Relaxation of a small local perturbation of the surface of a viscous fluid in the stokes approximation

Abstract

A study is made of the aperiodic regime of local relaxation of a small local perturbation of the plane surface of an incompressible fluid of infinite depth under the influence of gravity and surface tension. The description is given in the Stokes approximation [1]. It is shown in the paper that this imposes limitations on the three-dimensional spectrum of the considered perturbations. An equation is obtained which describes the damping of the individual Fourier harmonics of the perturbation to the form of the surface. It is shown that the volume of the perturbation becomes zero in times which are small by comparison with the characteristic time of damping of the perturbation. In the short wave limit, the law of evolution of the perturbation permits a simple geometric interpretation. For large times the surface acquires a self-similar partly ordered shape. This phenomenon is illustrated by means of a numerical experiment. The idea of using an inertialess approximation to calculate waves on the surface of a fluid with high viscosity was suggested by Lamb [2]. The relaxation of harmonic perturbations in this approximation was considered by Levich [3]. This approach has also been used in studying the flow of thin films of fluid [4]. The process of self-ordering of the shape of the surface of a viscous fluid in the presence of mass flow at the boundary is described in [5, 6].