Two‐Dimensional Carreau Law for a Quasi‐Newtonian Fluid Flow Through a Thin Domain With a Slightly Rough Boundary

ABSTRACT This study investigates the asymptotic behavior of the steady‐state quasi‐Newtonian Stokes flow with viscosity given by the Carreau law within a thin domain, focusing on the effects of a rough boundary of the domain. Employing asymptotic techniques with respect to the domain's thickness, we rigorously derive the effective nonlinear two‐dimensional Reynolds model describing the fluid flow. The mathematical analysis is based on deriving the sharp a priori estimates and proving the compactness results of the rescaled functions together with monotonicity arguments. The resulting limit model incorporates contributions of the oscillating boundary and thus, it could prove useful in the applications involving this lubrication regime.