Variational methods of solving dynamic problems for fluid-containing bodies

Abstract

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid-body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Introduction. The theory of motion of bodies fully or partially filled with a fluid, which is an important division of body mechanics, is widely used to solve a large variety of applied problems. Primarily, these are problems arising in designing modern vehicles for transportation of large amounts of fluid. Such problems include the determination of the mechanical interaction of a tank car and the fluid partially filling it, strength analysis of vessels with environmentally dangerous fluids in seismic areas, stability analysis and efficient control of air/space/sea craft, etc.This paper deals only with dynamic systems consisting of a perfectly rigid body and a perfect incompressible fluid. Formulations of problems for a body interacting with an internal or external fluid are detailed in [7, 13, 15, 52, 71, 79, 93-95, etc.].Mathematical models of such mechanical systems are based on the nonlinear dynamic equations of a body and the nonlinear equations of waves on the free surface of a finite fluid. Scientists working in this area of mechanics aimed their efforts at overcoming the chief difficulty of the theory-the necessity to describe the joint motion of two essentially different objects: a body and a fluid.The motion of a body is known to be described by nonlinear ordinary differential equations, whereas the description of the wave motions of the free liquid surface involves the formulation and solution of nonlinear initial-boundary-value problems for partial differential equations. 1 2 l l J J J J t t 12 1 / , (3.32) J J J J J J 1 0. (3.34)The generalized forces will appear on the right-hand sides of Eqs. (3.32) and (3.33), where

Deformation of a Bingham viscoplastic fluid in a plane confuser

“…Noteworthy are the detailed reviews [60,66] published recently and concerned with the dynamics and interaction of rigid bodies with the surrounding fluid, which is closely related to the problem being discussed here. In these reviews, particular emphasis is placed on the trajectory approach-tracing of solid particles moving in a fluid under the action of so-called acoustic radiation pressure [60], and development of variational methods to study the interaction of a system of rigid bodies with a fluid [66].…”

“…In these reviews, particular emphasis is placed on the trajectory approach-tracing of solid particles moving in a fluid under the action of so-called acoustic radiation pressure [60], and development of variational methods to study the interaction of a system of rigid bodies with a fluid [66].…”

Variational methods of solving dynamic problems for fluid-containing bodies

“…The papers [44,54] consider different special cases of the system of equations 2)-(2.6) and the derivatives of the velocity potential and the mode of the free surface with respect to y and z can be neglected. As a result, we have…”

Variational Derivative for Buckling Loads of Beams and Frames

“…This is because in the finite-difference method, some of the points, termed as fictitious points, fall outside the domain of solution and, therefore, cannot easily be defined for all kinds of supports. Many researchers overcome this difficulty by using the finite-difference approach with the principle of virtual work or minimum potential energy, referred to as the finite-difference energy (FDE) method and applied to static, vibration, and instability problems for plates [1,15,16].The variational method is a powerful method that can be used for both formulation and approximate solution of problems [11]. In the present study, we use this method to overcome the above-mentioned difficulty.…”

“…The variational method is a powerful method that can be used for both formulation and approximate solution of problems [11]. In the present study, we use this method to overcome the above-mentioned difficulty.…”