Laminar Pipe Flow With Injection and Suction Through a Porous Wall

Abstract

Abstract The effect of injection and suction at the wall on the two-dimensional steady-state laminar flow of a fluid in a porous-wall pipe has been investigated in detail by the solution of the Navier-Stokes equations in cylindrical co-ordinates. An exact solution of the dynamic equations, reduced to a third-order nonlinear differential equation with appropriate boundary conditions, is obtained. A perturbation method was used to solve the latter equation for both small and large flows through the porous wall. The velocity components are expressed as functions of the ratio of velocity through the porous wall to the maximum axial velocity at the pipe entrance, the co-ordinates of the pipe, and the physical properties of the fluid. The results show that the effect of injection at the porous wall of the pipe is to increase the friction coefficient at the wall. For an injection ratio Q/W = 0.01 (500 ≤ Re ≤ 2500) the friction coefficient at the wall is increased by 70–85 per cent over the zero injection case (Poiseuille case).