Variational formulation of nonlinear boundary value problems in the dynamics of two fluids performing a given motion in space

DOI: 10.34759/trd-2023-130-11

Аuthors

Win K. K.^*, Temnov A. N.^**

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: win.c.latt@gmail.com
**e-mail: antt45@mail.ru

Abstract

This article contains the derivation of the equations of motion of a solid case with a two-layer fluid, considered as one mechanical system. The dynamics is based on the principle of least action in the form of Hamilton-Ostrogradsky. The variational formulation of the problem of dynamics has certain advantages, for example, from the point of view of substantiating the necessity and sufficiency of the derived equations and boundary conditions, and considering the body and fluid as one system allows one to achieve a certain multiplicity. The mechanical meaning of the equations is interpreted, which are presented in several different forms, in particular, in the form of the Lagrange equations. The question of the integrals of the equations of motion and the conditions under which they take place is considered.

The article is devoted to the formulation of the variational principle for a multilayer ideal heavy fluid located in a cylindrical cavity of a solid body that performs specified angular oscillations around a fixed axis. A similar problem is due to the fact that the application of the variational principle in continuum mechanics is based on the Hamilton-Ostrogradsky principle, the mathematical form of which is written in Euler variables for hydrodynamic problems acquires significant mathematical differences. The article shows that with the help of the variational principle, written in a form different from the traditional one, it is possible to obtain a complete set of equations of nonlinear motions of liquids, including nonlinear kinematic and dynamic conditions on the interfaces of liquids filling the cavity of a solid body that performs a given movement.

Keywords:

variational principle, immiscible liquids, perturbed interface, nonlinear boundary value problem, comparison motions

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