Exact solutions to the unsteady two-phase Hele-Shaw problem

While many explicit solutions to the single-phase Hele-Shaw problem are known, solutions to the two-phase problem (also known as the ‘Muskat problem’) are scarce. This paper presents a new class of exact time-dependent solutions to the two-phase Hele-Shaw problem. It is demonstrated that an elliptical inclusion of one phase remains elliptical under evolution when immersed in any unsteady far-field linear flow of a second ambient phase. On the basis of this solution class, an ‘elliptical inclusion model ’ for interactions in inhomogeneous porous media is outlined. 1.