Multiple Phase Hele-Shaw Flows

Abstract: A one phase Hele-Shaw flow, described by a domain D(t) (t represents time) in the plane is the flow of a liquid injected at a constant rate in the separation between two narrowly separated parallel planes. This thesis deals with the formulation and proof of existence for a multiple phase Hele-Shaw flow in arbitrary dimension R^n exhibiting separation of the phases. A smooth version of the problem, depending on a small parameter epsilon, has been considered. Solutions to this smooth problem approximate the multiple-phase Hele-Shaw flow. We show that the smooth problem has a solution using a variational technique with functions u=u(t;eps) in the Sobolev space H_0^1 describing the Hele-Shaw flow with D(t)=support(u(t;eps)). As we let the parameter epsilon tend to zero we get that the solutions u(t;eps) converges weakly to a family of functions u(t) in the same Sobolev space which describe the desired Hele-Shaw flow. Furthermore the phases represented by the components of u(t) are separated in the sense that the overlap of any two distinct phases has vanishing n-dimensional Lebesgue measure.  We also touch upon a formulation of the multiple phase Hele-Shaw flow which would, beyond separation of the phases, provide freezing of the intersecting boundary of two phases. This formulation of the problem tries to incorporate memory in to the system via means of an integration over previous states.