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Axisymmetric Oscillations of MembraneSeparated Bilayer Fluid in a Closed Vessel
# 12, December 2016
DOI: 10.7463/1216.0852775
authors: D.A. Goncharov^{1,*}, A.A. Pozhalostin^{1}
 ^{1} Bauman Moscow State Technical University, Moscow, Russia 
The paper deals with the problem of small oscillations of bilayer liquid separated by a membrane in a closed vessel. This problem arises when studying the behavior of the liquid component of the fuel in the spacecraft tanks in their interaction with the elements of systems providing the startup of the propulsion system. The problem of small oscillations is considered in the linear formulation. We consider symmetrical motion of liquid. The membrane is assumed to be thin, elastic, and noninertial. The liquid completely fills the cylindrical vessel. We consider motion of an ideal and viscous incompressible liquid. On the vessel walls the flow tangency condition is fulfilled. The liquid container volume satisfies the condition of continuity, which in the area under consideration takes a form of the Laplace equation. The differential equation of membrane motion is written with the righthand side, which includes the hydrodynamic liquidinduced pressure. Restricted deflection of the membrane center and zero movements of the membrane contour stipulate boundary conditions for the equation of membrane motion. Integrating the Laplace equation we obtain the expression for the liquid velocity potential in the areas above and below membrane. Writing the equation of the membrane motion for deflection rate, representing the function of membrane deflection rate in accordance with the Fourier method as a product of functions of coordinates and time, we can integrate the expressions for the function of membrane deflection. Fulfilling the boundary conditions for the function of membrane deflection together with the boundary conditions for the Laplace equation, we can obtain an analytical expression for the function of membrane deflection and a frequency equation for the given boundary value problem. It can be seen that for a range of tensions typical for used materials, which are considered, for example, in V.M. Polyaev’s monograph, the frequency values of natural oscillations of the membrane will be much higher than the frequency of natural oscillations of the liquid that fills the cylinder of a given radius and height determined from the known relationship. References Ibrahim R.A. Liquid sloshing dynamics: theory and applications. Camb., N.Y., Cambridge Univ. Press, 2005. 948 p.
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