Dispersive Hydrodynamics in Viscous Fluid Conduits

TL;DR

This paper derives a precise mathematical model for the nonlinear dispersive wave dynamics of a viscous fluid conduit, advancing understanding of interfacial phenomena in two-fluid systems at small Reynolds numbers.

Contribution

A rigorous derivation of the interfacial equation for viscous fluid conduits using multiple scales perturbation analysis, clarifying the balance between nonlinearity and dispersion.

Findings

Derived a scalar nonlinear PDE for conduit cross-sectional area dynamics.

Characterized the regime of model validity and confirmed agreement with experiments.

Provided a systematic theoretical framework for viscous fluid conduit behavior.

Abstract

The evolution of the interface separating a conduit of light, viscous fluid rising buoyantly through a heavy, more viscous, exterior fluid at small Reynolds numbers is governed by the interplay between nonlinearity and dispersion. Previous authors have proposed an approximate model equation based on physical arguments, but a precise theoretical treatment for this two fluid system with a free boundary is lacking. Here, a derivation of the interfacial equation via a multiple scales, perturbation technique is presented. Perturbations about a state of vertically uniform, laminar conduit flow are considered in the context of the Navier-Stokes equations with appropriate boundary conditions. The ratio of interior to exterior viscosities is the small parameter used in the asymptotic analysis, which leads systematically to a maximal balance between buoyancy driven, nonlinear self-steepening and viscous, interfacial stress induced, nonlinear dispersion. This results in a scalar, nonlinear partial differential equation describing large amplitude dynamics of the cross-sectional area of the intrusive fluid conduit, in agreement with previous derivations. The leading order behavior of the two fluid system is completely characterized in terms of the interfacial dynamics. The regime of model validity is characterized and shown to agree with previous experimental studies. Viscous fluid conduits provide a robust setting for the study of nonlinear, dispersive wave phenomena.