Homogenization of a generalization of Brinkman's equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid
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In this paper we consider a generalization of Brinkman's equation governing the flow of an incompressible fluid through a porous medium to take into account the variation of the viscosity with the pressure. There is overwhelming evidence that the viscosity depends on pressure and that this dependence is exponential. We also consider the problem of a barotropic fluid in which both the viscosity and the density depend upon the pressure. We use a homogenization procedure, that exploits a multiple scale structure possessed by a solid porous medium, a 'micro-scale' comparable to the pore size and a 'macro-scale' associated with the global size of the body, to carry out a multiple-scale asymptotic analysis. The ratio of the micro-scale length to the macro-scale length provides a small parameter that is a natural parameter with which to carry out a perturbation analysis. The perturbation procedure leads to a systematic methodology by means of which we can obtain equations at different orders of the small parameter. Specifically, we study some simple boundary value problems corresponding to one-dimensional flows. We find that while the dependence of the viscosity on the pressure can have a significant effect on the nature of the solution, so much so that the solution is completely different in comparison to the constant viscosity case Brinkman's equation, we find that the dependence of the density on the pressure has little or no effect on the nature of the solution. Copyright 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
