A theoretical and numerical study was conducted to obtain approximate analytical equations for the flow field rearrangement at the entrance region of membrane channels in Stokes flow. A singular pressure field is predicted because of the discontinuity in velocity boundary conditions at the edge of the permeable wall. The singularity, which occurs on the order of the permeability lengthscale, is due to the rapid rearrangement in the flow field from an undisturbed Poiseuille flow profile to one that has fluid leaking through the porous membrane walls. This fine structure, is important to resolve in the context of membrane separations due to concentration polarization and the resulting limits on throughput which are sensitive to the pressure field.

The numerical results, obtained using the boundary singularity method, were used to determine that the singular behavior of the pressure field was a series solution with the leading order for the radial dependence being r^-1/2. This, in turn allowed one to determine that the pressure buildup at the permeable wall had the leading order radial behavior of r^1/2. The angular dependence of the pressure terms was determined as a consequence of the continuity condition which required the pressure field to satisfy Laplace's equation for Stokes flows. Subsequently, coupled expressions for the velocity field were obtained by solving the Stokes equations. The coupled expressions for the pressure and velocity in radial and angular coordinates were fit to excellent quantitative agreement with the numerical results.

In order to determine the numerical constants for the pressure terms in the inner series solution, the closed-form analytical solution on the outer lengthscale was examined. The outer solution varies from the inner solution in that the wall seepage boundary condition is modified from a pressure-driven flux to a constant wall flux, which corresponds to a lengthscale that is much bigger than the permeability lengthscale. The purpose of this analysis was to provide the theoretical basis for the numerical constants of the inner solution terms.

The outer solution for the pressure and flow fields had the following angular terms: sin q, cos q, q sin q, q cos q; whereas the inner solutions had the angular dependence in terms of sin(q/2), cos(q/2), sin(3q/2), cos(3q/2), etc. In order to patch the outer and inner solutions together, the angular functions of the outer solution were expanded in terms of basis functions that had the same angular functional dependence as the inner solution. Thus, by phrasing the inner and outer solutions in terms of the same angular basis functions, the individual radial coefficients were obtained using Fourier integration. The radial coefficients were subsequently fit to correlations that had the same asymptotic behavior as indicated by the inner solution. Thus, the approximate analytical equations for the pressure and velocity fields valid in both inner and outer lengthscales were obtained.