Dispersive Hydrodynamics in Viscous Fluid Conduits

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.

10 pages, 1 figure

References in corpus (5)

• Dispersive, superfluid-like shock waves in nonlinear optics
• On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
• Observation of a gradient catastrophe generating solitons
• A Multiscale Model of Partial Melts 1: Effective Equations
• Dispersive Shock Waves in Viscously Deformable Media

Cited by in corpus (10)

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• Dispersive Riemann problem for the Benjamin-Bona-Mahony equation
• Interactions of large amplitude solitary waves in viscous fluid conduits
• Solitary wave fission of a large disturbance in a viscous fluid conduit
• Experimental investigations of linear and nonlinear periodic traveling waves in a viscous fluid conduit
• Controlling Dispersive Hydrodynamic Wavebreaking in a Viscous Fluid Conduit