Title:

Wavemean flow interactions : from nanometre to megametre scales

Waves, which arise when restoring forces act on small perturbations, are ubiquitous in fluids. Their counterpart, mean flows, capture the remainder of the motion and are often characterised by a slower evolution and larger scale patterns. Waves and mean flows, which are typically separated by time or spaceaveraging, interact, and this interaction is central to many fluiddynamical phenomena. Wavemean flow interactions can be classified into dissipative interactions and nondissipative interactions. The former is important for smallscale flows, the latter for largescale flows. In this thesis these two kinds of interactions are studied in the context of microfluidics and geophysical applications. Viscous wavemean flow interactions are studied in two microfluidic problems. Both are motivated by the rapidly increasing number of microfluidic devices that rely on the meanflow generated by dissipating acoustic waves  acoustic streaming  to drive smallscale flows. The first problem concerns the effect of boundary slip on steady acoustic streaming, which we argue is important because of the high frequencies employed. By applying matched asympototics, we obtain the form of the mean flow as a function of a new nondimensional parameter measuring the importance of the boundary slip. The second problem examined is the development of a theory applicable to experiments and devices in which rigid particles are manipulated or used as passive tracers in an acoustic wave field. Previous work obtained dynamical equations governing the mean motion of such particles in a largely heuristic way. To obtain a reliable mean dynamical equation for particles, we apply a systematic multiscale approach that captures a broad range of parameter space. Our results clarify the limits of validity of previous work and identify a new parameter regime where the motion of particles and of the surrounding fluid are coupled nonlinearly. Nondissipative wavemean flow interactions are studied in two geophysical fluid problems. (i) Motivated by the open question of mesoscale energy transfer in the ocean, we study the interaction between a mesoscale mean flow and nearinertial waves. By applying generalized Lagrangian mean theory, Whitham averaging and variational calculus, we obtain a Hamiltonian wavemean flow model which combines the familiar quasigeostrophic model with the Young & Ben Jelloul model of nearinertial waves. This research unveils a new mechanism of mesoscale energy dissipation: nearinertial waves extract energy from the mesoscale ow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. (ii) We study the interaction between topographic waves and an unidirectional mean flow at an inertial level, that is, at the altitude where the Dopplershifted frequency of the waves match the Coriolis parameter. This interaction can be described using linear theory, using a combination of WKB and saddlepoint methods, leading to explicit expressions for the meanflow response. These demonstrate, in particular, that this response is switched on asymptotically far downstream from the topography, in contrast to what is often assumed in parameterisation.
