Чисельний аналіз деформації в’язкого тіла під діею сил поверхневого натягу

Abstract

Among the boundary value problems of hydrodynamics of viscous fluid, an important from a practical point of view the class of tasks in the region with a free (previously unknown) boundary, which is determined in the process of direct solving, is distinguished. One of the possible approaches to solving such problems in such an area is a method of hydrodynamic potentials, which translates the basic complexity of research and numerical calculations into certain boundary integral equations, which relate only to the boundary of a given region and take into account the boundary conditions directly. This approach allows you to immediately identify unknown values at the boundary without calculating them in the entire area. This advantage distinguishes the method of boundary integral equations from other methods. The aim of the research was to develop a method for numerical solution of the problems of the viscous fluid movement with a previously unknown boundary. The problem of deformation of a viscous elliptical cylinder under the action of forces of surface tension is formulated and solved in the article. The boundary integral equations are used, which are considered together with the kinematic contour condition. An algorithm for the steps of the numerical solution of the system of boundary integral equations for the problem of viscous fluid with a previously unknown boundary was implemented and computational features were analyzed. On the basis of the conducted research the regularities of the deformation process of the viscous elliptical cylinder under the influence of the forces of surface tension, depending on the viscosity of the liquid and the coefficient of surface tension , were established. A convex contour with continuous curvature deforms into a circle, carrying out fading oscillations around the equilibrium position. Zones of stable synchronous oscillations of points of the contour around the equilibrium position are observed, if , and when , ; in both cases, the density of the liquid was taken equal to one. The liquid body reaches the equilibrium position without oscillation, if . Key words: nucleus, contour discretization interval, curvature, value of the Cauchy integral, surface tension forces, elliptic cylinder