MULTIPLE SOLUTIONS OF THE STEADY FLOWS IN A RECTANGULAR CHANNEL WITH SLIP EFFECT ON TWO EQUALLY POROUS WALLS

Abstract

We study the boundary layer equation $f^{\prime\prime\prime}(\eta) + R((f^{\prime}(\eta))^2 - f(\eta) f^{\prime\prime}(\eta)) = K$, subjects to the boundary conditions $f(0) = f^{\prime\prime}(0) = 0$, $f(1) = 1$ and $f^{\prime}(1) + \varphi f^{\prime\prime}(1) = 0$. The given problem arises from the study of steady laminar flows in channels with two equally porous walls, where $R$ relates to the Reynold's number, and $K$ is an integration constant. We are able to obtain the homogeneity property and classify all types of solutions for the prescribed positive slip coefficient $\varphi$. In particular, the existence of the continuums in the $R-K$ plane has been verified, and this leads to the existence of multiple solutions for large $R$.