Homogenization of a Navier–Stokes-type system for electrorheological fluid

Abstract

Some fluids sharply change their rheological properties in the presence of electromagnetic field. The viscous stress tensor of such fluids becomes not just a nonlinear function of the strain rate tensor D, but acquires a strong dependence on the spatial argument x. An example is provided by the tensor D|D| ^p(x)−2 , where the exponent is determined by the applied electromagnetic field. But, in general, the viscous stress tensor has a more complex anisotropic structure. It is assumed that the electromagnetic field is periodic in x and its period is specified by a small parameter. In this situation, one has to deal with a homogenization problem, and the main task of homogenization is to find the effective (homogenized) viscous stress tensor that does not depend on the spatial variable. The corresponding homogenization theorem is formulated and a procedure of constructing the homogenized tensor is described.

Keywords:

• homogenization
• Navier–Stokes system
• Sobolev–Orlicz space
• electrorheological fluid

AMS Subject Classifications::

• 35B27
• 35Q10
• 76M50

Acknowledgement

This work was supported by the Russian Foundation for Basic Research, project nos. 08-01-00616 and 09-01-12157.