On a Camassa-Holm type equation describing the dynamics of viscous fluid conduits

TL;DR

This paper derives a new nonlocal nonlinear dispersive PDE modeling the interface dynamics of viscous fluid conduits, analyzing its well-posedness, traveling waves, and potential finite-time singularities.

Contribution

It introduces the g-model, a novel Camassa-Holm type equation with nonlocal effects for viscous fluid interfaces, and studies its mathematical properties and behaviors.

Findings

Well-posedness of the g-model established

Existence of periodic traveling wave solutions demonstrated

Numerical simulations indicate possible finite-time singularity formation

Abstract

In this note we derive a new nonlocal and nonlinear dispersive equations capturing the main dynamics of a circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. This equation that we termed the $g-$model shares some common structure with the Camassa-Holm equation but has additional nonlocal effects. For this new PDE we study the well-posedness together with the existence of periodic traveling waves. Furthermore, we also show some numerical simulations suggesting the finite time singularity formation.