Solution of filtration problems with a limiting gradient when the transformation is not single-sheeted

Abstract

Difficulties associated with the fact that the transform is in general not single-sheeted arise when a linearizing hodograph transformation is applied to the equations governing the nonlinear filtration of an incompressible liquid. In problems hitherto considered, it has always been possible to distinguish a symmetry element of the flow such as would allow a matually unambiguous transformation to the plane of the hodograph. The impression has thus been created that these cases exhaust all the situations in which the application of the hodograph transformation is effective. In this paper, we shall show that even in problems not allowing a single-sheeted transformation to the plane of the hodograph, the hodograph transformation may still be useful, thus, enabling the problem to be reduced to the solution of coupled boundary problems on several sheets of the hodograph plane w, θ at the same time. In this connection, we make use of the fact that the transformation of the region of flow (w>0) to the hodograph plane is quasiconformal [1–3], and topologically equivalent to a conformal transformation.