Acoustic streaming in a waveguide with slowly varying height

An analysis of acoustic streaming in a two-dimensional waveguide having slowly varying height is presented. Special attention is paid to waveguides with cross sections that are small compared to the acoustic and/or wall wavelengths. It is shown that the dynamic behavior of the enclosed fluid can be parameterized by the values of three small parameters, ɛ, 1/S, and 1/R, where ɛ is the ratio of the typical duct height H₀ to the wall wavelength L₀, 1/S is the ratio of the typical oscillatory particle displacement U₀ to the typical duct height H₀ and 1/R is the ratio of the oscillatory boundary layer thickness lᵥ to the typical duct height H 0. An analytical solution describing the streaming flow in the duct is given in terms of a regular perturbation sequence in ɛ. It is shown that the oscillatory pressure must satisfy the lossy Webster horn equation to O(ɛ²) if the no slip boundary condition is to be satisfied. Outside the boundary layer it is shown that the time averaged slip velocity is the sum of two terms. The first term is proportional to the product of the incident and reflected wave amplitudes. The second term is proportional to the difference between the incident and reflected acoustic intensity of the wave. For small values of 1/S, 1/R, and ɛ the streaming solution given is shown to be valid until R/S 2 becomes of O(1).