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Theoretical and numerical results for electrorheological fluids. (English) Zbl 1022.76001
Freiburg im Breisgau: Univ. Freiburg im Breisgau, Mathematische Fakultät, 156 p. (2002).
The author presents a fairly complete theory focusing on the mathematical background of behavior of electrorheological fluids. These fluids have very important property: their viscosity changes significantly when disposed to a strong electromagnetic field. The main purpose of the thesis is to provide theoretical results on existence and regularity of solutions as well as on stability of numerical discretizations. The full system of governing equations consists of the system of incompressible Navier-Stokes equations for velocity field $$u$$ and pressure $$p$$, and Maxwell’s equations for electric field $$\mathbf E$$ and polarization $$\mathbf P$$. If the electric field and polarization are given, one can study a simplified system $$\partial_t u -\text{div} (S(Du)) + (u\cdot\nabla) u +\nabla p = f$$, $$\operatorname {div}u = 0$$, where $$S(Du) = (1+|Du|^2)^{{m-2}\over 2} Du$$ and $$Du$$ denotes the symmetric gradient. Interestingly enough, in the case of electrorheological fluids the exponent $$m$$ may depend on the strength of electric field $$E$$, i.e. $$m= m(|E|^2)$$, and thus the exponent $$m$$ is no longer constant.
From a mathematical point of view, the natural question is the right functional space setting. The energy functional cannot be expressed within the frame of classical Sobolev spaces, and it requires the so-called generalized Orlicz-Lebesgue and Orlicz-Sobolev spaces of functions satisfying $$\int_\Omega |\nabla u|^{m(x)} dx <\infty$$. Unfortunately, many of standard results for the classical $$L^p$$ and $$W^{k,m}$$ spaces cannot be transferred to the generalized Orlicz-Sobolev space. For example, the translation operator is no longer continuous. Nevertheless, in this thesis the author proves boundedness of Hardy-Littlewood maximal operator in the generalized Orlicz-Lebesgue space $$L^{m(.)}$$.
The thesis consists of eight chapters. Chapter 1 deals with generalized Orlicz-Lebesgue and Orlicz-Sobolev spaces. The next two chapters are focused on the study of the potential and extra stress. Chapter 4 is devoted to the study of two-dimensional flows under pressure stabilization which replaces $$\operatorname {div}u = 0$$ by $$\operatorname {div} u = \varepsilon \Delta p$$, where $$0<\varepsilon\ll 1$$ is a small stabilization parameter. In chapter 5 the author examines the governing system of equations in the case of three space dimensions. Chapter 6 is devoted to numerical approximation of the governing system. Chapter 7 deals with stationary Stokes problem. The last chapter contains various auxiliary results needed in this dissertation. The thesis is well written and selfcontained. It can be used as a good study and reference material for the mathematical study of electrorheological fluids.

##### MSC:
 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q35 PDEs in connection with fluid mechanics 35Q60 PDEs in connection with optics and electromagnetic theory