Time-harmonic waves in Korteweg and nematic-Korteweg fluids

TL;DR

This paper derives and analyzes Helmholtz--Korteweg equations for acoustic waves in Korteweg and nematic-Korteweg fluids, predicting wave behavior and scattering effects with potential experimental verification.

Contribution

It introduces a nematic variant of the Helmholtz-Korteweg equation, extending dispersion analysis to include boundary effects and nematic orientation influences.

Findings

Dispersion relations match Virga's analysis for Euler-Korteweg equations.

Predictions on wave penetration depth and scattering in nematic fluids.

Proposals for experimental verification of boundary condition effects.

Abstract

We derive the Helmholtz--Korteweg equation, which models acoustic waves in Korteweg fluids. We further derive a nematic variant of the Helmholtz-Korteweg equation, which incorporates an additional orientational term in the stress tensor. Its dispersion relation coincides with that arising in Virga's analysis of the Euler-Korteweg equations, which we extend to consider imaginary wave numbers and the effect of boundary conditions. In particular, our extensions allow us to analyze the effect of nematic orientation on the penetration depth of evanescent plane waves, and on the scattering of sound waves by obstacles. Furthermore, we make new, experimentally-verifiable predictions for the effect of boundary conditions for a modification of the Mullen-L\"uthi-Stephen experiment, and for the scattering of acoustic waves in nematic-Korteweg fluids by a circular obstacle.