Steady-state fingering patterns for a periodic Muskat problem

TL;DR

This paper analyzes how steady-state fingering patterns in the Muskat problem behave as surface tension varies, revealing thresholds where fingers touch boundaries or develop infinite slopes.

Contribution

It establishes a bifurcation analysis of stationary solutions and identifies critical thresholds for boundary contact and slope blow-up related to surface tension.

Findings

Fingering patterns blow up as surface tension increases.

A threshold cell height determines whether fingers touch boundaries.

Maximal slope tends to infinity when the threshold is not met.

Abstract

We study global bifurcation branches consisting of stationary solutions of the Muskat problem. It is proved that the steady-state fingering patterns blow up as the surface tension increases: we find a threshold value for the cell height with the property that below this value the fingers will touch the boundaries of the cell when the surface tension approaches a finite value from below; otherwise, the maximal slope of the fingers tends to infinity.