Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section

Abstract

We analyse the steady streaming generated in an infinite elliptical tube containing a viscous, incompressible fluid when the boundary oscillates in such a way that the area and ellipticity of the cross-section vary with time but remain independent of the longitudinal coordinate. The parameters α⁻¹ = (ν/Ωa₀²)^½ and ε = U₀/a₀Ω, where ν is the kinematic viscosity, Ω is the oscillation frequency, a₀ is the undisturbed semi-major axis and U₀ is a typical wall velocity, are taken to be small, so that the Stokes layer is thin and the interaction which leads to the steady streaming can be analysed as a small perturbation. Coupled axial and transverse velocities, both oscillatory and steady, are generated. A complication is the need to specify the tangential as well as the normal velocity component on the tube wall, which requires an assumption concerning its elastic properties. We have considered two cases: (i) constant major axis, in which all boundary points move parallel to the minor axis, and (ii) an inextensible wall. The three-dimensional steady streaming in the core of the tube is computed only in the limit of small steady-streaming Reynolds number, R_s = ε²α².