Homogenization of the acoustic streaming in periodic rigid porous structures

Abstract

The paper presents a new model of the acoustic streaming (AS) in rigid porous media. The modelling is based on the classical perturbation approach combined with the periodic homogenization. The first one enables to linearize the Navier–Stokes equations for a barotropic fluid using the decomposition into the first and the second order subproblems governing the fluid dynamics in the rigid period scaffolds. The acoustic wave captured by the first order problem provides the Reynolds stress which appears in the second order problem as the streaming source term. Both the subproblems are treated by the homogenization resulting in the dynamic Darcy flow equations. Using spectral analysis of the characteristic microscopic dynamic Stokes flow and the associated spectral decomposition of the responses, the dynamic permeability is derived and also the driving force for the time-averaged permanent flow is evaluated. The AS can be observed at both the macroscopic and the microscopic levels. While the acoustics-driven microflows are observed for any microstructure, the macroscopic AS depends on the porous microstructure geometry, its nonsymmetry and boundary conditions. For porous particulate structures, the forces and force moments acting on the suspended particles are computed and the influence of the wave frequency and geometrical features is examined. All these effects are illustrated using 2D examples of periodic porous microstructures.