DROP ELECTROHYDRODYNAMICS A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Master of Science by Alejandro Agust´ın Carderera de Diego August 2016 c 2016 Alejandro Agust´ın Carderera de Diego ALL RIGHTS RESERVED ABSTRACT Electrohydrodynamics is the study of the interaction between fluids and electric fields, and is used to model phenomena like fuel atomization or the mixing of multiphase flows under the influence of electric fields. Increasing interest is being placed in using electric fields to vary mul- tiphase behaviour, one example is combustion processes, where finer droplets and wider sprays are created to increase engine efficiency. Another example can be seen in the pharmaceutical industry, where micro-encapsulation of com- pounds is achieved through the use of electrified coaxial liquid jets. In this work, the Ghost Fluid Method (GFM), and the Continuum Sur- face Force (CSF) approach will be used to discretize the electric potential Poisson equation for multiphase problems with arbitrary interfaces and discontinuous physical properties. A new scheme has also been derived to solve this problem, in the Finite Volume (FV) framework, and an extensive error analysis has been carried out to gauge the accuracy and properties of these schemes. These tools, coupled with NGA, the Computational Fluid Dynamics (CFD) code used in Dr. Olivier Desjardins’ research group will allow the study of, among others, the two phase mixing of two dielectric liquids under the influ- ence of an electric field, of interest to the chemical engineering industry, where an alternative non-mechanical way of mixing corrosive liquids is sought out, or the atomization of drops during fuel injection when an electric field is applied. BIOGRAPHICAL SKETCH Alejandro Carderera earned his Bachelors Degree in Industrial Engineering Technologies in 2014 from the Universidad Politecnica´ de Madrid (Escuela Tecnica´ Superior de Ingenieros Industriales). He joined the Master of Science program in Applied and Engineering Physics at Cornell University in August 2014. His interest in CFD sparked during his last year of undergraduate study, in which he was granted a Research Collaboration Scholarship by the Spanish Ministry of Education to conduct research in the department of fluid mechanics under the guidance of Dr. Jaime Carpio. This work formed the base of his undergraduate thesis, which aimed at improving current methods to solve CFD problems with mobile frontiers in the Finite Element Method (FEM) framework, and later allowed him to study the freefall regimes of circular discs according to their corresponding Froude and Reynolds number. As a graduate student at Cornell University, he joined the Computa- tional Thermo-Fluids Laboratory in August of 2015, undertaking research in electrohydrodynamics (EHD) supervised by Dr. Olivier Desjardins. iii This document is dedicated to my family, Kala, my friends and to my supervisor, Professor Olivier Desjardins. iv ACKNOWLEDGEMENTS I would like to thank my advisor Olivier for his untiring guidance during this project, as well as for the myriad of brilliant suggestions and advice that have led to this document and its content. A number of students have been instrumental or inspirational in the development of my research and among these are Robert Chiodi, Peter Ireland, Ravi Patel, John Palmore, Houssen Kasbaoui, Stephanie Firehammer, Sheng Wang and Meilin Dong. This thesis would not have been possible without the help, support and patience of my family, Kala and my friends, as well as to all the people who have mentored and counselled me, which include Ruth Siboza and Dr. Jaime Carpio. v TABLE OF CONTENTS Biographical Sketch . iii Dedication . iv Acknowledgements . .v Table of Contents . vi List of Tables . vii List of Figures . viii 1 Introduction 1 1.1 Background and motivation . .1 1.2 Previous Work . .2 1.2.1 Experimental and theoretical work . .2 1.2.2 Numerical work . .5 2 Mathematical Model 8 2.1 Navier Stokes Equations . .8 2.1.1 Conservation of Mass . .8 2.1.2 Conservation of Momentum . .9 2.2 Electromagnetic Equations . 10 2.3 Interface Conditions . 11 3 CFD Implementation 15 3.1 General Aspects . 15 3.2 Electric Potential Poisson Equation . 17 3.2.1 GFM implementation . 17 3.2.2 Finite Volume Implementation . 20 3.2.3 Continuum Surface Force Implementation . 24 4 Validation 29 4.1 1D Study . 29 4.1.1 Horizontal Liquid-Gas Interface . 29 4.2 2D Study . 36 4.2.1 Method of Manufactured Solutions (MMS) . 36 4.2.2 Dielectric rod in a uniform electric field . 41 4.3 3D Study . 56 4.3.1 Dielectric drop in a uniform electric field . 56 5 Conclusions and Perspectives 68 Bibliography 70 vi LIST OF TABLES 4.1 Error (%) in the calculation of the pressure jump p at the interface. 36 4.2 Percentage error (%) in the electric field in the vicinity of the liquid-gas interface with the GFM. 53 4.3 Percentage error (%) in the pressure jump in the vicinity of the liquid-gas interface with the GFM method. 54 4.4 Error (%) in the calculation of the total pressure jump p at the pole. 55 vii LIST OF FIGURES 2.1 Multiphase interface configuration. 11 3.1 GFM Schematic. 19 3.2 Finite Volume Schematic. 20 3.3 Finite Volume Schematic with fluid ”-” at the cell face. 24 3.4 GFM and FV vs. CSF representation of EHD effects. 25 4.1 Horizontal interface with charge density. 29 4.2 φ/φ0 as a function of y for the horizontal flat interface. 31 4.3 Ey=Eg as a function of y for the horizontal flat interface. 31 4.4 Error in the L2 norm for the horizontal interface potential. 32 4.5 Normalized error in the L1 norm for the horizontal interface po- tential. 33 4.6 Normalized error in the L1 norm for the horizontal potential. 33 4.7 Normalized error in the L2 norm for the horizontal interface elec- tric field. 34 4.8 Normalized error in the L1 norm for the horizontal interface elec- tric field. 34 4.9 Normalized error in the L1 norm for the horizontal interface electric field. 35 4.10 MMS geometry for 2D Validation. 37 4.11 φ error in the L2 norm for the 2D MMS validation. 38 4.12 φ error in the L1 norm for the 2D MMS validation. 38 4.13 φ error in the L1 norm for the 2D MMS validation. 39 2 4.14 Ey error in the L norm for the 2D MMS validation. 39 1 4.15 Ey error in the L norm for the 2D MMS validation. 40 1 4.16 Ey error in the L norm for the 2D horizontal MMS validation. 40 4.17 Dielectric rod placed in a uniform electric field. 42 ◦ 4.18 Electric field Ey along θ = 0 for dielectric rod (GFM vs.FV). 43 ◦ 4.19 Electric field Ey along θ = 90 for dielectric rod (GFM vs.FV). 43 4.20 Percentage error for the normal (left) and tangential (right) com- ponent of the electric field using the FV method. 44 4.21 Percentage error for the normal (left) and tangential (right) com- ponent of the electric field using the CSF approach. 45 4.22 Percentage of the normal electric field with respect to the total electric field for the dielectric rod. 46 4.23 [p]Γ due to the electric field along the normal direction for a di- electric rod (GFM). 48 4.24 [p]Γ due to the electric field along the normal direction for a di- electric rod (CSF). 48 4.25 [p]Γ due to the electric field along the tangential direction for a dielectric rod (GFM). 49 viii 4.26 [p]Γ due to the electric field along the tangential direction for a dielectric rod (CSF). 49 4.27 Total [p]Γ due to the electric and surface tension effects for a di- electric rod (GFM). 50 4.28 [p]Γ due to the electric field along the tangential direction for a dielectric rod (CSF). 50 4.29 Pressure jump [p] due to surface tension versus angle for a di- electric rod. 51 ◦ 4.30 Electric field Ey along θ = 0 for the dielectric rod (5 vs. 20 Cells per radius). 52 ◦ 4.31 Electric field Ey along θ = 90 for the dielectric rod (5 vs. 20 Cells per radius). 53 4.32 Deviation from the analytical solution for the jump due to the normal field (GFM vs. FV). 55 4.33 Deviation from the analytical solution for the jump due to the tangential field (GFM vs. FV). 56 4.34 Spherical liquid drop suspended in a gas and subject to an elec- tric field E ................................ 57 0 4.35 p Γ due to the normal electric field En for the spherical drop (FV & GFM). 59 4.36 p due to the normal electric field En for the spherical drop (CSF). 59 Γ 4.37 p Γ due to the normal electric field Et for the spherical drop (FV & GFM). 60 4.38 p due to the normal electric field Et for the spherical drop (CSF). 60 Γ 4.39 Total p Γ due to electric and surface tension effects for the spher- ical drop (FV & GFM). 61 4.40 Total p Γ due to electric and surface tension effects for the spher- ical drop (CSF). 61 4.41 Total p due to surface tension effects for the spherical drop (CSF). 62 Γ 4.42 Absolute error comparison in the pressure jump p due to the normal electric field En for the dielectric drop. 63 4.43 Absolute error comparison in the pressure jump p due to the tangential electric field Et for the dielectric drop.