Title:

Asymptotic methods applied to problems of steadystreaming flows and acoustic radiation forces

Smallamplitude, highfrequency (ultrasound) forcing of fluid/particle systems is being used in a number of applications associated with nondestructive fluid mixing and the movement/manipulation of particles in suspension. Of most importance in this context are the secondorder, steady, effects arising from the nonlinear interaction of a leadingorder oscillatory field with itself. In this thesis we consider some of these steady effects in both incompressible and compressible fluids. We first consider the axisymmetric steady streaming generated in an incompressible, viscous fluid contained between two (radially) infinite parallel plates, each oscillating in a direction normal to its own plane. In the limit of smallamplitude, highfrequency oscillations, we show that the steadystreaming flow in the fluid bulk is driven by thin streaming sublayers at the plates, at which the normal velocity is zero and the radial velocity varies linearly with distance from the axis of rotational symmetry. Effectively, in the bulk flow, the bounding plates appear as (noslip) impermeable walls that stretch radially. This bulkflow problem is extended to allow for the analogous steady flow of two immiscible, incompressible, viscous fluids, each undergoing a radialstretching motion appropriate to highfrequency steady streaming. For a flat interface between the fluids, a selfsimilar solution reduces the NavierStokes equations to a nonlinear boundaryvalue problem, the solution of which exhibits an interesting structure in the limit of large Reynolds number. In this limit, solutions can be found using matched asymptotic expansions, but the location of the interface between the fluids can only be determined if terms that are exponentially small in the Reynolds number are included. It is shown that for fluids of almostequal densities, exponentiallysmall differences can have a leadingorder effect on the observed flow. The second part of the thesis is concerned with the (steady) acoustic radiation force on a rigid sphere submerged in a compressible, inviscid fluid, when the wavelength of the incident acoustic field is large compared to the radius of the sphere. In this limit, a matched asymptotic expansion method is used to derive an expression for the acoustic radiation force, on both fixed and free rigid spheres, due to a range of incident fields. For incident acoustic fields that are appropriate to planar and circular waveguides/channels, expressions are derived for the scattered field and the radiation force on a rigid sphere in the longwavelength limit. Fixed and free spheres located both on and off the axis of symmetry of these incident fields are considered. This is an extension to the current literature, in which numerical methods are used to examine the scattering from spheres in an offaxis position, and problems are restricted to the consideration of fixed spheres only. It is shown that there are stable and unstable positions within the waveguide where any offaxis acoustic radiation force vanishes, leaving only an alongchannel component. For free spheres, these positions are shown to be dependent on the relative particle density and it is suggested that this may allow for a mechanism to sort such small particles radially in a circular waveguide, if secondary scattering effects are neglected.
