On the steady viscous flow of a nonhomogeneous asymmetric fluid

TL;DR

This paper investigates the steady flow of a nonhomogeneous asymmetric viscous fluid in a bounded domain, using a stream-function approach to establish the existence of solutions for the simplified system.

Contribution

It introduces a stream-function formulation for nonhomogeneous asymmetric fluids that simplifies the governing equations and proves the existence of solutions using fixed point methods.

Findings

Existence of solutions for the boundary value problem.

Stream-function formulation effectively reduces the system.

Dropping the continuity equation simplifies analysis.

Abstract

We consider a boundary value problem for the system of equations describing the stationary motion of a viscous nonhomogeneous asymmetric fluid in a bounded planar domain having a $C^2$ boundary. We use a stream-function formulation after the manner of N. N. Frolov [Math. Notes, \textbf{53}(5-6), 650--656, 1993] in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation. With a fixed point argument we show the existence of solutions to the resulting system.