Effets d’un gradient de pression externe constant sur un écoulement laminaire incompressible dans une conduite annulaire semi - poreuse

Abstract

We are, examine a steady laminar flow for an incompressible fluid located in 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 velocity, orthogonally to wall. The problem understudy depends on three parameters: the pipe gap ratio, the dimensionless external pressure gradient, and Reynolds number defined from the sum of the suction or injection velocity and the maximal Hagen-Poiseuille velocity. The conservation of mass induces the zero-divergence velocity field which allows replacing the steady-flows Navier-Stokes equation 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 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.