Viscous Fluid Flow in a Layer with a Free Boundary
E. N. Zhuravleva1,2
1Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: Navier-Stokes equations, problems with a free boundary, self-similar solution, solution blow up
Abstract
A partially invariant solution of a three-dimensional problem with a free boundary for the Navier-Stokes equations is studied. The flow domain under consideration is a horizontal layer bounded by a solid plane from below and by a flat free surface from above. The vertical velocity and pressure are independent of the x and y coordinates. Three flow modes can be formed for different initial velocities of the flow: stabilization to the quiescent state with time, solution blow up within a finite time, and self-similar regime in which the layer thickness unboundedly increases with time.
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