Abstract
The motion of a viscous incompressible capillary liquid with the free boundary in the absence of external forces is considered. Suppose that initially the liquid is bounded by a cylindrical free surface and that the longitudinal component of the initial velocity field depends linearly on a longitudinal coordinate, while the other components and the pressure are independent of this coordinate. Then the Navier-Stokes equations have a solution where the velocity field keeps the same structure and the free surface remains a cylindrical one. The solution gives a pithy example of the partially invariant solution for the Navier-Stokes equations describing a free boundary flow. As a corollary, the primary 3-D problem is reduced to a 2-D one.
The analogy between the problem of the motion of a “flat drop with distributed mass sources” is drawn. The question of a local solvability in time to this problem is discussed. A sufficient condition for blowing-up of the solution is formulated.
A further reduction of the problem occurs for rotationally symmetric and planar flows. In both cases, sufficient conditions for a global solvability in time are obtained and the asymptotic behaviour of solutions for a large time is analyzed. Besides, the family of exact solutions in the 1-D problem is found.
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Pukhnachov, V.V. (1999). Non-stationary Viscous Flows with a Cylindrical Free Surface. In: Escher, J., Simonett, G. (eds) Topics in Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 35. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8765-6_23
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DOI: https://doi.org/10.1007/978-3-0348-8765-6_23
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