(156a) Mathematical Analysis of Boundary Singularity at the Entrance Region of Membrane Channels

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.