Channel flow of electrorheological fluids under an inhomogeneous electric field Vom Fachbereich Mechanik der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation vorgelgt von Ana Ursescu, M. Sc. aus Cˆampina, Romania Referent: Prof. K. Hutter, Ph.D. Korreferent: Prof. Dr. M. R˚uˇziˇcka Tag der Einreichung: 16. September 2004 Tag der m¨undlichen Pr¨ufung: 21. Januar 2005 D 17 Aknowledgments This work was done during my stay at the Institute of Mechanics of the Darmstadt University of Technology within the framework of the Graduiertenkolleg “Modelling and Numerical Description of Technical Flows” with the financial support of DFG. I express my sincere gratitude to my supervisor, Professor Kolumban Hutter, for his instructive guidance, for his generous advice, for his constant encouragement and his continuous care to provide me everything I need for my research work, especially for the financial support for conferences and graduate schools and also for his prompt effort and his precious time spent to correct my manuscripts. Thank you for the fresh spirit which made the work at the University to be sunny. I would like to thank my co-referee, Professor Michael R˚uˇziˇcka for his interest in my work, for his kind invitation at his department for discussing the modeling of the ER- fluids under inhomogeneous electric fields. I gained a lot of insight from these meetings. I am grateful to him for accepting to be co-referee and for the effort to come in Darmstadt for my examination. I am very much thankful to Dr. Winfried Eckart for initiating me in the domain of electrorheological fluids, for introducing me in this fascinating subject and for helpful discussions and suggestions during my study. He offered me valuable support to find a flat and to go through all the formalities at the beginning of my stay in Darmstadt. I am grateful to Dr. Hubert Marschall for fruitful discussions concerning the analytical solution of the electrical problem, for helping me to get rid of the difficulties I encountered in applying the Wiener-Hopf technique and for providing me access to a Femlab license. The author is indebted also to Prof. Dr. J. Boersma from Eindhoven University of Technology (The Netherlands) for his careful examination and constructive criticism as a referee of a paper containing an earlier version of the actual Chapter 3 which have resulted in the removal of several inaccuracies. Special thanks go to my colleagues: Dr. Bernd Muegge, Dr. Shiva Prasad Pudasaini, Dr. Angelika Humbert, Mahmoud Reza Maneshkarimi, Dr. Sergio Faria, Dr. Nina Kirchner, Dr. Harald Ehrentraut, Dr. Ralf Greve, Dr. Yongqi Wang, Dr. Agnieszka Chrzanowska, Niklas Mellgren, Timo Schary, Dr. Vasily Vlasenko and the whole AG3 for their constant help. I will always remember the stimulating working atmosphere from our group and the nice time spent together. I want to thank further Dr. Steffen Eckert and his colleagues from AG4 for answer- ing my questions about Matlab and related stuff, Dr. Magnus Weiss and Dr. Thomas 4 Zwinger for providing me useful references about Finite Element Method. I express my appreciation to Dr. Thomas Wunderlich for providing me his doctoral work. Thanks are due also to Professor Corneliu Balan for sharing me his experience as a former PhD student in the same department, for teaching me important notions of rheol- ogy and giving me the opportunity to visit a laboratory of rheology and to see interesting experiments, for his instructive guidance during the European Rheology Conferences I attended twice and for the useful advice he gave me during my stay here in Darmstadt. I am grateful to my friend, Dr. Agnieszka Swierczewska for her constant help, espe- cially in understanding some matters of not standard mathematics and to her colleagues from the Mathematics Department, AG6, Dr. Andreas Frohlich, Dr. Piotr Gwiazda and Dr. Stefan Ebenfeld for the fruitful discussions we had about the formulation of the electrical problem. I must thank Dr. Amsini Sadiki for his interest in my work and for valuable discussions. Special thanks go to Professor Paul Flondor and Dr. Laurentiu Leustean for con- tributing to the demonstration from Appendix C.2. I gratefully appreciate the help gained in numerous discussions from Dr. Ioana Luca, guest scientist in our group. Besides, I would like to thank the people who taught me and encouraged me before I started the doctoral work: Dr. Victor Tigoiu, Professor Cristian Dascalu and all my colleagues from the Applied Mathematical Institute from Bucharest. Special thanks go to the directorate of the institute who allowed me to interrupt my research activity there and to accomplish these doctoral studies. I am extremely grateful to my husband Daniel for his support, but also for helpful discussions and suggestions. Further I want to thank my two mothers (who left their duties in Romania in order to come here and take care of their granddaughter) and my whole family for helping me in my efforts to finish the thesis and for their encouragement. Abstract This thesis is a theoretical study of the steady pressure driven channel flow of electrorhe- ological fluids (ERF) under a space dependent electric field generated by finite electrodes. Chapter 1 consists in a general description of ERF and their engineering applications and presents also the motivation, the goal and the borders of this work. Chapter 2 summarizes the governing equations of electrorheology (based on the phe- nomenological approach presented in [28, 29] with the corresponding jump conditions. It is assumed that the flow does not affect the electric field and consequently, the electrical problem is decoupled from the mechanical one. Both electrical and mechanical boundary value problems are formulated for various configurations of finite electrodes with different potentials placed along the channel walls. The simple case of two infinite electrodes which generate a homogeneous electric field is solved analytically. In Chapter 3 analytical solutions for different mixed boundary value problems arising from the electrical problem formulated in Chapter 2 are found by use of the Wiener- Hopf method. The solutions are given in terms of infinite series involving Gamma func- tions. The results can be used to describe the electric field generated between two infinite grounded electrodes by either one long electrode or two long electrodes charged in an anti-symmetric or a non-symmetric way. The electric field in the vicinity of the electrode edges is asymptotically evaluated. Some parametric studies are made with respect to the ratio between the permittivity of the electrorheological fluid and the permittivity of the isolating material outside the channel. We compare the analytical with numerical solutions and find good agreement which is considered as a validation of the numerical method. Chapter 4 treats the mechanical problem in more detail. First a review of the constitu- tive models used to describe the ER fluids in the literature is given. Then two-dimensional alternative constitutive laws appropriate for numerical simulations originating from the Casson-like and power law models are introduced using a parameter δ. In the end we non-dimensionalize the problem in both cases. In the last Chapter, we simulate numerically the flow of the Rheobay TP AI 3565 ER-fluid using the alternative Casson-like model and the EPS 3301 ER-fluid using the alternative power-law model by applying a finite element program. The behaviour of different fields such as velocity, pressure, generalized viscosity and the second invariant of the strain rate tensor near the electrode edges is studied for both fluids. A comparison with the experimental data is performed, validating the simulations. In order to investigate how 6 the numerical solution depends on the constitutive model we perform a parallel analysis of the two rheological models by applying them to the same material (Rheobay). Then we optimize the configuration of the electrodes by using the inhomogeneities caused by the end effects of the electrodes in order to obtain an enhancement of the ER-effect. Zusammenfassung Diese Arbeit ist eine theoretische Studie einer durch station¨aren Druck angetriebenen ebenen Poiseuille Str¨omung von elektrorheologischen Fl¨ussigkeiten (ERF) zwischen zwei parallelen Platten in einem r¨aumlich variablen elektrischen Feld, das durch endliche Elek- troden generiert wird. Kapitel 1 besteht aus einer generellen Beschreibung von ERFs und ihren technischen Anwendungen und pr¨asentiert die Motivation, das Ziel und die Grenzen dieser Arbeit. Kapitel 2 fasst die grundlegenden Gleichungen der Elektrorheologie (basierend auf ph¨anomenologischen Ans¨atzen, die in [28, 29] pr¨asentiert wurden mit den dazugeh¨orenden Sprungbedingungen) zusammen. Es wurde angenommen, dass die Str¨omung nicht das elektrische Feld beeinflusst und folglich das elektrische Problem vom mechanischen Prob- lem entkoppelt ist. Beide, elektrische und mechanische Randwertprobleme, sind in ver- schiedenen Konfigurationen von endlichen Elektroden mit unterschiedlichen Potentialen entlang der Kanalw¨ande formuliert. Der einfache Fall von zwei unendlichen Elektroden, die ein homogenes elektrisches Feld generieren, ist analytisch gel¨ost. In Kapitel 3 wurden analytische L¨osungen f¨urunterschiedliche gemischte Randwert- probleme, die durch das in Kapitel 2 formulierte elektrische Problem auftreten, durch die Anwendung der Wiener-Hopf Methode gefunden. Die L¨osung wurde mittels un- endlicher