On a two-phase Hele-Shaw problem with a time-dependent gap and distributions of sinks and sources

TL;DR

This paper investigates a two-phase Hele-Shaw problem with a time-varying gap, analyzing interface evolution driven by sink/source distributions and providing exact solutions for specific algebraic interface shapes.

Contribution

It introduces a Schwarz function approach to derive exact solutions for a Hele-Shaw problem with a dynamic gap and complex sink/source distributions, expanding understanding of interface dynamics.

Findings

Exact solutions for algebraic interface curves without cusps.

Demonstration of interface evolution influenced by sink/source distributions.

Analytical framework for time-dependent gap Hele-Shaw problems.

Abstract

A two-phase Hele-Show problem with a time-dependent gap describes the evolution of the interface, which separates two fluids sandwiched between two plates. The fluids have different viscosities. In addition to the change in the gap width of the Hele-Shaw cell, the interface is driven by the presence of some special distributions of sinks and sources located in both the interior and exterior domains. The effect of surface tension is neglected. Using the Schwarz function approach, we give examples of exact solutions when the interface belongs to a certain family of algebraic curves and the curves do not form cusps. The family of curves are defined by the initial shape of the free boundary.