Experimental investigations of linear and nonlinear periodic traveling waves in a viscous fluid conduit

TL;DR

This study experimentally investigates the dispersion and profiles of linear and nonlinear periodic traveling waves in a viscous fluid conduit, validating the conduit equation and revealing a critical frequency threshold for wave propagation.

Contribution

It provides the first experimental measurements of wave dispersion relations and profiles in viscous fluid conduits, confirming the conduit equation's predictions and identifying a frequency threshold for wave propagation.

Findings

Quantitative agreement with the conduit equation for wave profiles and dispersion relations.

Identification of a critical frequency separating propagating and non-propagating waves.

Observation of nonlinear cnoidal waves matching theoretical nonlinear dispersion relations.

Abstract

Conduits generated by the buoyant dynamics between two miscible, Stokes fluids with high viscosity contrast exhibit rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small and large amplitude periodic traveling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic traveling wave profiles. Measurements are favorably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber-frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is hypothesized to be explained by incorporating a weak recirculating flow into the full two-Stokes flow model which we observe to be operable in the experiment. When the boundary condition corresponds to the temporal profile of a nonlinear periodic traveling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics.