Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

Mathematical Modelling of the Elliptical Vortex Ring in a Viscous Fluid with the Vortex Filament Method

The article deals with modelling an elliptical vortex ring in a viscous fluid using the Lagrangian vortex filament method. The novelty is that earlier only inviscid flows restricted vortex filament method application. The proposed viscosity model uses an analogue of the diffusion rate method, which is widely applied to simulate plane-parallel and axisymmetric flows of viscous fluid. A transfer of the formula of a diffusion rate from two-dimensional flows to the model of spatial vortex filament is due to assumption that swirling of vortex lines (helicity of vorticity) is unavailable. Despite the laxity of the diffusion rate model for general spatial flows, its application enables taking into account the effect of viscous diffusion of vorticity, which provides expansion of vortex tubes in space. The paper formulates the vortex filament method in which the filaments are broken into the vortex segments. Such discretization enables turning from the equation of vorticity evolution in partial derivatives to a system of ordinary differential equations with respect to the parameters of the segments. Formulas to calculate a filament system-induced flow rate as well as formulas to perform approximate calculation of an analogue of the diffusion rate are given.The objective is to propose the viscosity model as an application to the vortex filament method by the example of modelling the evolution of an elliptical vortex ring in viscous fluid. The calculation results obtained by the vortex method are compared with the existing experiment and with the calculation performed by the grid method in the OpenFOAM package. A feature of the problem is that there are zones of nonzero helicity of vorticity where the proposed model of viscosity, strictly speaking, is not correct. It is shown that the results of calculations are in good agreement with each other and are in complete agreement with experiment. This allows saying that the effects of swirling vortex lines do not significantly affect the results of modelling a specific example of the spatial flow of viscous fluid by the proposed modification of the vortex filament method.

Local existence and uniqueness of skew mean curvature flow

The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.