Laminar Flow in the Entrance Region of a Porous Pipe

Abstract

The equation of motion for steady, incompressible, axially symmetric flow in a pipe with uniform wall suction or injection is expressed in terms of a ``dimensionless stream function,'' F(ζ, η) = ψ(x, r)/ψ(x, a), where ψ is the usual stream function, x and r are cylindrical polar coordinates, a is the radius of the pipe, ζ = 1 − [2Rx/(aN)], η = (r/a)2. R and N are Reynolds numbers which specify the flows through the pipe wall and through the inlet, respectively. This equation, which includes as a special case the equation previously obtained by Yuan and Finkelstein for similar (affine) velocity profiles, is greatly simplified by passing to the limit which is approached as N increases. Nonsimilar, ζ‐dependent solutions are approximated by a generalization of the method applied by Morduchow to the ζ‐independent equation of Yuan and Finkelstein. The results show the manner in which the condition of similarity of the velocity profiles is approached for a prescribed (e.g., parabolic) inlet profile. They also indicate the development of reverse flow at the wall of the pipe in the range of R values for which no similar solutions exist.