . Neither does 𝛽𝛽 have an influence on steady streaming on 𝑟𝑟 - 𝜃𝜃 plane. However, to order 𝜀𝜀 , various values of steady part of the azimuthal velocity are seen with variations of the phase difference 𝛽𝛽 , as shown in Figure 4.9. The magnitude of 𝑢𝑢 1 𝜙𝜙 ( 𝑠𝑠 ) is the greatest for cases in which the phase difference 𝛽𝛽 is around 𝑛𝑛 𝜋𝜋 . For low frequency oscillations, azimuthal velocity 𝑢𝑢 1 𝜙𝜙 ( 𝑠𝑠 ) vanishes at phase difference 𝛽𝛽 = 1 2 (2 𝑛𝑛 + 1) 𝜋𝜋 , and no recirculation appears on the 𝑟𝑟 - 𝜙𝜙 plane for all phase difference 𝛽𝛽 values (Figure 4.9(a)). For higher frequency, take | 𝑀𝑀 |=10 for 63 example (Figure 4.9(b)), azimuthal velocity 𝑢𝑢 1 𝜙𝜙 (𝑠𝑠 ) still exists at phase difference 𝛽𝛽 = 1 2 (2 𝑛𝑛 + 1) 𝜋𝜋 . However, under high frequency oscillations (Figure 4.9(c)), steady streaming has a recirculation on the 𝑟𝑟 - 𝜙𝜙 plane when phase difference 𝛽𝛽 is approximately in the range from 1 2 (2 𝑛𝑛 + 1) 𝜋𝜋 to 1 2 (2 𝑛𝑛 + 1) 𝜋𝜋 + 1 4 𝜋𝜋 . Since we have results for unrestricted | 𝑀𝑀 | in Eqns. (4.24)-(4.27), we are in a position to compare these with asymptotic analysis for | 𝑀𝑀 | ≫1. A comparison for various | 𝑀𝑀 | values has been made. Specifically, we have plotted the velocity components 𝑢𝑢 1 𝑟𝑟 (𝑠𝑠 ) , 𝑢𝑢 1 𝜃𝜃 (𝑠𝑠 ) , 𝑢𝑢 1 𝜙𝜙 (𝑠𝑠 ) as a function of 𝑟𝑟 for various values of 𝜃𝜃 at | 𝑀𝑀 |=25, 50 and 100 in Figures 4.10-4.12. These plots all correspond to 𝛼𝛼 =1, 𝛽𝛽 =0 and 𝛾𝛾 =0.5. The asymptotic results from Eqn. (4.103) show good agreement with the unrestricted | 𝑀𝑀 | result from Eqns. (4.24)-(4.27), particularly for | 𝑀𝑀 |=100. 64 Figure 4.3 Leading order shear stress profiles [Eqns.(4.14) and (4.32)] on the equatorial plane over one time period with 𝛾𝛾 =0.5, 𝛼𝛼 =1 and 𝛽𝛽 = 𝜋𝜋 4 ⁄ . The dashed curve shows the time average of absolute shear stress values over one period. (a) low frequency, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=5, (d) | 𝑀𝑀 |=20. 65 Figure 4.4 The streaming flow pattern on 𝑟𝑟 -𝜃𝜃 plane for low frequency [Eqn. (4.36)] under different amplitude ratio with 𝛾𝛾 =0.5. (a) 𝛼𝛼 =0, (b) 𝛼𝛼 =1, (c) 𝛼𝛼 =3. 66 Figure 4.5 The streaming flow pattern on 𝑟𝑟 -𝜃𝜃 plane for | 𝑀𝑀 |=10 [Eqn. (4.28)] under different amplitude ratio with 𝛾𝛾 =0.5. (a) 𝛼𝛼 =0, (b) 𝛼𝛼 =1, (c) 𝛼𝛼 =5. 67 Figure 4.6 The streaming flow pattern on 𝑟𝑟 -𝜃𝜃 plane for high frequency (| 𝑀𝑀 |=25) [Eqn. (4.102)] under different amplitude ratio with 𝛾𝛾 =0.5. (a) 𝛼𝛼 =0, (b) 𝛼𝛼 =1, (c) 𝛼𝛼 =8. 68 Figure 4.7 Example of the streaming flow pattern change on 𝑟𝑟 -𝜃𝜃 plane with increasing amplitude ratio (| 𝑀𝑀 |=10, 𝛾𝛾 =0.5). (a) 𝛼𝛼 =3.9, (b) 𝛼𝛼 =4.0, (c) 𝛼𝛼 =4.15. 69 Figure 4.8 Velocity profiles of steady streaming for (a) low frequency, (b) | 𝑀𝑀 |=10 and (c) high frequency (| 𝑀𝑀 |=25) with 𝛾𝛾 =0.5, 𝛼𝛼 =1 and 𝛽𝛽 =0. 70 Figure 4.9 Steady part of first order azimuthal velocity profiles on the plane 𝜃𝜃 = 𝜋𝜋 4 ⁄ under different phase difference 𝛽𝛽 with 𝛾𝛾 =0.5, 𝛼𝛼 =1 for (a) low frequency, (b) | 𝑀𝑀 |=10 and (c) high frequency (| 𝑀𝑀 |=25). 71 Figure 4.10 Steady part of first order velocity profiles with 𝛾𝛾 =0.5, 𝛼𝛼 =1, 𝛽𝛽 =0 and | 𝑀𝑀 |=25. (a) asymptotic analysis [Eqn. (4.103)], (b) Eqns. (4.24)-(4.27) 72 Figure 4.11 Steady part of first order velocity profiles with 𝛾𝛾 =0.5, 𝛼𝛼 =1, 𝛽𝛽 =0 and | 𝑀𝑀 |=50. (a) asymptotic analysis [Eqn. (4.103)], (b) Eqns. (4.24)-(4.27) 73 Figure 4.12 Steady part of first order velocity profiles with 𝛾𝛾 =0.5, 𝛼𝛼 =1, 𝛽𝛽 =0 and | 𝑀𝑀 |=100. (a) asymptotic analysis [Eqn. (4.103)], (b) Eqns. (4.24)-(4.27) 74 CHAPTER 5 TORSIONAL OSCILLATION OF SPHERICAL CAVITY FILLED WITH POROUS MEDIUM 5.1 Introduction In this chapter we consider flow generated in a viscous fluid as porous medium in a rigid spherical container performing torsional oscillation. In Section 5.2, the model is introduced. The details of deriving and solving the governing equations to get both the first order solutions and the steady streaming are provided. In Section 5.3, the theoretical results are discussed. 5.2 Model and theoretical development 5.2.1 The first order solutions Let ( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ) be the spherical coordinates. We consider a spherical container with radius 𝐷𝐷 , located at the origin. An incompressible Newtonian fluid of density 𝜌𝜌 and kinematic viscosity 𝜈𝜈 is filling the domain and is treated as porous medium. The spherical container is undergoing torsional oscillation with amplitude 𝑈𝑈 0 and frequency 𝜔𝜔 along 𝑧𝑧 -axis. The dimensional governing equations are 𝜕𝜕 𝒖𝒖 𝜕𝜕 𝑡𝑡 + 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 1 𝜌𝜌 𝛁𝛁 𝑝𝑝 + ν ∇ 2 𝒖𝒖 − 𝜈𝜈 𝜅𝜅 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0, (5.1) 75 with boundary conditions 𝑟𝑟 = 𝐷𝐷 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 0, 𝑢𝑢 𝜙𝜙 = 𝑈𝑈 0 sin( 𝜔𝜔 𝑡𝑡 ) sin 𝜃𝜃 . (5.2) The velocity, time, coordinates, pressure and the gradient operators are scaled as 𝒖𝒖 ∗ = 𝒖𝒖 𝜔𝜔𝜔𝜔 , 𝑡𝑡 ∗ = 𝜔𝜔 𝑡𝑡 , 𝑟𝑟 ∗ = 𝑟𝑟 𝜔𝜔 , 𝑝𝑝 ∗ = 𝑝𝑝 𝜌𝜌 𝜈𝜈 𝜔𝜔 , 𝛁𝛁 ∗ = 𝐷𝐷 𝛁𝛁 , (5.3) respectively, along with 𝜀𝜀 = 𝑈𝑈 0 𝜔𝜔𝜔𝜔 , | 𝑀𝑀 | = � 𝜔𝜔 𝜔𝜔 2 𝜈𝜈 , and 𝐷𝐷𝐷𝐷 = 𝜅𝜅 𝜔𝜔 2 , (5.4) where 𝐷𝐷𝐷𝐷 is the Darcy number, which represents the relative effect of the permeability 𝜅𝜅 of the medium versus its cross-sectional area. With stream function and angular circulation to relate to the velocity components are introduced as same as Eqn. (4.4), the dimensionless form of governing equations becomes 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜓𝜓 ) + 1 𝑟𝑟 2 � 𝜕𝜕 � 𝜓𝜓 , 𝐷𝐷 2 𝜓𝜓 � 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) + 2 𝐷𝐷 2 𝜓𝜓 𝐿𝐿 𝜓𝜓 + 2 Ω 𝐿𝐿 Ω � = 1 | 𝑀𝑀 | 2 𝐷𝐷 4 𝜓𝜓 − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 𝐷𝐷 2 𝜓𝜓 , 𝜕𝜕 Ω 𝜕𝜕 𝑡𝑡 + 1 𝑟𝑟 2 � 𝜕𝜕 ( 𝜓𝜓 , Ω) 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) � = 1 | 𝑀𝑀 | 2 𝐷𝐷 2 Ω − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 Ω , (5.5) together with the boundary conditions 𝑟𝑟 = 1 𝜓𝜓 = 𝜕𝜕 𝜓𝜓 𝜕𝜕 𝑟𝑟 = 0, Ω = 𝜀𝜀 (1 − 𝜇𝜇 2 ) sin 𝑡𝑡 . (5.6) Still, in these equations, 𝜇𝜇 = cos 𝜃𝜃 and operators are 𝐷𝐷 2 = 𝜕𝜕 2 𝜕𝜕 𝑟𝑟 2 + 1 − 𝜕𝜕 2 𝑟𝑟 2 𝜕𝜕 2 𝜕𝜕 𝜕𝜕 2 and 𝐿𝐿 = 𝜕𝜕 1 − 𝜕𝜕 2 𝜕𝜕 𝜕𝜕 𝑟𝑟 + 1 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 . (5.7) We perturb the stream function and angular circulation as 𝜓𝜓 = 𝜀𝜀 𝜓𝜓 1 + H.O.T. , Ω = 𝜀𝜀 Ω 1 + H.O.T. , (5.8) where H.O.T. denotes higher order terms. 76 Substitution into the governing equation (5.5) results in the following equations for 𝑂𝑂 ( 𝜀𝜀 ), 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜓𝜓 1 ) = 1 | 𝑀𝑀 | 2 𝐷𝐷 4 𝜓𝜓 1 − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 𝐷𝐷 2 𝜓𝜓 1 , 𝜕𝜕 Ω 1 𝜕𝜕 𝑡𝑡 = 1 | 𝑀𝑀 | 2 𝐷𝐷 2 Ω 1 − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 Ω 1 . (5.9) According to the boundary conditions 𝑟𝑟 = 1 𝜓𝜓 1 = 𝜕𝜕 𝜓𝜓 1 𝜕𝜕 𝑟𝑟 = 0, Ω 1 = (1 − 𝜇𝜇 2 ) sin 𝑡𝑡 . (5.10) The solutions are 𝜓𝜓 1 = 0, Ω 1 = ( 1 − 𝑚𝑚 𝑟𝑟 ) 𝑒𝑒 𝑚𝑚 𝑟𝑟 −( 1 + 𝑚𝑚 𝑟𝑟 ) 𝑒𝑒 − 𝑚𝑚 𝑟𝑟 2 𝑖𝑖 𝑟𝑟 [( 1 − 𝑚𝑚 ) 𝑒𝑒 𝑚𝑚 −( 1 + 𝑚𝑚 ) 𝑒𝑒 − 𝑚𝑚 ] (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , (5.11) where 𝑚𝑚 2 = 𝑠𝑠 | 𝑀𝑀 | 2 + 1 𝐷𝐷 𝜔𝜔 . Then from Eqn. (4.4), the velocities are 𝑢𝑢 1 𝜙𝜙 = ( 1 − 𝑚𝑚 𝑟𝑟 ) 𝑒𝑒 𝑚𝑚 𝑟𝑟 −( 1 + 𝑚𝑚 𝑟𝑟 ) 𝑒𝑒 − 𝑚𝑚 𝑟𝑟 2 𝑖𝑖 𝑟𝑟 2 [( 1 − 𝑚𝑚 ) 𝑒𝑒 𝑚𝑚 −( 1 + 𝑚𝑚 ) 𝑒𝑒 − 𝑚𝑚 ] (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + 𝑐𝑐 . 𝑐𝑐 . = ℎ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. , 𝑢𝑢 1 𝑟𝑟 = 𝑢𝑢 1 𝜃𝜃 = 0, (5.12) and shear stresses are 𝜏𝜏 1 𝑟𝑟 𝜙𝜙 = ��−3 1 𝑚𝑚 𝑟𝑟 3 + 3 1 𝑟𝑟 2 − 𝑚𝑚 𝑟𝑟 � 𝑒𝑒 𝑚𝑚 𝑟𝑟 + �3 1 𝑚𝑚 𝑟𝑟 3 + 3 1 𝑟𝑟 2 + 𝑚𝑚 𝑟𝑟 � 𝑒𝑒 − 𝑚𝑚 𝑟𝑟 � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝜏𝜏 1 𝑟𝑟 𝜃𝜃 = 𝜏𝜏 1 𝜃𝜃 𝜙𝜙 = 0. (5.13) 5.2.2 The second order solutions The variables to order 𝜀𝜀 2 are decomposed into steady part and unsteady part, and thus can be written as, 𝒖𝒖 2 = 𝒖𝒖 2 0 + � 𝒖𝒖 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 𝑝𝑝 2 = 𝑝𝑝 2 0 + � 𝑝𝑝 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 77 𝒖𝒖 1 ⋅ 𝛁𝛁 𝒖𝒖 1 = 𝓕𝓕 0 + � 𝓕𝓕 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � . (5.14) Then the steady component of order 𝜀𝜀 2 satisfies ∇ 2 𝒖𝒖 2 0 − 𝛁𝛁 𝑝𝑝 2 0 − 1 𝐷𝐷 𝜔𝜔 𝒖𝒖 2 0 = | 𝑀𝑀 | 2 𝓕𝓕 0 , 𝛁𝛁 ⋅ 𝒖𝒖 2 0 = 0 (5.15) with the boundary conditions 𝑟𝑟 = 1 𝒖𝒖 2 0 = 0, (5.16) and the solution are finite at 𝑟𝑟 = 0. Again, the steady part of nonlinear term is expanded in term of the vector spherical harmonics, 𝓕𝓕 0 = ℱ 𝑝𝑝 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝑝𝑝 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝐵𝐵 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 ( 𝜃𝜃 , 𝜙𝜙 ), (5.17) where ℱ 0 = − 2 ℎ( 𝑟𝑟 ) ℎ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 �sin 2 𝜃𝜃 𝒓𝒓 � + sin 𝜃𝜃 cos 𝜃𝜃 𝜽𝜽 � � , ℱ 𝑃𝑃 0 = − 8 3 √ 𝜋𝜋 ℎ( 𝑟𝑟 ) ℎ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 , ℱ 𝑃𝑃 2 = 8 1 5 √5 𝜋𝜋 ℎ( 𝑟𝑟 ) ℎ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 , ℱ 𝐵𝐵 2 = 4 1 5 √30 𝜋𝜋 ℎ( 𝑟𝑟 ) ℎ( 𝑟𝑟 ) � � � � � � 𝑟𝑟 . (5.18) The pressure 𝑝𝑝 2 0 and velocity 𝑢𝑢 2 0 = (𝑢𝑢 2 0 𝑟𝑟 , 𝑢𝑢 2 0 𝜃𝜃 , 𝑢𝑢 2 0 𝜙𝜙 ) can be written in the form, 𝒖𝒖 2 0 = 𝑢𝑢 2 0 𝑟𝑟 , 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 + 𝑢𝑢 2 0 𝑟𝑟 , 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 + 𝑢𝑢 2 0 𝜃𝜃 , 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 , 𝑝𝑝 2 0 = 𝑝𝑝 2 0, 0 𝑌𝑌 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑝𝑝 2 0, 2 𝑌𝑌 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) . (5.19) The resulting differential equations to order 𝜀𝜀 2 become d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 0 − 2 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 − d d 𝑟𝑟 𝑝𝑝 2 0, 0 − 1 𝐷𝐷 𝜔𝜔 𝑢𝑢 2 0 𝑟𝑟 , 0 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 , 1 𝑟𝑟 2 d d 𝑟𝑟 � 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 � = 0, d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − 8 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 − d d 𝑟𝑟 𝑝𝑝 2 0, 2 − 1 𝐷𝐷 𝜔𝜔 𝑢𝑢 2 0 𝑟𝑟 , 2 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 2 , d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 − 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑝𝑝 2 0, 2 − 1 𝐷𝐷 𝜔𝜔 𝑢𝑢 2 0 𝜃𝜃 , 2 = | 𝑀𝑀 | 2 ℱ 𝐵𝐵 2 , 78 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 = 0, (5.20) with homogeneous boundary conditions at 𝑟𝑟 = 1 and regularity conditions at 𝑟𝑟 = 0. The solution is then obtained as follows, 𝑢𝑢 2 0 𝑟𝑟 , 0 = 0 , 𝑢𝑢 2 0 𝑟𝑟 , 2 = 𝐷𝐷 1 𝑟𝑟 + 𝐷𝐷 2 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 4 + 𝐻𝐻 3 ( 𝑟𝑟 ) 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐻𝐻 4 ( 𝑟𝑟 ) 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ , 𝑢𝑢 2 0 𝜃𝜃 , 2 = 1 √ 6 �3 𝐷𝐷 1 𝑟𝑟 + 𝐷𝐷 2 �3 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 1 2 ⁄ � + 3 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 − 2 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 4 + 𝐻𝐻 3 ( 𝑟𝑟 ) �3 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 1 2 ⁄ � + 𝐻𝐻 4 ( 𝑟𝑟 ) �3 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐼𝐼 − 7 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 1 2 ⁄ �� , (5.21) where 𝐷𝐷 1 = − 𝑙𝑙 1 2 ⁄ 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) � 𝐻𝐻 1 (1) 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ + 𝐻𝐻 2 (1) �5 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 3 2 ⁄ + 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ � + 𝐻𝐻 4 (1) � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 ) 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) − 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 ) 𝐼𝐼 − 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 2 �� , 𝐷𝐷 2 = − 𝑙𝑙 1 2 ⁄ 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) � −5 𝐻𝐻 2 (1) + 𝐻𝐻 3 (1) 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ + 𝐻𝐻 4 (1) 𝐼𝐼 − 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ � , 𝐻𝐻 1 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 1 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ 𝑟𝑟 0 , 𝐻𝐻 2 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 2 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ 𝑟𝑟 0 , 𝐻𝐻 3 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 3 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ 𝑟𝑟 0 , 𝐻𝐻 4 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 4 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ 𝑟𝑟 0 , 𝑙𝑙 2 = 1 𝐷𝐷 𝜔𝜔 , (5.22) with 79 𝑊𝑊 ( 𝑟𝑟 ) = � � 𝑟𝑟 1 𝑟𝑟 4 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 1 − 4 𝑟𝑟 5 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d d 𝑟𝑟 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 2 0 𝑟𝑟 6 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 2 d 𝑟𝑟 2 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 − 1 2 0 𝑟𝑟 7 d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 3 d 𝑟𝑟 3 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 1 ( 𝑟𝑟 ) = � � 0 1 𝑟𝑟 4 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 0 − 4 𝑟𝑟 5 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d d 𝑟𝑟 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 2 0 𝑟𝑟 6 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 2 d 𝑟𝑟 2 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 𝐹𝐹 ( 𝑟𝑟 ) − 1 2 0 𝑟𝑟 7 d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 3 d 𝑟𝑟 3 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 2 ( 𝑟𝑟 ) = � � 𝑟𝑟 0 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 1 0 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d d 𝑟𝑟 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 0 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 2 d 𝑟𝑟 2 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 𝐹𝐹 ( 𝑟𝑟 ) d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 3 d 𝑟𝑟 3 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 3 ( 𝑟𝑟 ) = � � 𝑟𝑟 1 𝑟𝑟 4 0 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 1 − 4 𝑟𝑟 5 0 d d 𝑟𝑟 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 2 0 𝑟𝑟 6 0 d 2 d 𝑟𝑟 2 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 − 1 2 0 𝑟𝑟 7 𝐹𝐹 ( 𝑟𝑟 ) d 3 d 𝑟𝑟 3 � 𝐼𝐼 − 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 4 ( 𝑟𝑟 ) = � � 𝑟𝑟 1 𝑟𝑟 4 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 0 1 − 4 𝑟𝑟 5 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 0 2 0 𝑟𝑟 6 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 0 − 1 2 0 𝑟𝑟 7 d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 𝐹𝐹 ( 𝑟𝑟 ) � � , 𝐹𝐹 ( 𝑟𝑟 ) = −3| 𝑀𝑀 | 2 � 1 𝑟𝑟 2 ℱ 𝑃𝑃 2 ( 𝑟𝑟 ) − 1 𝑟𝑟 ℱ 𝑃𝑃 2 ′ ( 𝑟𝑟 ) � , (5.23) and 𝐼𝐼 𝑛𝑛 ( 𝑟𝑟 ) is the modified Bessel function of the first kind of order 𝑛𝑛 . 80 The expression for 𝑝𝑝 2 0, 0 and 𝑝𝑝 2 0, 2 can be obtained from the above equations, and thus is omitted here. Therefore, the steady streaming velocity and the stream function have the expressions as, 𝒖𝒖 2 0 = 1 4 � 5 𝜋𝜋 � 𝑢𝑢 2 0 𝑟𝑟 , 2 (3 cos 2 𝜃𝜃 − 1), − √6 𝑢𝑢 2 0, 𝜃𝜃 2 cos 𝜃𝜃 sin 𝜃𝜃 , 0 � , 𝜓𝜓 2 0 = 1 4 � 5 𝜋𝜋 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 sin 2 𝜃𝜃 cos 𝜃𝜃 . (5.24) 5.3 Theoretical results and discussion Figure 5.1 shows the first order azimuthal velocity profiles on the equatorial plane, where 𝜃𝜃 = 𝜋𝜋 2 , in one time period under different values of Womersley number | 𝑀𝑀 |. At low Womersley number | 𝑀𝑀 |, profiles of 𝑢𝑢 1 𝜙𝜙 tend to increase in a linear relation with radius 𝑟𝑟 , which is similar to rigid-body flow. On the other hand, at high Womersley number | 𝑀𝑀 |, the oscillation can induce leading-order flow only close to the wall. In Figure 5.2, the first order azimuthal velocity profiles on equatorial plane under different values of Darcy number 𝐷𝐷𝐷𝐷 have been plotted. For low Darcy number 𝐷𝐷𝐷𝐷 , oscillations have a deeper influence than higher Darcy number. But this difference is not significantly obvious. The streaming flow pattern is depicted in Figure 5.3, 5.4 and 5.5. A pair of vortices, symmetrical about the equatorial plane, is shown in the spherical region. At low Womersley number | 𝑀𝑀 |, the streaming vortex in the region shown in figures is counter-clockwise. With increasing | 𝑀𝑀 |, a new vortex with the opposite direction appears and grows close to the oscillating boundary. At the transient point, the old vortex vanishes, and the new vortex with the reversed direction dominates the region. The value of Womersley number | 𝑀𝑀 | corresponding to the transient 81 point decreases with an increase in 𝐷𝐷𝐷𝐷 . The | 𝑀𝑀 | values are around 5, 2.85 and 1.6 when 𝐷𝐷𝐷𝐷 equals to 0.1, 1 and 7.5, respectively. 82 Figure 5.1 Leading order velocity profiles [Eqn.(5.12)] on the equatorial plane over one time period with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=0.1, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=10. 83 Figure 5.2 Leading order shear stress profiles [Eqn.(5.12)] on the equatorial plane over one time period for different Darcy number with | 𝑀𝑀 |=5. (a) 𝐷𝐷𝐷𝐷 =0.1, (b) 𝐷𝐷𝐷𝐷 =1, (c) 𝐷𝐷𝐷𝐷 =10, (d) 𝐷𝐷𝐷𝐷 ≫ 1. 84 Figure 5.3 The streaming flow pattern [Eqn.(5.24)] with 𝐷𝐷𝐷𝐷 =0.1. (a) | 𝑀𝑀 |=4.5, (b) | 𝑀𝑀 |=5, (c) | 𝑀𝑀 |=5.5. 85 Figure 5.4 The streaming flow pattern [Eqn.(5.24)] with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=2.7, (b) | 𝑀𝑀 |=2.85, (c) | 𝑀𝑀 |=3.2. 86 Figure 5.5 The streaming flow pattern [Eqn.(5.24)] with 𝐷𝐷𝐷𝐷 =7.5. (a) | 𝑀𝑀 |=1.25, (b) | 𝑀𝑀 |=1.6, (c) | 𝑀𝑀 |=2.1. 87 88 CHAPTER 6 TRANSVERSE OSCILLATION OF SPHERE IN POROUS MEDIUM 6.1 Introduction In this chapter, the flow induced by the small-amplitude transverse oscillation of a sphere in porous medium is considered. In Section 6.2, with the governing equations and characteristic dimensionless parameters defined for the introduced mathematical model, I derived and solved the leading and second order governing equations for the oscillatory flow and steady streaming. A discussion of theoretical results follows in Section 6.3. 6.2 Model and theoretical development 6.2.1 The first order solutions We consider a spherical porous medium with radius 𝐷𝐷 , located at the origin. The porous medium is undergoing torsional oscillation. A sphere of radius 𝐷𝐷 , placed in an unbounded fluid domain, is performing transverse oscillation with amplitude 𝑈𝑈 0 and frequency 𝜔𝜔 along 𝑧𝑧 -axis. The domain fills with an incompressible Newtonian fluid of density 𝜌𝜌 and kinematic viscosity 𝜈𝜈 . The fluid is treated as porous medium 89 with permeability 𝜅𝜅 . The axisymmetric problem is formulated in the spherical coordinates( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ), which are fixed on the sphere. The dimensional governing equations are 𝜕𝜕 𝒖𝒖 𝜕𝜕 𝑡𝑡 + 𝒖𝒖 ∙ 𝛁𝛁 𝒖𝒖 = − 1 𝜌𝜌 𝛁𝛁 𝑝𝑝 + ν ∇ 2 𝒖𝒖 − 𝜈𝜈 𝜅𝜅 𝒖𝒖 , 𝛁𝛁 ∙ 𝒖𝒖 = 0, (6.1) with boundary conditions 𝑢𝑢 𝑟𝑟 = 𝑢𝑢 𝜃𝜃 = 𝑢𝑢 𝜙𝜙 = 0 on 𝑟𝑟 = 𝐷𝐷 , 𝑢𝑢 𝑟𝑟 = 𝑈𝑈 0 cos( 𝜔𝜔 𝑡𝑡 ) cos 𝜃𝜃 , 𝑢𝑢 𝜃𝜃 = − 𝑈𝑈 0 cos( 𝜔𝜔 𝑡𝑡 ) sin 𝜃𝜃 , 𝑢𝑢 𝜙𝜙 = 0 as 𝑟𝑟 → ∞. (6.2) The dimensionless scaling, 𝒖𝒖 ∗ = 𝒖𝒖 𝜔𝜔𝜔𝜔 , 𝑡𝑡 ∗ = 𝜔𝜔 𝑡𝑡 , 𝑟𝑟 ∗ = 𝑟𝑟 𝜔𝜔 , 𝑝𝑝 ∗ = 𝑝𝑝 𝜌𝜌 𝜈𝜈 𝜔𝜔 , 𝛁𝛁 ∗ = 𝐷𝐷 𝛁𝛁 , (6.3) is applied for velocity, time, coordinates, pressure and the gradient operators, respectively, along with 𝜀𝜀 = 𝑈𝑈 0 𝜔𝜔𝜔𝜔 , | 𝑀𝑀 | = � 𝜔𝜔 𝜔𝜔 2 𝜈𝜈 , and 𝐷𝐷𝐷𝐷 = 𝜅𝜅 𝜔𝜔 2 , (6.4) where 𝐷𝐷𝐷𝐷 is the Darcy number, which represents the relative effect of the permeability 𝜅𝜅 of the medium versus its cross-sectional area. With stream function and angular circulation to relate to the velocity components are introduced as before in Eqn. (4.4), the resulting dimensionless form of governing equations becomes 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜓𝜓 ) + 1 𝑟𝑟 2 � 𝜕𝜕 � 𝜓𝜓 , 𝐷𝐷 2 𝜓𝜓 � 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) + 2 𝐷𝐷 2 𝜓𝜓 𝐿𝐿 𝜓𝜓 + 2 Ω 𝐿𝐿 Ω � = 1 | 𝑀𝑀 | 2 𝐷𝐷 4 𝜓𝜓 − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 𝐷𝐷 2 𝜓𝜓 , 𝜕𝜕 Ω 𝜕𝜕 𝑡𝑡 + 1 𝑟𝑟 2 � 𝜕𝜕 ( 𝜓𝜓 , Ω) 𝜕𝜕 ( 𝑟𝑟 , 𝜕𝜕 ) � = 1 | 𝑀𝑀 | 2 𝐷𝐷 2 Ω − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 Ω , (6.5) with boundary conditions 𝜓𝜓 = 𝜕𝜕 𝜓𝜓 𝜕𝜕 𝑟𝑟 = Ω = 0 on 𝑟𝑟 = 1 , 90 𝜓𝜓 ~ 1 2 𝜀𝜀 𝑟𝑟 2 (1 − 𝜇𝜇 2 ) cos 𝑡𝑡 , Ω = 0 as 𝑟𝑟 → ∞. (6.6) As before, in these equations, 𝜇𝜇 = cos 𝜃𝜃 and the operators are 𝐷𝐷 2 = 𝜕𝜕 2 𝜕𝜕 𝑟𝑟 2 + 1 − 𝜕𝜕 2 𝑟𝑟 2 𝜕𝜕 2 𝜕𝜕 𝜕𝜕 2 and 𝐿𝐿 = 𝜕𝜕 1 − 𝜕𝜕 2 𝜕𝜕 𝜕𝜕 𝑟𝑟 + 1 𝑟𝑟 𝜕𝜕 𝜕𝜕𝜕𝜕 . (6.7) We perturb stream function and angular circulation as 𝜓𝜓 = 𝜀𝜀 𝜓𝜓 1 + H.O.T. , Ω = 𝜀𝜀 Ω 1 + H.O.T. , (6.8) where H.O.T. denotes higher order terms. Substitution into the governing equation results in the following equations for 𝑂𝑂 ( 𝜀𝜀 ), 𝜕𝜕 𝜕𝜕 𝑡𝑡 ( 𝐷𝐷 2 𝜓𝜓 1 ) = 1 | 𝑀𝑀 | 2 𝐷𝐷 4 𝜓𝜓 1 − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 𝐷𝐷 2 𝜓𝜓 1 , 𝜕𝜕 Ω 1 𝜕𝜕 𝑡𝑡 = 1 | 𝑀𝑀 | 2 𝐷𝐷 2 Ω 1 − 1 | 𝑀𝑀 | 2 1 𝐷𝐷 𝜔𝜔 Ω 1 . (6.9) According to the boundary conditions 𝜓𝜓 1 = 𝜕𝜕 𝜓𝜓 1 𝜕𝜕 𝑟𝑟 = Ω 1 = 0 on 𝑟𝑟 = 1 , 𝜓𝜓 1 ~ 1 2 𝜀𝜀 𝑟𝑟 2 (1 − 𝜇𝜇 2 ) cos 𝑡𝑡 , Ω 1 = 0 as 𝑟𝑟 → ∞. (6.10) The solutions are 𝜓𝜓 1 = � 1 4 𝑟𝑟 2 − � 𝑚𝑚 2 + 3 𝑚𝑚 + 3 4 𝑚𝑚 2 � 1 𝑟𝑟 + 3 4 𝑚𝑚 � 1 𝑚𝑚 𝑟𝑟 + 1 � 𝑒𝑒 − 𝑚𝑚 ( 𝑟𝑟 − 1) � (1 − 𝜇𝜇 2 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , Ω 1 = 0 , (6.11) where 𝑚𝑚 2 = 𝑠𝑠 | 𝑀𝑀 | 2 + 1 𝐷𝐷 𝜔𝜔 . Then given Eqn. (4.4), the velocities are 𝑢𝑢 1 𝑟𝑟 = 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 cos 𝜃𝜃 + c.c. = � 1 2 − � 𝑚𝑚 2 + 3 𝑚𝑚 + 3 2 𝑚𝑚 2 � 1 𝑟𝑟 3 + 3 2 𝑚𝑚 𝑟𝑟 2 � 1 𝑚𝑚 𝑟𝑟 + 1 � 𝑒𝑒 − 𝑚𝑚 ( 𝑟𝑟 − 1) � 𝜇𝜇 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝑢𝑢 1 𝜃𝜃 = 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑒𝑒 𝑖𝑖 𝑡𝑡 sin 𝜃𝜃 + c.c. 91 = �− 1 2 − � 𝑚𝑚 2 + 3 𝑚𝑚 + 3 4 𝑚𝑚 2 � 1 𝑟𝑟 3 + 3 4 𝑚𝑚 2 𝑟𝑟 3 (1 + 𝑚𝑚 𝑟𝑟 + 𝑚𝑚 2 𝑟𝑟 2 ) 𝑒𝑒 − 𝑚𝑚 ( 𝑟𝑟 − 1) � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝑢𝑢 1 𝜙𝜙 = 0. (6.12) The shear stress on the surfaces of ultrasounds contrast agents may damage them. Considering the safety issue, the shear stress should be investigated, which is 𝜏𝜏 1 𝑟𝑟 𝜃𝜃 = �� 3 𝑚𝑚 2 + 9 𝑚𝑚 + 9 2 𝑚𝑚 2 � 1 𝑟𝑟 4 − 3 4 𝑚𝑚 2 𝑟𝑟 4 (6 + 6𝑚𝑚 𝑟𝑟 + 3 𝑚𝑚 2 𝑟𝑟 2 + 𝑚𝑚 3 𝑟𝑟 3 ) 𝑒𝑒 − 𝑚𝑚 ( 𝑟𝑟 − 1) � (1 − 𝜇𝜇 2 ) 1 2 𝑒𝑒 𝑖𝑖 𝑡𝑡 + c.c. , 𝜏𝜏 1 𝑟𝑟 𝜙𝜙 = 𝜏𝜏 1 𝜃𝜃 𝜙𝜙 = 0. (6.13) 6.2.2 The second order solutions The variables to order 𝜀𝜀 2 are decomposed into steady part and unsteady part, and thus can be written as, 𝒖𝒖 2 = 𝒖𝒖 2 0 + � 𝒖𝒖 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 𝑝𝑝 2 = 𝑝𝑝 2 0 + � 𝑝𝑝 2 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � , 𝒖𝒖 1 ⋅ 𝛁𝛁 𝒖𝒖 1 = 𝓕𝓕 0 + � 𝓕𝓕 2 𝑒𝑒 2 𝑖𝑖 𝑡𝑡 + c.c. � . (6.14) Then the steady component of order 𝜀𝜀 2 satisfies ∇ 2 𝒖𝒖 2 0 − 𝛁𝛁 𝑝𝑝 2 0 − 1 𝐷𝐷 𝜔𝜔 𝒖𝒖 2 0 = | 𝑀𝑀 | 2 𝓕𝓕 0 , 𝛁𝛁 ⋅ 𝒖𝒖 2 0 = 0 (6.15) with the boundary conditions 𝒖𝒖 2 0 = 0 on 𝑟𝑟 = 1, (6.16) and the solution are finite as 𝑟𝑟 → ∞. Again, the steady part of nonlinear term is expanded in term of the vector spherical harmonics, 𝓕𝓕 0 = ℱ 𝑝𝑝 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝑝𝑝 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) + ℱ 𝐵𝐵 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 ( 𝜃𝜃 , 𝜙𝜙 ), (6.17) where 𝓕𝓕 0 = � � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � 𝑓𝑓 2 ′ ( 𝑟𝑟 ) � cos 2 𝜃𝜃 − 1 𝑟𝑟 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � � sin 2 𝜃𝜃 � 𝒓𝒓 � 92 + � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ′ ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 1 𝑟𝑟 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � � � sin 𝜃𝜃 cos 𝜃𝜃 𝜽𝜽 � , ℱ 𝑃𝑃 0 = 2 3 √ 𝜋𝜋 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � 𝑓𝑓 2 ′ ( 𝑟𝑟 ) − 2 𝑟𝑟 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � � � , ℱ 𝑃𝑃 2 = 4 1 5 √5 𝜋𝜋 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � 𝑓𝑓 2 ′ ( 𝑟𝑟 ) + 1 𝑟𝑟 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � � � , ℱ 𝐵𝐵 2 = − 2 1 5 √30 𝜋𝜋 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ′ ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ′ ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 1 𝑟𝑟 � 𝑓𝑓 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � + 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑓𝑓 2 ( 𝑟𝑟 ) � � � � � � � + 2 𝑠𝑠 2 ( 𝑟𝑟 ) 𝑠𝑠 2 ( 𝑟𝑟 ) � � � � � � � � � . (6.18) The pressure 𝑝𝑝 2 0 and velocity 𝑢𝑢 2 0 = (𝑢𝑢 2 0 𝑟𝑟 , 𝑢𝑢 2 0 𝜃𝜃 , 𝑢𝑢 2 0 𝜙𝜙 ) can be written in the form, 𝒖𝒖 2 0 = 𝑢𝑢 2 0 𝑟𝑟 , 0 ( 𝑟𝑟 ) 𝑷𝑷 0 0 + 𝑢𝑢 2 0 𝑟𝑟 , 2 ( 𝑟𝑟 ) 𝑷𝑷 2 0 + 𝑢𝑢 2 0 𝜃𝜃 , 2 ( 𝑟𝑟 ) 𝑩𝑩 2 0 , 𝑝𝑝 2 0 = 𝑝𝑝 2 0, 0 𝑌𝑌 0 0 ( 𝜃𝜃 , 𝜙𝜙 ) + 𝑝𝑝 2 0, 2 𝑌𝑌 2 0 ( 𝜃𝜃 , 𝜙𝜙 ) . (6.19) The resulting differential equations to order 𝜀𝜀 2 become d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 0 − 2 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 − d d 𝑟𝑟 𝑝𝑝 2 0, 0 − 1 𝐷𝐷 𝜔𝜔 𝑢𝑢 2 0 𝑟𝑟 , 0 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 0 , 1 𝑟𝑟 2 d d 𝑟𝑟 � 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 0 � = 0, d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − 8 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 − d d 𝑟𝑟 𝑝𝑝 2 0, 2 − 1 𝐷𝐷 𝜔𝜔 𝑢𝑢 2 0 𝑟𝑟 , 2 = | 𝑀𝑀 | 2 ℱ 𝑃𝑃 2 , d 2 d 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 𝑟𝑟 d d 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 − 6 𝑟𝑟 2 𝑢𝑢 2 0 𝜃𝜃 , 2 + 2 √ 6 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑝𝑝 2 0, 2 − 1 𝐷𝐷 𝜔𝜔 𝑢𝑢 2 0 𝜃𝜃 , 2 = | 𝑀𝑀 | 2 ℱ 𝐵𝐵 2 , d d 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 + 2 𝑟𝑟 𝑢𝑢 2 0 𝑟𝑟 , 2 − √ 6 𝑟𝑟 𝑢𝑢 2 0 𝜃𝜃 , 2 = 0, (6.20) with homogeneous boundary conditions at 𝑟𝑟 = 1 and regularity conditions as 𝑟𝑟 → ∞. The solution is then obtained as follows, 𝑢𝑢 2 0 𝑟𝑟 , 0 = 0 , 𝑢𝑢 2 0 𝑟𝑟 , 2 = 𝑏𝑏 1 1 𝑟𝑟 4 + 𝑏𝑏 2 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 + 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 4 + 𝐻𝐻 3 ( 𝑟𝑟 ) 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐻𝐻 4 ( 𝑟𝑟 ) 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ , 𝑢𝑢 2 0 𝜃𝜃 , 2 = 1 √ 6 � −2 𝑏𝑏 1 𝑟𝑟 + 𝑏𝑏 2 �3 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ − 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 1 2 ⁄ � + 3 𝐻𝐻 1 ( 𝑟𝑟 ) 𝑟𝑟 − 2 𝐻𝐻 2 ( 𝑟𝑟 ) 1 𝑟𝑟 4 + 𝐻𝐻 3 ( 𝑟𝑟 ) �3 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ + 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 1 2 ⁄ � + 𝐻𝐻 4 ( 𝑟𝑟 ) �3 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ − 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 1 2 ⁄ �� , (6.21) 93 where 𝑏𝑏 1 = 𝑙𝑙 3 2 ⁄ 5 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 ) − 𝑙𝑙 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 ) � 𝐻𝐻 1 (1) 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ − 𝐻𝐻 2 (1) �5 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 3 2 ⁄ − 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ � + 𝐻𝐻 3 (1) � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 ) 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 ) + 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 2 �� , 𝑏𝑏 2 = 𝑙𝑙 3 2 ⁄ 5 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 ) − 𝑙𝑙 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 ) � −5 𝐻𝐻 1 (1) − 𝐻𝐻 3 (1) �5 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 3 2 ⁄ + 𝐼𝐼 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ � − 𝐻𝐻 4 (1) �5 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 3 2 ⁄ − 𝐾𝐾 7 2 ⁄ ( 𝑙𝑙 ) 𝑙𝑙 1 2 ⁄ �� , 𝐻𝐻 1 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 1 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ ∞ 𝑟𝑟 , 𝐻𝐻 2 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 2 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ ∞ 𝑟𝑟 , 𝐻𝐻 3 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 3 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ ∞ 𝑟𝑟 , 𝐻𝐻 4 ( 𝑟𝑟 ) = ∫ 𝐶𝐶 4 � 𝑟𝑟 ′ � 𝑊𝑊 ( 𝑟𝑟 ′ ) d𝑟𝑟 ′ ∞ 𝑟𝑟 , 𝑙𝑙 2 = 1 𝐷𝐷 𝜔𝜔 , (6.22) with 𝑊𝑊 ( 𝑟𝑟 ) = � � 𝑟𝑟 1 𝑟𝑟 4 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 1 − 4 𝑟𝑟 5 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d d 𝑟𝑟 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 2 0 𝑟𝑟 6 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 2 d 𝑟𝑟 2 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 − 1 2 0 𝑟𝑟 7 d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 3 d 𝑟𝑟 3 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 1 ( 𝑟𝑟 ) = � � 0 1 𝑟𝑟 4 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 0 − 4 𝑟𝑟 5 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d d 𝑟𝑟 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 2 0 𝑟𝑟 6 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 2 d 𝑟𝑟 2 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 𝐹𝐹 ( 𝑟𝑟 ) − 1 2 0 𝑟𝑟 7 d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 3 d 𝑟𝑟 3 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 94 𝐶𝐶 2 ( 𝑟𝑟 ) = � � 𝑟𝑟 0 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 1 0 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d d 𝑟𝑟 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 0 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 2 d 𝑟𝑟 2 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 𝐹𝐹 ( 𝑟𝑟 ) d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � d 3 d 𝑟𝑟 3 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 3 ( 𝑟𝑟 ) = � � 𝑟𝑟 1 𝑟𝑟 4 0 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 1 − 4 𝑟𝑟 5 0 d d 𝑟𝑟 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 2 0 𝑟𝑟 6 0 d 2 d 𝑟𝑟 2 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 − 1 2 0 𝑟𝑟 7 𝐹𝐹 ( 𝑟𝑟 ) d 3 d 𝑟𝑟 3 � 𝐾𝐾 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � � � , 𝐶𝐶 4 ( 𝑟𝑟 ) = � � 𝑟𝑟 1 𝑟𝑟 4 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ 0 1 − 4 𝑟𝑟 5 d d 𝑟𝑟 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 0 2 0 𝑟𝑟 6 d 2 d 𝑟𝑟 2 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 0 0 − 1 2 0 𝑟𝑟 7 d 3 d 𝑟𝑟 3 � 𝐼𝐼 5 2 ⁄ ( 𝑙𝑙 𝑟𝑟 ) ( 𝑙𝑙 𝑟𝑟 ) 3 2 ⁄ � 𝐹𝐹 ( 𝑟𝑟 ) � � , 𝐹𝐹 ( 𝑟𝑟 ) = −6| 𝑀𝑀 | 2 � 1 𝑟𝑟 2 ℱ 𝑃𝑃 2 ( 𝑟𝑟 ) − 1 √ 6 � 1 𝑟𝑟 2 ℱ 𝐵𝐵 2 ( 𝑟𝑟 ) + 1 𝑟𝑟 ℱ 𝐵𝐵 2 ′ ( 𝑟𝑟 ) � � , (6.23) with 𝐼𝐼 𝑛𝑛 ( 𝑟𝑟 ) and 𝐾𝐾 𝑛𝑛 ( 𝑟𝑟 ) as the modified Bessel function of the first and second kind of order 𝑛𝑛 , respectively. The expression for 𝑝𝑝 2 0, 0 and 𝑝𝑝 2 0, 2 can be obtained from the above equations, and thus is omitted here. Therefore, the steady streaming velocity and the stream function have the expressions as, 𝒖𝒖 2 0 = 1 4 � 5 𝜋𝜋 � 𝑢𝑢 2 0 𝑟𝑟 , 2 (3 cos 2 𝜃𝜃 − 1), − √6 𝑢𝑢 2 0, 𝜃𝜃 2 cos 𝜃𝜃 sin 𝜃𝜃 , 0 � , 𝜓𝜓 2 0 = 1 4 � 5 𝜋𝜋 𝑟𝑟 2 𝑢𝑢 2 0 𝑟𝑟 , 2 sin 2 𝜃𝜃 cos 𝜃𝜃 . (6.24) 95 6.3 Theoretical results and discussion In Figure 6.1, the leading order velocity profiles on the 𝜃𝜃 = 𝜋𝜋 4 ⁄ plane in one time period at different values of Womersley number | 𝑀𝑀 | are compared. At low frequency, the velocity tends to linearly increase with radius 𝑟𝑟 until it reaches the outflow velocity. It is obvious that higher frequency oscillations have a greater influence on velocity profiles in the regions close to the sphere, compared to low frequency. Therefore the shear stress value at the sphere surface increases as | 𝑀𝑀 | goes up, as shown in Figure 6.2. Figure 6.3 presents the leading order velocity profiles on the 𝜃𝜃 = 𝜋𝜋 4 ⁄ plane in one time period at different values of Darcy number 𝐷𝐷𝐷𝐷 . The corresponding shear stress profiles on the equatorial plane are plotted in Figure 6.4. When Darcy number is small, as 𝐷𝐷𝐷𝐷 =0.01 in Figure 6.3(a), velocity profiles increase quickly to match the outflow velocity along the radius 𝑟𝑟 . When 𝐷𝐷𝐷𝐷 is pretty small, the resistance imposed by porous medium is large, which leads to large shear stress at the sphere surface (Figure 6.4(a)). As Darcy number increases to some level ( 𝐷𝐷𝐷𝐷 > 0.1 in Figure 6.3(b)-(d) and Figure 6.4(b)-(d)), the influence of Darcy number on leading order velocity and shear stress profiles is not obvious any more. In ultrasonography, the frequency usually used is 1-20 MHz, and the diameter of microbubble is 1-20 𝜇𝜇 𝑚𝑚 . For blood flow, the kinetic viscosity at 35 ℃ is 2.74 𝑚𝑚 𝑚𝑚 2 𝑠𝑠 ⁄ . In this case, the Womersley number | 𝑀𝑀 | ranges from 1.9 to 54. The streamlines for the streaming are presented in Figure 6.5 ( 𝐷𝐷𝐷𝐷 =0.1), Figure 6.6 ( 𝐷𝐷𝐷𝐷 =1) and Figure 6.7 ( 𝐷𝐷𝐷𝐷 =10). The results show that, at low frequency, the streaming vortex outside the sphere is clockwise. With increasing | 𝑀𝑀 |, this clockwise vortex is squeezed by a vortex in the opposite direction in the outer region. 96 Eventually, the clockwise vortex ceases and the counter-clockwise vortex dominates the region. The transition and cessation occur at lower values of Womersley number | 𝑀𝑀 | for high 𝐷𝐷𝐷𝐷 values than low 𝐷𝐷𝐷𝐷 values. The transition process takes place in a wide range of Womersley number | 𝑀𝑀 |. 97 Figure 6.1 Leading order velocity profiles [Eqn.(6.12)] on the 𝜃𝜃 = 𝜋𝜋 4 ⁄ plane over one time period with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=0.1, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=10. 98 Figure 6.2 Leading order shear stress profiles [Eqn.(6.13)] on the equatorial plane over one time period with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=0.1, (b) | 𝑀𝑀 |=1, (c) | 𝑀𝑀 |=10. 99 Figure 6.3 Leading order velocity profiles [Eqn.(6.12)] on the 𝜃𝜃 = 𝜋𝜋 4 ⁄ plane over one time period for different Darcy number with | 𝑀𝑀 |=5. (a) 𝐷𝐷𝐷𝐷 =0.01, (b) 𝐷𝐷𝐷𝐷 =0.1, (c) 𝐷𝐷𝐷𝐷 =1, (d) 𝐷𝐷𝐷𝐷 ≫1. 100 Figure 6.4 Leading order shear stress profiles [Eqn.(6.13)] on the equatorial plane over one time period for different Darcy number with | 𝑀𝑀 |=5. (a) 𝐷𝐷𝐷𝐷 =0.01, (b) 𝐷𝐷𝐷𝐷 =0.1, (c) 𝐷𝐷𝐷𝐷 =1, (d) 𝐷𝐷𝐷𝐷 ≫1. 101 Figure 6.5 The streaming flow pattern [Eqn.(6.24)] with 𝐷𝐷𝐷𝐷 =0.1. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=15, (c) | 𝑀𝑀 |=50. 102 Figure 6.6 The streaming flow pattern [Eqn.(6.24)] with 𝐷𝐷𝐷𝐷 =1. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=10, (c) | 𝑀𝑀 |=45. 103 Figure 6.7 The streaming flow pattern [Eqn.(6.24)] with 𝐷𝐷𝐷𝐷 =10. (a) | 𝑀𝑀 |=1, (b) | 𝑀𝑀 |=7.5, (c) | 𝑀𝑀 |=40. 104 CHAPTER 7 DISCUSSION AND FUTURE DIRECTIONS 7.1 Summary The objective of this work was to determine the flow due to oscillation of boundaries, which may serve as a theoretical model for study of protein aggregation and ultrasound contrast agents. In Chapter 1, a brief background of protein aggregation, ultrasound contrast agents and the streaming was given, with reference to existing work in the literature. A theoretical study was carried out to provide sufficient information on shear stress distribution and streaming flow properties for the fluid between two concentric hemispheres with the inner one undergoing torsional oscillation in Chapter 2 and transverse oscillation in Chapter 3. The analysis shows that the flow is determined by two important dimensionless parameters, Womersley number | 𝑀𝑀 | and perturbation parameter 𝜀𝜀 . The transverse oscillation case has greater amplitude of shear stress and deeper influence to the outer boundary, thus is better way to create shear stress than torsional oscillation. In addition, steady streaming pattern doesn’t change with the increase of Womersley number | 𝑀𝑀 | , for both types of oscillations. In Chapter 4, we developed a theoretical model of supposed oscillations with phase lag. The profiles of leading order oscillatory flow and steady streaming are given for unrestricted values of Womersley number | 𝑀𝑀 |, as well as for low and high frequency. The results are in close agreement with those of single oscillation in Chapter 2 and Chapter 3. 105 The model of flow in a spherical container with torsional oscillation is introduced in Chapter 5. Since the flow is through porous medium, an additional body force term is introduced into the momentum equation to consider the resistance imposed by porous medium. Darcy number 𝐷𝐷𝐷𝐷 is introduced, which represents the relative effect of the permeability of the medium versus it s cross-sectional area. Both the first order oscillatory flow and second order steady streaming are obtained with detailed derivation. High frequency oscillations induce leading-order oscillatory flow closer to the wall than low frequency ones. Oscillation at small Darcy number values has a deeper influence on leading order velocity profiles than large Darcy number values, though this effect is not obvious. Increase of Womersley number | 𝑀𝑀 | changes the direction of streaming vortex from counter-clockwise to clockwise. This transient point with larger Darcy number value occurs at lower frequency. In Chapter 6, the ultrasound contrast agents were modelled as a sphere in porous medium undergoing transverse oscillation. The analytical solutions show that large Womersley number | 𝑀𝑀 | value and small Darcy number 𝐷𝐷𝐷𝐷 lead to oscillatory flow closer to the wall region, and cause large shear stress at the sphere surface. For the streaming, increasing | 𝑀𝑀 | also caused reverse of vortex direction. This transition process occurs in a wide range of | 𝑀𝑀 |. 7.2 Future work Based on the results of the current work, there are still some unresolved issues that need to be addressed in the future work. Pan et al[86] investigated elasticity of protein lysozyme solution, and found that solutions of protein present viscoelasticity in high concentration. The oscillatory flow and steady streaming 106 of a viscoelastic fluid within or outside a periodically rotating sphere are examined by Bohme[87] and Repetto et al[56, 57]. Their works show that there is a considerably more complex set of possible flow regimes in viscoelastic fluid than in purely viscous fluid. Therefore, the issue of viscoelasticity needs to be addressed in the future. Due to ultrasound, high compressible microbubble contrast agents also have radial oscillations, which would also result in the microstreaming. Therefore, we shall include radial oscillations besides transverse ones. For a more complicated model, the factor of non-spherical oscillation could be investigated. 107 108 BIBLIOGRAPHY [1] Ross, C. A., and Poirier, M. A., 2004, "Protein Aggregation and Neurodegenerative Disease," Nature Medicine, 10, pp. S10-S17. [2] Irvine, G. B., El-Agnaf, O. M., Shankar, G. M., and Walsh, D. M., 2008, "Protein Aggregation in the Brain: The Molecular Basis for Alzheimer’s and Parkinson’s Diseases," Molecular Medicine, 14(7-8), pp. 451. [3] Cromwell, M. E., Hilario, E., and Jacobson, F., 2006, "Protein Aggregation and Bioprocessing," The AAPS Journal, 8(3), pp. E572-E579. [4] Philo, J. 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Abstract (if available)

Abstract In this dissertation, some fundamental fluid flow problems, due to oscillations in spherical geometry have been studied. It consists of three parts. In the first part, the flow in the annular region between two rigid hemispheres induced by the transverse and torsional oscillations of the inner solid hemispherical boundary has been studied. This work was originally motivated by the need to design an experiment system in which we can effectively apply and control the shear stress and correlate it to the protein aggregation rate. In the second part, we consider combined transverse and torsional oscillations in the annular region between two spheres, as a fundamental development in streaming phenomenon. In the third part, we aim to mathematically model the flow around bubbles in porous media as ultrasound contrast agents in ultrasonography. However, we consider only solid particulates in the present work. We also consider the basic problem of torsional oscillation within a porous fluid-filled sphere. ❧ First, the flow properties of fluid between two concentric hemispheres with inner hemisphere undergoing torsional oscillation and transverse oscillation are explored in detail in Chapters 2 and 3 respectively, using perturbation method. The Womersley number |M|, which expresses oscillation inertia forces in relation to the shear forces is introduced to determine the flow, along with perturbation parameter ε, which is amplitude of the oscillation in radians, scaled with the oscillation frequency. With mathematical analysis, the analytical solutions for the velocity field, shear rate, and the flow pattern of steady streaming are obtained, which can be applied to unrestricted Womersley number |M| values. In Chapter 4, we consider the same system with combined oscillations with phase difference β and amplitude ratio α. The leading order velocity field and shear stress profiles, and the steady streaming are discussed not only for unrestricted |M| values, but also in the low frequency (|M|≪1) and high frequency (|M|≫1) limits. Especially in high frequency limit, the flow field has been divided into three regions, two boundary layers and the outer region. The streaming flow field in determined for the limiting case of the streaming Reynolds number R_s ≪1. ❧ In Chapter 5, a mathematical model of flow for fluid through porous medium in a sphere with torsional oscillation is described. Darcy number Da is defined to represent the relative effect of the permeability κ of the medium versus its cross-sectional area. The mathematical analysis of both the oscillatory flow and the steady streaming is performed. Finally, the flow outside a sphere in porous medium due to transverse oscillation is analyzed in Chapter 6. The effects of Darcy number Da and Womersley number |M| on the flow are provided.

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Creator Kong, Dejuan (author)

Core Title Oscillatory and streaming flow due to small-amplitude vibrations in spherical geometry

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 07/25/2016

Defense Date 05/26/2016

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag fluid dynamics,OAI-PMH Harvest,perturbation method,porous medium,protein aggregation,steady streaming,ultrasound contrast agents

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Language English

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Advisor Sadhal, Satwindar S. (committee chair), Eliasson, Veronica (committee member), Redekopp, Larry G. (committee member), Shing, Katherine (committee member)

Creator Email dejuanko@usc.edu,kongkongpopo@gmail.com

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