Effects of an external constant pressure gradient on a steady incompressible laminar flow through a semi-porous annular pipe

Abstract

Abstract In this paper, we examine a steady laminar flow for an incompressible fluid located in a semi porous annular pipe and subjected to a favorable constant pressure gradient applied between the two borders of the pipe. The inner wall is impermeable and the fluid is sucked or injected at the outer wall at constant and uniform velocity, orthogonally to the wall. The problem under study depends on three parameters: the pipe gap ratio, the dimensionless external pressure gradient, and the Reynolds number defined from the sum of the suction or injection velocity and the maximum Hagen–Poiseuille velocity. The conservation of mass induces the zero-divergence velocity field which allows replacing the steady-flow Navier–Stokes equations with a single equation satisfied by the stream function and called the vorticity equation. Assuming the similarity-solution hypothesis, the problem under consideration is reduced to a fourth-order nonlinear ordinary differential equation with two boundary conditions at each wall. The numerical shooting technique including the Runge–Kutta algorithm and the Newton–Raphson optimization method is applied to obtain the solution for the steady flow. For various values of the dimensionless external pressure gradient, the profiles of the velocity components are found and investigations on the wall shear stress for both walls are performed. The results obtained are discussed and physical understandings for the problem studied are derived.