Pressure-exerted steady laminar flow of an incompressible fluid along a porous parallel-walled channel with an impermeable wall

Abstract

Abstract This paper is devoted to the pressure-exerted steady laminar flow of an incompressible Newtonian fluid along a parallel-walled horizontal channel with a porous upper wall and an impermeable lower wall. The fluid is sucked or blown through the porous wall, at constant and uniform velocity, orthogonally to the wall. At the same time, an external pressure gradient constant in time is applied between the two ends of the channel. The aim of this work is to determine and analyze the effects of the external pressure gradient on the flow, the suction/blowing velocity being kept constant. The two-dimensional configuration of the flow with zero-divergence velocity field allows the existence of the stream function given by a single nonlinear partial differential equation which replaces the Navier–Stokes equations and is called the vorticity equation. This latter equation is demonstrated by applying an unusual approach which uses the vector momentum equation in its general form. From the similarity-solutions assumption, it is shown that the vorticity equation leads to a two-point boundary value problem whose solutions are computed by means of a numerical shooting technique including the Newton–Raphson optimization algorithm. Physical understandings of the flow under consideration are derived from the results obtained.