| Abstract: | The problem under consideration is the unsteady motion of an ideal fluid with constant density in an unbounded volume when the velocity divergence is nonzero and is specified by the sink density a which depends on the coordinates r and the time t. It is well known that the introduction of such idealized hydrodynamic objects as a point vortex, a source, or a sink and the related studies of fluid flows are useful in solving a number of specific hydrodynamic problems [1, 2]. There have been many studies of point vortices, and some of the earliest are reviewed in [3], whereas the motion of free point sinks or sources has not been studied. The reason for this situation is that it is hard to find the appropriate hydrodynamic counterparts. The aim of the present paper is to study the basic laws governing the motion of a system of sinks and sources, both point and distributed, and then apply the results obtained to a simulation of thermal convection in a plane horizontal fluid layer consisting, for example, of periodic convective cells. Special attention is given to the asymptotic behavior of  as t  . Conservation laws for a system of N point sinks are derived and discussed. The qualitative behavior of the system for large t is investigated. Under the assumption of a frozen sink density in the velocity field of the fluid, an evolution equation for is obtained for an arbitrary initial distribution of the velocity divergence. In the case of a finite integrated intensity of the sink density in an unbounded volume, an exact solution of the evolution equation is given for a cylindrically symmetric initial distribution. The asymptotic behavior of this solution as t is studied in three qualitatively different cases. Finally, a steady-state solution of the evolution equation is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 21–27, July–August, 1976.The author thanks A. A. Zaitsev for his interest in the work, valuable advice, and discussion of the results. |
