Scenarios of passive particle transport in the velocity field of a vortex pair in shear flow

The purpose of the work is to analyze the transport of passive particles in the velocity field of a two-vortex configuration on a plane with a possible presence of a shear flow. We model the system using two point vortices and a shear flow, where the velocity components depend linearly on one coordinate. Scenarios of particle transport and mixing are studied depending on the intensity of one vortex (in the region of [-1, 1]∖{0}) and various shear flows with fixed initial positions of the vortices and an intensity of the second equal to unity. In the investigation, we mainly use numerical methods of dynamical systems analysis. We apply 8th-order of accuracy integrators to solve the Cauchy problem for a system of ordinary differential equations. The study also involved constructing Poincare sections and fields of local Lyapunov exponents, as well as studying transformations of marker circles (fluid contours) on a plane. Results. Depending on the signs of the vortex intensities and the direction of the shear flow, the following scenarios were found: mixing of particles near the vortex structure; movement of a vortex pair along closed orbits with the transfer of particles from its vicinity and mixing near the orbits; mixing of particles in a large area on the plane; movement of a vortex pair to infinity with the transfer of particles from the vicinity of its initial position over long distances; disintegration of the pair and movement of vortices in different directions to infinity with the transfer of particles from the vicinity of their initial positions. In the presence of a shear flow, stochastic scattering of passive particles is typical, which is because of their chaotic dynamics. Conclusion. We show that depending on the signs of intensities and parameters of the shear flow, a vortex pair can be a ’carrier’ moving particles from the vicinity of its initial position over long distances, a ’mixer’ of particles in a limited area of the plane, a ’scatterer’ of particles from a certain area along its path to infinity. The results of the article can be useful in explaining the complexity of transfer processes in fluids and gas flows when vortex pairs arise in them.  

system of point vortices, particle transfer, advection, Nonlinear systems