Collapse of Point Vortex Dipoles in a Bounded Fluid Layer and the Hydrodynamic Mechanism of Mutual Attraction of Like-Charged Micro-Particles in the Colloid or Dusty Plasma Systems

Abstract

The new exact weak solution of the equations for the ideal incompressible fluid dynamics in the finite layer inside two plates with solid boundaries is obtained. The solution meets a devised non-linear finite dimension Hamiltonian dynamic system for the coordinates and the Lamb impulses of N point vortex dipoles (PVD), i.e. extremely small solid spherical particles, moving with respect to the fluid. For N=2 the necessary condition for the collapse (or the converging in one point during the finite time) of two PVD is stated. On the base of the proposed theory, the new hydrodynamic mechanism of converging for two small spherical particles (of the same radius) is introduced and used for interpreting the observed paradoxical effects of attraction for micro particles with the same sign of electrical charge in the colloid and dusty plasma systems. The correspondence of the condition for collapse of two PVD with the experimentally observed data where convergence of two like-charged micro-spheres moving in the fluid layer is observed only when definite necessary restrictions on the fluid layer depth are imposed from above and below (J. C. Crocker, D. G. Grier, 1996) is stated. We also state the possibility of correspondence between the obtained condition of collapse for two PVD in the limit of unbounded fluid and known numerical results on the threshold of the impact parameter for the realization of resonance bond state (and corresponding stochastic dynamic regime) for two finite size vortex dipoles during their scattering in unbounded fluid (S. V. Manakov, L. N. Shchur, 1983; L. Tophoj, H. Aref, 2008). Here we show that in the latter process the size of dipoles is not so important as it was thought before.