Theoretical Analysis of Laminar Pipe Flow in a Porous Wall Cylinder

Abstract

A theoretical investigation was conducted to obtain velocity, pressure, and shear stress distributions for incompressible, steady, fully developed, laminar flow through a cylinder with a uniformly porous wall. Ejection/injection at the walls results from the pressure difference across the porous wall. Fluid flow phenomena in porous tubes and ducts have previously been investigated with the velocity prescribed as the boundary condition at the wall. An accurate wall condition must account for the variable wall velocity being dependent upon the pressure difference across the wall, the properties of the fluid, the thickness and the permeability of the structure. An integral momentum technique was employed to reduce the axisymmetric Navier-Stokes equations in cylindrical coordinates to a nonlinear, second-order ordinary differential equation with appropriate boundary conditions. The velocity condition at the wall was established for the ejection/injection at the surface resulting from the pressure difference across the porous wall derived from Darcy’s law. Numerical solutions were obtained for a range of axial flow Reynolds numbers, wall permeabilities, and initial pressure difference across the porous wall. The calculated static pressure variation in the axial flow direction, the velocity components, and the wall shear stress are presented. For the case of fluid ejection, the results of the analysis show that the wall shear stress and static pressure decrease in the axial flow direction. The rates of decrease are functions of the wall porosity, initial pressure gradient across the wall, and inlet flow Reynolds number. The present analysis treats the realistic problem of flow adjustment to the condition where zero pressure differential across the porous wall occurs (the normal wall velocity vanishes). Previous models are based upon the assumptions of constant radial velocity at the wall and/or prescribed wall shear stress without taking into account the pressure drop through the wall. Such assumptions imply that a variable pressure exists external to the pipe, or that the pipe has walls of variable permeability and thickness rather than the hypothesized condition that the pipe has uniformly porous walls. For one set of boundary conditions it is shown that the outflow through the walls completely discharges the entering flow. As a result no far downstream axial flow occurs. Such effects were not previously discussed by other investigators. For other sets of boundary conditions reductions in centerline velocity and shear stress occur.