Variational approach to compiling nonlinear equations of motion for a body-two fluids system


Аuthors

Win K. K.

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

e-mail: win.c.latt@gmail.com

Abstract

The dynamics of motion of a solid body interacting with a fluid is an important field of research in mechanics and physics. Particularly interesting are cases where several immiscible liquids are present in the cavity of a solid. The introduction briefly introduces the main aspects of this topic, covering both linear and nonlinear approximations. The article is devoted to the formulation of the basic equations of nonlinear dynamics of a solid body undergoing complex motion and having a cavity filled with a two-layer ideal heavy liquid. In the formulation of the problem, consideration of quasi-velocities was introduced instead of generalized velocities and the Euler-Lagrange equation for the motion of a rigid body was used. The article shows that using the Ostrogradsky principle, a complete set of equations of motion of a rigid body relative to quasi-velocities Voi , ωi and the motion of a two-layer liquid relative to generalized coordinates Bj and Qn was obtained. The article is devoted to the definition of differential equations for the generalized coordinate motions of a two-layer liquid in the cavity of a solid body performing a given motion in space. In the article, the formulation of a nonlinear problem about the motions of immiscible incompressible ideal liquids that completely fill a cylindrical cavity is formulated, and velocity potentials are given for each liquid. 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, variations of quasi-coordinates, variations of quasi-velocities

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