| Abstract: | A study is made of the process of relaxation to the equilibrium configuration of an isolated volume of a viscous incompressible Newtonian fluid under the influence of capillary forces. The fluid has the form of an infinite cylinder of arbitrary shape with a smooth compact and, in general, multiply connected boundary. In the course of relaxation, internal cavities collapse, and the cylinder acquires asymptotically a circular configuration. The quasisteady Stokes approximation 1] is used to describe the flow. First proposed by Frenkel' 2], this approximation has been used in the calculation of a dynamic boundary angle 3], the collapse of a circular cylinder 4], and the collapse of a hollow cylinder 5]. The analogy between the hydrodynamic equations in the Stokes approximation and the equations of elasticity theory made it possible 6] to describe the relaxation of a simply connected cylinder by a method close to the one employed by Muskheleshvili 7]. In the present paper, the approach of the author 8] based on work of Grinberg 9] and Vekua 10] is developed. It is shown that the true pressure distribution gives a minimum of the integral of the square of the pressure over the region for fixed integral of the pressure over the boundary. An explicit expression for the pressure is obtained in the form of the projection of a generalized function with support on the boundary onto the subspace of harmonic functions. The velocity field on the boundary of the region is calculated. An upper bound is found for the law of decrease of the perimeter of the region and for the time during which the number of connected components of the boundary remains unchanged.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–122, January–February, 1992. |
