Forced fluid fluctuations in cylindrical reservoirs under vertical excitation

Abstract

The shell and shell structures containing various types of liquid fillers can be exposed to intense dynamic effects during the exploitation. In order to analyze the strength of structures in these conditions, it is necessary to take into account nonlinear phenomena in fluid motion, since the application of linear equations does not provide an adequate assessment for the determination of the pressure and amplitude of the splashing. In this paper, a study of fluid fluctuations in a rigid cylindrical reservoir partially filled with the liquid under condition of vertical agitation has been carried out. The systems of differential equations that correspond to the linear and nonlinear formulation of the problem are presented. The fluid is believed to be perfect and incompressible, and its movement induced by external influences is non-vortex. Under these conditions there is a velocity potential that satisfies the Laplace equation. The conditions of non-leakage on the wetted surfaces of the shell are chosen as the boundary conditions for solving the boundary value problem. The kinematic and static conditions are specified on a free surface. The static condition consists in the equality of pressure on the liquid surface with atmospheric pressure. The pressure is determined from the Cauchy-Lagrange integral. In this case the linearization of the Cauchy-Lagrange integral leads to the linear formulation of the problem. Quadratic components are taken into account for the nonlinear formulation. To formulate the kinematic condition an additional unknown function describing the motion of the free surface is introduced. The kinematic condition is the equality of the liquid velocity described by the velocity potential and the velocity of the free surface itself. If there is a vertical agitation, an additional acceleration will be present. Therefore for the linear formulation we obtain a system of unbounded differential equations, each of which is the equation of Mathieu. This allows us to investigate the phenomena of parametric resonance. When analyzing differential equations which occur in case of a nonlinear problem, it has been found that the solutions of such equations depend essentially on the initial conditions. The phase portraits of a dynamic system with indication of resonances are presented. A numerical analysis of the differential equation corresponding to nonlinear formulation has been carried out.