Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration

Abstract

Well-known results on the filtration laws for the Newtonian fluids, obtained by the homogenization technique, served as a motivation to derive similar, analogous counterparts for the specific type of a non-Newtonian fluid. We studied a stationary filtration of the polymer fluid (Ostwald-de Waele model) through the periodic porous medium. Dimensions in the periodic domain are controlled by the two small parameters, one representing the period and the other controlling the ratio of a solid region and a fluid region of the porous medium. We found the specific critical size of the obstacles representing the impermeable part for which homogenization of the nonlinear Stokes flow results in the nonlinear Brinkman-type law by means of -convergence.

Keywords:

• homogenization
• Γ-convergence
• polymer fluid
• filtration
• Brinkman’s law
• drag force function

AMS Subject Classifications:

• 76S05
• 76D07
• 35Q35