Faraday waves in radial outflow

Faraday waves are a type of free surface standing waves that occur, by the common definition, when a fluid is subject to an oscillating body force in the vertical direction. The scales on which this phenomenon occurs in nature span an enormous range. At the centimetre level, they can be found on the backs of alligators during their water dance‚ÄövÑvp, or in the cup of an unlucky coffee drinker. On the decametre scale, Faraday waves can be involved in the stability of ships and tankers; and on the thousands and millions of kilometers scale, they can affect the planets and stars. It is this last topic that is the motivation of this thesis. In this work, we extend the definition of Faraday waves to cover waves that are excited by oscillating radial outflow in the presence of a gravitational sink. Waves covered by this definition, unlike their classical counterparts, have seen little study. Before examining this extended problem, we begin with a review of the original formulation, as well as some new results attained by adding damping terms to boundary conditions of the ideal fluid model. We study the linearized theory and stability of this model then give some nonlinear results found with numerical methods extended in this thesis. We use these nonlinear results to find the maximum amplitude of Faraday waves in this system. We then go on to give our first radial outflow model consisting of a fluid line-source surrounded by a cylindrical free surface in an ideal fluid. We again study the linear theory of this model, which is complicated by the new geometry, and produce plots of the stability of the system. We produce nonlinear results with numerical methods developed in this thesis, as well as estimates of the maximum amplitude of these waves. Before continuing on to study Faraday waves in more complex fluid models, we take an aside to study the topic of Bessel function cross products and phase functions. In this aside, the contents of which are used in a later section, we develop new asymptotic expansions and inequalities for the Bessel phase functions of both theoretical and practical interest. We use these to produce new efficient and robust algorithms to calculate the roots of linear and cross-product combinations of Bessel functions. With these tools in-hand, we return to the study of Faraday waves excited by a line-source, but this time in a Boussinesq fluid with non-zero viscosity. We use our new Bessel function cross-product root finder to assist in calculating the eigenfunctions of an annulus, which we use as our domain for a numerical method to solve the Boussinesq model. We conclude with an exploration of the parameter space of this Boussinesq model, guided by the ideal model, and present some detailed plots of numerical solutions.