Analytical solution to a sharp interface problem in a vortex-generated flow

Abstract

Conformal mappings and integral representations of the Dirichlet boundary value problem for analytic functions are employed to solve explicitly the problem of steady, two-dimensional, Darcian seepage from a reservoir with fresh water to a sea in a confined aquifer of a finite thickness. A sharp interface between moving fresh and stagnant saline water forming a wedge is determined depending on one dimensionless parameter, which includes the difference in water elevations between the reservoir and the sea, the contrast in water densities, and the aquifer thickness: If the acting head reaches some critical(minimal) value, saline water will always stay at some depth; that is, the wedge will have an infinite width. In this case the interface coincides with the Saffman-Taylor shape of a finger in a Hele-Shaw apparatus.