Институт теоретической физики им. Л.Д. Ландау РАН

L.D. Landau Institute for Theoretical Physics RAS

L.D. Landau Institute for Theoretical Physics RAS

Breaking of slipping flows

Date/Time: 12:45 01-Jul-2021

Abstract:

The mixed Lagrangian-Eulerian description is applied to the inviscid Prandtl equation. This description for two-dimensional flows coincides with the application of the so-called Crocco transformation. For the zeroth value of the pressure gradient, the velocity component u parallel to the wall for the Prandtl equation plays the role of Lagrangian coordinate and the second - Eulerian - variable coincides with the Cartesian coordinate x along the wall. In new variables x, u, t the continuity equation yields linear relation between stream function and the coordinate y = y(x, u, t) which defines the level line u = const. This allows constructing the exact solution in the implicit form by introducing the generating function. It is shown that in the case of the slipping boundary condition the singularity formation of the gradient catastrophe type takes place at the wall due to the compressible character of the flow there. The same character for the singularity formation remains for arbitrary dependence of pressure on x. The velocity gradient near the singular point behaves like 1/(t0-t) where t0 is the collapse time. Such singularity formation occurs also for the three-dimensional Prandtl equation in the case of the slipping boundary condition at the wall. The breaking, in this case, generates the singular vorticity perpendicular to the wall.

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