Newtonian and non-newtonian fluids through permeable boundaries

Abstract

We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interface Γ of a domain Ω, where Ω ⊂ R 3. Γ is defined as a smooth two-dimensional (at least class C 2) manifold in Ω. The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interface Γ. We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normal n, where the tangential components are assumed to be zero. These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape of Ω; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a "shear flow" is assumed.