## Velocity induced by a plane uniform vortex having the Schwarz function of its boundary with two simple poles.(English)Zbl 1162.30008

Summary: The velocity induced by a plane, uniform vortex is investigated through the use of an integral relation between the Schwarz function of the vortex boundary and the conjugate of the velocity. The analysis is restricted to a certain class of vortices, the boundaries of which are described through conformal maps onto the unit circle. The corresponding Schwarz functions possess two poles in the plane of the circle. The dependence of the velocity field on the vortex shape is investigated by comparing the velocity and the streamfunction with the ones of the equivalent Rankine vortex (which has the same vorticity, area, and center of vorticity). By changing the parameters of the Schwarz function (poles and corresponding residues), rather complicated vortex shapes can be easily analyzed, some of them mimicing an incipient filamentation of the vortex boundary.

### MSC:

 30C20 Conformal mappings of special domains

### Keywords:

Schwarz function; velocity
Full Text:

### References:

 [1] P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, NY, USA, 1992. · Zbl 0777.76004 [2] P. J. Davis, The Schwarz Function and Its Applications, The Carus Mathematical Monographs, no. 17, The Mathematical Association of America, Washington, DC, USA, 1974. · Zbl 0293.30001 [3] J. R. Kamm, Shape and stability of two-dimensional uniform vorticity regions, Ph.D. thesis, California Institute of Technology, Pasadena, Calif, USA, 1987. [4] G. Riccardi, “Intrinsic dynamics of the boundary of a two-dimensional uniform vortex,” Journal of Engineering Mathematics, vol. 50, no. 1, pp. 51-74, 2004. · Zbl 1073.76015 [5] D. Crowdy, “A class of exact multipolar vortices,” Physics of Fluids, vol. 11, no. 9, pp. 2556-2564, 1999. · Zbl 1149.76353 [6] D. Crowdy, “Multipolar vortices and algebraic curves,” Proceedings of the Royal Society of London. Series A, vol. 457, no. 2014, pp. 2337-2359, 2001. · Zbl 1058.76019 [7] D. Crowdy and M. Cloke, “Stability analysis of a class of two-dimensional multipolar vortex equilibria,” Physics of Fluids, vol. 14, no. 6, pp. 1862-1876, 2002. · Zbl 1185.76096 [8] D. Crowdy, “The construction of exact multipolar equilibria of the two-dimensional Euler equations,” Physics of Fluids, vol. 14, no. 1, pp. 257-267, 2002. · Zbl 1184.76114 [9] D. Crowdy, “Exact solutions for rotating vortex arrays with finite-area cores,” Journal of Fluid Mechanics, vol. 469, pp. 209-235, 2002. · Zbl 1019.76011 [10] D. Crowdy and J. Marshall, “Growing vortex patches,” Physics of Fluids, vol. 16, no. 8, pp. 3122-3130, 2004. · Zbl 1186.76117 [11] H. Aref and D. L. Vainchtein, “Point vortices exhibit asymmetric equilibria,” Nature, vol. 392, no. 6678, pp. 769-770, 1998. [12] D. Crowdy and J. Marshall, “Analytical solutions for rotating vortex arrays involving multiple vortex patches,” Journal of Fluid Mechanics, vol. 523, pp. 307-337, 2005. · Zbl 1065.76030 [13] I. Gned, D. Durante, G. Riccardi, and L. Zannetti, “The self-induced dynamics of vortex patches,” in Proceedings of IUTAM Symposium on 150 Years of Vortex Dynamics, Lyngby, Denmark, October 2008. [14] N. J. Zabusky, M. H. Hughes, and K. V. Roberts, “Contour dynamics for the Euler equations in two dimensions,” Journal of Computational Physics, vol. 30, no. 1, pp. 96-106, 1979. · Zbl 0405.76014 [15] D. G. Dritschel, “Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows,” Computer Physics Reports, vol. 10, no. 3, pp. 77-146, 1989. [16] D. I. Pullin, “Contour dynamics methods,” Annual Review of Fluid Mechanics, vol. 24, pp. 89-115, 1992. · Zbl 0743.76021 [17] M. V. Melander, J. C. McWilliams, and N. J. Zabusky, “Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation,” Journal of Fluid Mechanics, vol. 178, pp. 137-159, 1987. · Zbl 0633.76023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.