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Electrohydrodynamics of Particles and Drops in Strong Electric Fields

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Electrohydrodynamics of Particles and Drops in Strong Electric Fields UC San Diego UC San Diego Electronic Theses and Dissertations Title Electrohydrodynamics of Particles and Drops in Strong Electric Fields Permalink https://escholarship.org/uc/item/335049s5 Author Das, Debasish Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrohydrodynamics of Particles and Drops in Strong Electric Fields A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Engineering Sciences (Mechanical Engineering) by Debasish Das Committee in charge: Professor David Saintillan, Chair Professor Juan C. Lasheras Professor Bo Li Professor J´er´emiePalacci Professor Daniel M. Tartakovsky 2016 Copyright Debasish Das, 2016 All rights reserved. The Dissertation of Debasish Das is approved, and it is ac- ceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii EPIGRAPH If you keep proving stuff that others have done, getting confidence, increasing the complexities of your solutions - for the fun of it - then one day you’ll turn around and discover that nobody actually did that one! —Richard P. Feynman iv TABLE OF CONTENTS Signature Page................................... iii Epigraph...................................... iv Table of Contents..................................v List of Figures................................... viii List of Tables....................................x Acknowledgements................................. xi Vita......................................... xiv Abstract of the Dissertation............................ xv Chapter 1 Introduction.............................1 1.1 Low Reynolds number regime: Stokes flow.........3 1.2 Fundamental singularities..................6 1.3 Boundary element method.................. 10 1.3.1 Laplace’s equation.................. 10 1.3.2 Stokes equation.................... 12 1.4 Overview of current work................... 14 Chapter 2 Electrohydrodynamic interaction of spherical particles under Quincke rotation............................... 19 2.1 Introduction.......................... 19 2.2 Theoretical model....................... 24 2.2.1 Single sphere in a nonuniform field......... 24 2.2.2 Two spheres in a uniform field............ 34 2.3 Linear stability analysis.................... 40 2.4 Numerical simulations.................... 44 2.4.1 Fixed spheres..................... 45 2.4.2 Freely-suspended spheres............... 49 2.5 Conclusion.......................... 53 Chapter 3 Collective motion of Quincke rollers................ 58 3.1 Introduction.......................... 58 3.2 Experimental Setup...................... 60 3.3 An isolated Quincke roller.................. 61 3.4 Electrohydrodynamic interactions of Quincke rollers.... 65 3.5 Results............................. 68 3.6 Conclusion........................... 74 v Chapter 4 Electrohydrodynamics of axisymmetric drops........... 77 4.1 Introduction.......................... 77 4.2 Problem formulation..................... 80 4.3 Problem solution by domain perturbation.......... 85 4.3.1 Shape parametrization and expansion....... 85 4.3.2 Electric problem.................... 87 4.3.3 Flow problem: streamfunction formulation..... 92 4.3.4 Kinematic boundary condition............ 94 4.3.5 Dynamic boundary condition............ 95 4.3.6 Nonlinear charge convection............. 98 4.4 Summary of the small-deformation theory......... 99 4.4.1 Taylor deformation parameter............ 100 4.4.2 First-order theory................... 100 4.4.3 Second-order theory.................. 102 4.5 Results and discussion.................... 104 4.5.1 Effect of transient charge relaxation and shape defor- mation......................... 105 4.5.2 Effect of nonlinear charge convection........ 106 4.6 Conclusion........................... 113 Chapter 5 Electrohydrodynamics of drops under Quincke rotation: Numerical simulations............................. 116 5.1 Introduction.......................... 116 5.2 Problem definition...................... 119 5.2.1 Governing equations................. 119 5.2.2 Non-dimensionalization................ 122 5.3 Boundary integral formulation................ 125 5.3.1 Electric problem.................... 125 5.3.2 Flow problem..................... 127 5.3.3 Numerical implementation.............. 129 5.4 Results and discussion.................... 131 5.4.1 Taylor regime..................... 132 5.4.2 Quincke regime.................... 137 5.5 Conclusion........................... 144 Chapter 6 Conclusions and directions for future work............ 147 6.1 Conclusions.......................... 147 6.2 Directions for future work.................. 149 Appendix A Pair interactions........................... 151 A.1 Asymptotic estimate of the steady-state angular velocity. 151 A.2 Contact algorithm....................... 154 vi Appendix B Small deformation theory...................... 156 B.1 Electric field, charge and jump in Ohmic current...... 156 B.2 Interfacial velocity and hydrodynamic stress........ 158 B.3 Axisymmetric boundary element method.......... 161 Appendix C Boundary element method..................... 166 C.1 Discrete surface parametrisation............... 166 C.2 Regularisation of hypersingular integral........... 168 C.3 Weilandt’s deflation technique................ 169 Bibliography.................................... 172 vii LIST OF FIGURES Figure 2.1: Polarization of a spherical particle in an applied electric field E0. 20 Figure 2.2: Isolated sphere undergoing Quincke rotation in a nonuniform ex- ternal field Ee(x)........................... 24 Figure 2.3: Stability diagram for the angular velocity magnitude of a single sphere................................. 33 Figure 2.4: Interaction of two identical spheres undergoing Quincke rotation in a uniform field E0.......................... 33 Figure 2.5: Results of the linear stability analysis................ 43 Figure 2.6: Angular velocities in a simulation of two fixed interacting spheres separated by a distance R = 10 undergoing Quincke rotation in an 0 applied electric field of magnitude E0 = 1.5 Ec ........... 44 Figure 2.7: Angular velocities as functions of time in two simulations with R = 10 at different field strengths.................. 46 Figure 2.8: Onset of Quincke rotation of two interacting spheres........ 47 2 2 Figure 2.9: Dependence of the deviation (Ω − Ω0) of the steady-state angular from the isolated sphere value.................... 48 2 2 3 Figure 2.10: Dependence of (Ω − Ω0) × R on electric field strength E0 above the onset of Quincke rotation, for different values of Θ and R... 49 Figure 2.11: Dynamics of freely suspended spheres................ 50 Figure 2.12: Probability of the two spheres pairing up (cases 1, 2, and 3) vs separating (case 4)........................... 52 0 Figure 2.13: Typical particle trajectories for E0/Ec = 6.0, ε21 = −0.1097 and σ21 = −0.5............................... 54 Figure 3.1: Experimental Setup: (a) Sketch of the setup. (b)Superimposed fluorescence pictures of a dilute ensemble of rollers......... 60 Figure 3.2: Single spherical particle placed on an electrode and its image flow singularities.............................. 62 Figure 3.3: An isolated Quincke roller: (a) Coupling between rotational and translational velocity of the Quincke roller (b) In the plane of the surface, the direction of the translation velocity is defined by the angle θ................................. 65 Figure 3.4: Electrohydrodynamic interactions between two Quincke rollers and their image flow singularities..................... 65 Figure 3.5: Collective dynamics of Quincke rollers confined in a ring and square confinement (φ0 = 0.1)........................ 69 Figure 3.6: Collective dynamics experiments in circular confinement...... 70 Figure 3.7: Collective dynamics simulations in circular confinement...... 72 Figure 4.1: Problem definition: a liquid drop is placed in a uniform electric field E0................................. 81 viii Figure 4.2: Deformation parameter D as a function of time for the parameters of system 1b in the absence of charge convection.......... 107 Figure 4.3: (a) Deformation parameter D as a function of time for the parame- ters of system 1a. (b) Steady interfacial charge profile....... 108 Figure 4.4: (a) Deformation parameter D as a function of time for the parame- ters of system 1b. (b) Steady interfacial charge profile....... 109 Figure 4.5: (a) Deformation parameter D as a function of time for the parame- ters of system 1c. (b) Steady interfacial charge profile....... 110 Figure 4.6: (a) Deformation parameter D as a function of time for the parame- ters of system 4, which correspond to a steady prolate shape. (b) Steady interfacial charge profile................... 111 Figure 4.7: Steady drop deformation D as a function of electric capillary number CaE for the parameters of: (a) system 2a, (b) system 2b, and (c) system 3................................ 112 Figure 5.1: Problem definition: A liquid droplet with surface S and outward pointing unit normal n in an unbounded domain is placed in a uniform electric field E0 pointing in the vertical direction..... 119 Figure 5.2: Discretized mesh: N4 = 1280 curved elements with 6 nodes... 129 Figure 5.3: Deformation parameter D as a function of time for the parameters of: a) system 1a, b) system 1b, c) system 1c, and d) system 3... 133 Figure 5.4: Time evolution profiles of interfacial charge and
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