Одна із задач руху рідини в областях з вільною границею

Abstract

UDC 517.53

ONE OF THE GOALS OF FLUID MOTION IN FIELDS

FREE BOUNDARY

Alexey Zinkevych, Alexander Neshchadym, Volodymyr Safonov

e-mail: oleksandr_neshchadym@mail.ru

        In theory, viscous fluid secreted practically important class of problems unknown in advance (free boundary), which is determined in the same solution. One of the possible methods to solving this class of problems is hydrodynamics hydrodynamic potentials method that switches the main difficulty studies and numerical calculations for some boundary integral equations, which relate only to the border area and take into account boundary conditions directly.     

        Consider uncompressed viscous fluid (liquid body) that fills in time  region , limited loop  in coordinates , , .

        Each parameter value  which corresponds to an elementary (infinitesimal) fluid particle, located in time  the curve  the plane.

        Speed  of fluid at  consider zero:

        Let’s say, beginning with the time  the border liquids  the normal force exerted pressure                                          

                                      .                                        (1)

        Massive power in  and tangent tension on the circuit  equals zero.

        At the boundary conditions (1)  – the surface tension;  – curvature curve ;  – radius vector fluid path points;  – external normal to .

        Under the action of surface tension of the liquid particles set in motion, creating a lively path that is described by equations

        It is necessary to define the border  Liquid body and speed  of the liquid body and hydrodynamic pressure, satisfying to  system Navier-Stokes

                                                           (2)

        Here  and  – in accordance with the viscosity and density of the liquid.

        Functions  linked to the solution of system (2) kinematic ratio

                                      ,                                   (3)

where  – the velocity vector of fluid circuit ;  – velocity vector contour points ;  – free path and there are forces of tension that keep the path one and the same fluid particles.     

        The paper  established a general integral representation of solutions of equations (2) in the case of moving boundaries, based on hydrodynamic formulas Green. Integral representation is a combination of double and potentials of a single layer.

        Go in the integral representation, ,. Using the hydrodynamic properties of potentials  (including (3)), we reduce the problem to solving systems of nonlinear integral-differential equations.

List of references.

1. Белоносов, С.М. Краевые задачи для уравнений Навье-Стокса / С.М. Белоносов, К.А. Черноус – М. Наука, 1985. – 312 с.