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Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form. (English) Zbl 0674.76013
Summary: The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value problem can be found in closed form using Lagrangian coordinates. These solutions exhibit singular behaviour in finite time, because of the infinite domain and unbounded initial vorticity. Thus, the blow-up found by J. T. Stuart [Nonlinear partial differential equations: Singularities and their solution. Symp. Honor C. C. Lin, World Scientific, D. J. Benney, F. H. Schu and C. Yuan (eds.) (1987)] in three dimensions using the stagnation-point form, also occurs in two. The singularity vanishes under a discrete, finite-dimensional ‘point vortex’ approximation, but is recovered as the number of vortices tends to infinity. We find that a small positive viscosity does not arrest the breakdown, but does strongly alter its form. Similar results are summarized for certain Boussinesq stratified flows.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76M99 Basic methods in fluid mechanics
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