Abstract Summary
The paper is devoted to the acoustic streaming (AS) in periodic porous structures constituted by rigid, or elastic scaffolds saturated by Newtonian barotropic fluid. The AS in porous media has been studied only very rarely so far in spite of many potential applications, such as tissue engineering, or thermoacoustic engines. We focus on analytical and numerical methods developed to solve efficiently the acoustic streaming problem in the homogenized porous medium. We use the asymptotic method of the periodic homogenization and the classical perturbation approach with respect to a small parameter proportional to the inverse Strouhal number. This yields the first and the second order sub-problem enabling to linearize the Navier-Stokes (N-S) equations governing the barotropic viscous fluid dynamics in pores of a periodic structure. Subsequent treatment by the asymptotic homogenization leads to a two scale problem where the macroscopic model of the porous medium describes the acoustic streaming (AS) phenomenon. For the rigid scaffolds, the first order macroscopic problem attains the form of the dynamic Darcy flow model defined in terms of the frequency-dependent permeability. This model governs the acoustic wave propagation. Its amplitude provides the streaming source vector in the form of the time averaged divergence of the Reynolds stress. The second order problem governs the AS described by the time averaged pressure satisfying a Darcy flow problem involving a mean permeability and the divergence of the streaming source vector. It appears, that the micro streaming is seen even when the macroscopic acoustic jet is zero (depending on the boundary conditions). We propose a solution method based on the spectral analysis of the characteristic microscopic dynamic Stokes flow. For the elastic scaffolds, the modelling procedure follows much the same way. The vibro-acoustic analysis of the first order problem yields the streaming source term for the second order problem for a Biot-type homogenized medium. By virtue of the homogenization and the sensitivity analysis of the effective medium parameters w.r.t the deformation, nonlinear effects can be handled which may lead to the acoustic wave modulation. This is important in cases of soft elastic materials. Our study provides a basis for the multiscale modelling of acoustic metamaterials with the AS phenomenon. The models are implemented in our in-house developed finite element based software SfePy. Numerical illustrations are presented.
