In this paper we prove two results regarding the large-time behavior of vortex dynamics in the full plane. In the first result we show that the total integral of vorticity is confined in a region of diameter growing at most like the square-root of time. In the second result we show that if a dynamic rescaling of the absolute value of vorticity with spatial scale growing linearly with time conve...

We study numerically a recently introduced formulation of incompressible Newtonian fluid equations in vorticity – helical density and velocity – Bernoulli pressure variables. Unlike most numerical methods based on vorticity equations, the current approach provides discrete solutions with mass conservation, divergencefree vorticity, and accurate kinetic energy balance in a simple and natural way...

We derive boundary conditions for the vorticity equation with solid wall boundaries. The formulation uses a Dirichlet condition for the normal component of vorticity, and Neumann type conditions for the tangential components. In a Galerkin (integral) formulation the tangential condition is natural, i.e. it is enforced by a right-hand side functional and does not impose a boundary constraint on ...

In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data. We formulate our results for flows outside a finite number of smooth obstacles. Our po...

In the following, we prove new regularity results for two-dimensional steady periodic capillary water waves with vorticity, in the absence of stagnation points. Firstly, we prove that if the vorticity function has a Hölder-continuous first derivative, then the free surface is a smooth curve and the streamlines beneath the surface will be real analytic. Furthermore, once we assume that the vor...

Wavetrain modulations upon arbitrary vertically sheared currents were thoroughly studied by Benny and his group. If the Benny asymptotic expansion becomes invalid for the range of small values of vorticity the present work is focusing upon, then it has to be shown and explained what is wrong with the Benny expansion. The same question applies to Jonhson (1976) results. The dependence of the cub...

We study in detail, solving the Bogoliubov-de Gennes equations, the magnetic field, supercurrent and order parameter profiles originated by a solenoid or magnetic whisker inserted in a type-II superconductor. We consider solutions of different vorticities, n, in the various cases. The results confirm the connection between the vorticity, the internal currents and the boundstates in a self-consi...

A control-volume analysis of a hydraulic jump is used to obtain the mean vorticity downstream of the jump as a function of the Froude number. To do this it is necessary to include the conservation of angular momentum. The mean vorticity increases from zero as the cube of Froude number minus one, and, in dimensionless form, approaches a constant at large Froude number. Digital particle imaging v...

This paper demonstrates that analytical solutions to the steady two-dimensional Euler equations possessing all the qualitative properties of multipolar vortices observed experimentally and numerically can be constructed using the theory of algebraic curves and quadrature domains. The solutions consist of a finite set of line vortices superposed on finite-area patches of uniform vorticity. By wa...