Computation of Axisymmetric Suction Flow through Porous Media in the Presence of Surface Tension

Abstract

The effect of small surface tension on a class of axisymmetric flows with suction is studied numerically. The dynamic evolution of a blob of incompressible viscous fluid, surrounded by air and drawn into an interior sink, is considered. The velocity field of the viscous fluid is assumed to satisfy Darcy's law and thus the motion is that of a flow through porous media. The fluid interface motion is computed using a highly accurate boundary integral method. This method combines recent numerical techniques to achieve efficient and high-order space and time discretizations for this type of flow. Through accurate computations, it is shown that, in the presence of small surface tension, the dynamic behavior of the axisymmetric flows is very similar to that of the (two-dimensional) Hele–Shaw counterparts. A long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out. The finger bulges and develops a neck for sufficiently small surface tension. The bulge–neck formation is enhanced by the additional component of the mean curvature in the three-dimensional flow. However, its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here.