Exact solutions to a Muskat problem with line distributions of sinks and sources

TL;DR

This paper derives exact solutions for a two-phase Hele-Shaw (Muskat) problem with line distributions of sinks and sources, using Schwarz functions, focusing on algebraic curve interfaces without cusp formation.

Contribution

It introduces a novel approach with line distributions of sinks and sources for exact solutions in the Muskat problem, avoiding cusp formation.

Findings

Exact solutions for algebraic interface shapes

Use of Schwarz function approach for two-phase flows

No cusp formation in the solutions

Abstract

The evolution of a two-phase Hele-Shaw problem, a Muskat problem, under assumption of a negligible surface tension is considered. We use the Schwarz function approach and allow the sinks and sources to be line distributions with disjoint supports located in the exterior and the interior domains, a two-phase mother body. We give examples of exact solutions when the interface belongs to a certain family of algebraic curves, defined by the initial shape of the boundary, and the cusp formation does not occur.