Consistent continuum equations and numerical benchmarks for a perturbation-based, variable-coefficient acoustofluidic solver

TL;DR

This paper develops a consistent continuum framework and a numerical solver for acoustic streaming in fluids with variable density, clarifying boundary condition assumptions and verifying accuracy through benchmarks.

Contribution

It introduces a systematic formulation of second-order mass balance equations and a high-accuracy numerical scheme for variable-coefficient acoustofluidic problems.

Findings

Boundary condition choice significantly affects flow fields.

The solver achieves second-order accuracy and verified convergence.

Variable density systems show notable differences from constant density cases.

Abstract

We present a consistent continuum framework and a variable-coefficient acoustofluidic solver to analyze acoustic streaming. A perturbation approach is used to split the compressible Navier-Stokes equations into two sub-systems: a first-order harmonic system and a time-averaged second-order mean system. Prior acoustofluidic numerical studies have typically employed simplifying assumptions regarding the second-order mass balance equation and boundary conditions. These assumptions -- frequently left unjustified -- result in an incongruent problem statement where the boundary conditions are not consistent with the governing equations. To clarify these assumptions and mitigate the associated confusion, we systematically formulate the second-order mass balance equation into two analytically equivalent but numerically distinct forms by introducing the fluid's Lagrangian and mass transport velocity. We clarify the relation between these quantities to illustrate that a zero mass transport velocity does not necessarily imply a zero Lagrangian velocity. A second-order accurate finite difference/volume scheme is used to discretize the governing equations and boundary conditions expressed in Eulerian form. We solve the first-order coupled Helmholtz system via a sparse direct solver while the second-order system is solved via a Krylov (FGMRES) solver with a projection method-based preconditioner. To ascertain the spatial accuracy and consistency of numerical implementation of the split system of equations and boundary conditions, we demonstrate verification results and convergence rates using manufactured solutions. We provide benchmark test cases to demonstrate the impact of the choice of boundary conditions on second-order flow field. Lastly, we examine a system with spatially varying density to illustrate the difference in flow fields between constant and variable density systems.