Asymptotic models of acoustic resonators and Leidenfrost flows

Abstract

In this thesis, we develop and analyse mathematical models of acoustic resonators and Leidenfrost flows through the use of scaling arguments and asymptotic methods.

Part I is devoted to the study of acoustic resonators, particularly narrow slits and Helmholtz resonators. A novel feature of our modelling is that dissipative effects are included starting from the fundamental equations of thermoviscous acoustics. In the case of Helmholtz resonators, we begin by analysing the neck region, whose geometry is that of a cylindrical orifice in a rigid plate. We use the method of matched asymptotic expansions to derive analytical formulae for the acoustic impedance of the orifice. Building on this, we present an asymptotic model of a Helmholtz resonator embedded in a wall, as well as `metasurfaces’ formed of arrays of such resonators. Following a similar approach, we also investigate the problem of extraordinary wave transmission through narrow slits in an infinite plate, focusing on the effects of dissipative boundary layers.

Part II is devoted to the study of Leidenfrost flows. We are mainly motivated by recent experiments showing that Leidenfrost drops levitated above a solid substrate can exhibit symmetry-breaking spontaneous dynamics. Focusing on drops much smaller than the capillary length, we begin by developing a simplified model of a two-dimensional drop. Our model couples the equations of motion of the drop, which flows like a rigid wheel, and the lubricating flow in the vapour film. In addition to predicting that a stationary drop is unstable above a critical radius, the model also rationalises several experimental observations. We then extend our model to three dimensions and compare linear stability predictions with existing experimental data. Last, we asymptotically investigate the morphology of the vapour film beneath a stationary spherical particle levitating above a liquid bath, describing the evolution of that morphology with particle size.