Publication Date:
2018-03-06
Description:
The Kelvin–Helmholtz model for the evolution of an infinitesimally thin vortex sheet in an inviscid fluid is mathematically ill-posed for general classes of initial conditions. However, if the initial data, say imposed at t = 0, are in a certain class of analytic functions then the problem is well-posed for a finite time until a singularity forms, say at t = t s , on the vortex-sheet interface, e.g. as illustrated by Moore ( 1979 , The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. Lond. A , 365 , 105–119). However, if the problem is analytically continued into the complex plane, then the singularity, or singularities, exist for t 〈 t s away from the physical real axis. More specifically, Cowley et al. ( 1999 , On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech. , 378 , 233–267) found that for a class of analytic initial conditions, singularities can form in the complex plane at t = 0+. They posed asymptotic expansions in the neighbourhood of these singularities for 0 〈 t ≪ 1 and found numerical solutions to the governing similarity differential equations. In this paper we obtain new exact solutions to these equations, show that the singularities always correspond to local ${\textstyle \frac {3}{2}}$-power singularities and determine both the number and precise locations of all branch points. Further, our analytical approach can be extended to a more general class of initial conditions. These new exact solutions can assist in resolving the small-time behaviour for the numerical solution of the Birkhoff–Rott equations.
Print ISSN:
0272-4960
Electronic ISSN:
1464-3634
Topics:
Mathematics
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