Acoustic wave propagation in permeable media with rotating inclusions

This work investigates long-wavelength acoustic wave propagation in active permeable media consisting of a visco-thermal fluid with periodically distributed rotating solid inclusions. The macroscopic description of wave propagation in the studied media is established using two-scale asymptotic homogenisation. The steady background fluid flow generated by the rotating inclusions is first upscaled. The resulting velocity field is shown to influence the dynamic convective forces appearing in the oscillatory transport problems, which are subsequently upscaled in the harmonic regime. For small Mach numbers, the solutions of the linearised local oscillatory fluid flow and heat transfer problems, arising from the homogenisation process, yield the dynamic conductivity and effective compressibility of the equivalent homogenised fluid. Theoretical analyses and numerical results show that the dynamic conductivity is strongly affected by the rotational motion of the inclusions. Furthermore, when oscillatory convective forces balance viscous or dynamic inertial forces in the fluid flow, phenomena such as an extremely low or negative real part of the dynamic conductivity, pseudo-resonant behaviour and an acoustic analogue of the Hall effect are identified. In terms of wave propagation characteristics, the results reveal atypical phenomena, including sound wave amplification, regressive sound waves, supersonic wave propagation and weak non-reciprocal wave propagation. These depend strongly on the microstructure of the medium and constitute a macroscopic manifestation of local oscillatory convective effects. This work offers new perspectives for the control of long acoustic waves using active permeable media with internal momentum sources.