On the streaming in a microfluidic Kundt's tube

Abstract

We derive an analytical solution for the acoustic streaming inside a rigid tube resulting from a pseudo-standing wave field, generated by two counterpropagating travelling waves. We solve the second-order axisymmetric problem that follows from the perturbation expansion of the governing equations. In the process, we impose no restriction on the diameter of the tube with respect to the thickness of the viscous boundary layer and acoustic wavelength. The derived solution is then used to study the evolution of streaming patterns inside the tube when geometrical and material parameters are varied. We show how the Schlichting streaming torus at the wall bounds the Rayleigh streaming near the axis of the tube. Decreasing the ratio (Xi) of the tube radius to the viscous boundary layer thickness gradually expands the Schlichting streaming, suppressing the Rayleigh streaming. Considering the average mass transport velocity, the Rayleigh streaming vanishes at the critical ratio Xi(M)(S) = 6.2 . The critical ratio is independent of fluid properties in the limit of large acoustic wavelength relative to the radius of the tube (Lambda). When decreases towards unity, large-scale Eckart-like streaming develops near the axis, superseding the Rayleigh streaming, while the Schlichting streaming remains at the wall. In addition, we demonstrate the relevance of the compressibility of the streaming flow and of the full inclusion of the spatial variation of the Reynolds stress that acts as the streaming source. The study is especially relevant for microfluidic systems, wherein the viscous boundary layer can reach significant thicknesses.