On the existence of finite collapsing systems of plane vortices

• Autorzy: Lewkowicz M.
• Języki publikacji: EN

Abstrakty

EN

A rigorous criterion for a finite system of plane vortices to be close to the set of all collap- sing configurations is given. The criterion is a consequence of a general theorem (based on the topological degree of a map) on the existence of solutions of the nonlinear systems of equations, combined with some analytical bounds for derivatives of the equations satisfied by collapsing configurations. The criterion can be used to prove the existence of collapsing configurations of vortices for many circulations for which the existence theory of O’Neil do- es not work. A detailed application of the criterion to a sample collapsing system of seven vortices is given.

Słowa kluczowe

EN

collapsing vortex configurations

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2014
• Tom: Vol. 52 nr 4
• Strony: 1047--1059
• Opis fizyczny: Bibliogr. 8 poz., rys.

Twórcy

autor

Lewkowicz M.

• Wrocław University of Technology, Department of Numerical Flow Modeling, Wrocław, Poland

Bibliografia

• 1. Aref H., Rott N., Thomann H., 1992, Groebli’s solution of the three-vortex problem, Annual Review of Fluid Mechanics, 24, 1-21
• 2. Frommer A., Hoxha F., Lang B., 2007, Proving the existence of zeros using the topological degree and interval arithmetic, Journal of Computational and Applied Mathematics, 199, 397-402
• 3. Garduno A., Lacomba E.A., 2007, Collisions and regularization for the 3-vortex problem, Journal of Mathematical Fluid Mechanics, 9, 75-86
• 4. Groebli W., 1877, Spezielle Probleme uber die Bewegung geradliniger paralleler Wirbelfaden, Vierteljahrsschr. Naturforsch. Ges. Z¨urich, 22, 129
• 5. Helmholtz H., 1858, Uber die Integrale der hydrodynamischen Gleichungen, welche Wirbelbewegungen entsprechen, Journal f¨ur die reine und angewandte Mathematik, 55, 25-55
• 6. Lewkowicz M., Kudela H., 2012, Collapse of N vortices, [In:] XX Fluid Mechanics Conference, Gliwice, Poland, 17-20 September, Silesian University of Technology. Institute of Power Engineering and Turbomachinery
• 7. Novikov E.A., Sedov Yu.B., 1979, Vortex collapse, Soviet Physics – Journal of Experimental and Theoretical Physics, 50, 297
• 8. O’Neil K.A., 1987, Stationary configurations of point vortices, Transactions of the American Mathematical Society, 302, 2, 383-425