(142ct) An Exact Solution for the Capillary Driven Transport in Porous Media in the Presence of Sudden Expansion

Theoretical analysis of the capillary driven transport of
liquid in porous media is given for special cases of sudden expansion.  Elliptical and oblate spheroid
coordinate transforms where used to find transient analytic solutions for two
and three-dimensional expansions (Figures 1 and 2).  The fluid flow is simplified to
potential functions, which reduces the equations to single dimensional, time
dependent problems.  The asymptotic
expression of the two-dimensional case is compared with the circular result of Hyvaluoma.  In three dimensions a spherical
expansion is derived which is compared to the asymptotic solution of the oblate
spheroid.  In both two and three
dimensions, the expressions for expansion near the entrance are compared to the
one-dimensional Lucas-Washburn equation and are shown to deviate from the
classical result due to the sudden expansion.  The system in the two-dimensional case additionally
includes the time dependent effects of a constant evaporation rate, allowing a
steady state to be reached.  Two
characteristic time-scales, one related to fluid capillary pressure and viscous
resistance and the other to the evaporation rate, give a dimensionless
evaporation number which shows the transport mechanism that dominates the
time-dependent flow.  

Figure 1:  Elliptic coordinates, used to describe
the flow in an expanding 2D porous domain. The points –a and a are the foci of the ellipses.

Figure 2:  Oblate spheroid coordinates, used to
describe the flow in an expanding 3D porous domain. The points –a and a are the foci of the oblate
surface corresponding to eta. Only systems with axial symmetry are considered.