On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r²/a² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B_K Bernoulli numbersParticular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g_c()=g() near centre of pipe - * point where g()=0Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()–₀ - ₀/ - ₀ a constant - * point where f()=0