Unstable Interface Dynamics for Gravity Stokes Flow

TL;DR

This paper analyzes the unstable behavior of fluid interfaces under gravity in Stokes flow, demonstrating infinite growth of interface features in unstable configurations and finite-time transition to instability.

Contribution

It provides analytical proofs of interface growth phenomena and numerical simulations confirming these effects in gravity-driven Stokes flow.

Findings

Unstable interface configurations exhibit infinite growth in length or curvature.

Numerical simulations confirm analytical predictions of growth phenomena.

Stable configurations can evolve into unstable regimes in finite time.

Abstract

We investigate some unstable behavior of the interface given by two incompressible fluids of different densities evolving by the regular Stokes law with gravity force. In the unstable scenario, where the denser fluid lies above the lighter fluid, we prove infinite-in-time growth of the length or the curvature of the interface. We support these analytical results with numerical simulations that confirm the predicted growth phenomena. In the stable configuration, where the denser fluid lies below the lighter fluid, we show that certain initial configurations evolve into the unstable regime in finite time.