# Your search: "author:"Llewellyn Smith, Stefan""

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## Scholarly Works (22 results)

The trajectory of a non-isolated monopole on the beta-plane is calculated as an asymptotic expansion in the ratio of the strength of the vortex to the beta-effect. The method of matched asymptotic expansions is used to solve the equations of motion in two regions of the flow: a near field where the beta-effect enters as a first-order forcing in relative vorticity, and a wave field in which the dominant balance is a linear one between the beta-effect and the rate of change of relative vorticity. The resulting trajectory is computed for Gaussian and Rankine vortices.

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

The scattering of plane acoustic waves by a vortex in a two-dimensional superfluid is examined for small Mach number M. The solution is developed from a systematic expansion of the governing equations in three separate regions: an inner vortical region which scales as the healing length, an interaction region governed by irrotational hydrodynamics and an outer wave region. The solutions in the different regions are matched together. The leading-order scattered field occurs at O(M2δ) in the wave region, where δ is the small non-dimensional amplitude of the incoming acoustic wave. The far-field behaviour of the wave-region solution shows that a different form of the expansion is required in the forward scatter direction: this corresponds to the expression previously derived for the acoustic Magnus force.

The dispersion relation is found for edge waves in a rotating stratified fluid over a constant sloping bottom. The dispersion relation is then extended to the case of arbitrary gentle bottom bathymetry. Superinertial trapped modes do not exist in the rigid-lid Boussinesq case. The effect of some of the approximations that have been made in this problem is discussed.

The point vortex has been used as a simple model for flows with circulation, and has been desingularized into vortex patches and vortex sheets. In this work, we investigate the steady states of a combination of these two formulations, the Sadovskii vortex. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. Numerical continuation is used to follow families of solutions. In the limiting cases of the vortex patch and vortex sheet cases, we confirm previous research, and in the vortex patch case show new solutions.

In the regime where both sources of circulation exist, we show the relationship between the vortex patch and vortex sheet solution families. The more complicated vortex patch solution families lead to the simpler vortex sheet solution family due to a splitting of the vortex patch families at bifurcation points in the presence of the vortex sheet. The more circular elliptical family remains attached to the family with a single pinch off, and this family extends all the way to the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, and these families do not exist in the vortex sheet regime.

In the presence of surface tension, vortex patch shapes are deformed where the background straining flow is into the vortices. This leads to oscillations on the boundary, similar to those found by Tanveer (1986) for bubbles in Hele-Shaw cells. The pinch off cusp for the patch vortex becomes desingularized in the presence of surface tension, and no longer touches. The effects of surface tension on the vortex sheet and Sadovskii vortex shapes are shown to be similar but smaller.

In an incompressible inviscid flow system, helical symmetry means invariance though combined axial translation and rotation about the same axis. In helical symmetry, the axial vorticity is materially conserved if the velocity components along the helical lines are proportional to 1/(1+epsilon^2r^2), where e is the pitch and r is the distance from the z-axis. Linear instability analysis shows that a circular helical vortex patch centered at the origin is neutrally stable. We present the evolution of a family of helically symmetric vortices using contour dynamics, a Lagrangian technique to compute the motion of vortices via contour integrals. For contours perturbed by both lower and high modes, the first mode always becomes the most unstable mode for large time. We can inspect the features induced by the lower perturbed mode. We take mode 4 and mode 9 as examples in this work. Adding a vortex sheet on the boundary of the shifted contour accelerates the twisting and rotating process. The distribution of vortex sheet forms a sharpening shock in the evolution and may lead to the discontinuity.