Laminar flow with injection through a long dead-end cylindrical porous tube: Application to a hollow fiber membrane

Effects of membrane length and hydraulic resistance on the steady-state laminar flow of a fluid with injection in a dead-end cylindrical porous tube have been investigated using the perturbation approach of the Navier-Stokes and continuity equations. Analytic solutions of the dynamic equations, reduced to non linear differential equations, were obtained and applied to submerged, dead-end hollow fiber membranes of large resistances and small wall Reynolds numbers. The velocity and pressure profiles were solved using the method of separation of variables as with physical properties of the membrane and fluid and the wall velocity at the dead end. The constant flux approximation was found to be valid only for a short membrane with a large hydraulic resistance. The constant permeability approximation used in this study is universal for a membrane of an arbitrary length, through which the fluid velocities and pressure increase exponentially from the dead end to the open end.