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U is the velocity of the fluid.
FROM:
Sergey A. Bochkarev, Sergey V. Lekomtsev, Valery P. Matveenko (Institute of continuous media mechanics UB RAS, Perm, Russia bochkarev@icmm.ru, lekomtsev@icmm.ru, mvp@icmm.ru),
“Natural Vibrations and stability of shells with arbitrary cross-section, containing flowing fluid”, 11th International Conference on Vibration Problems, Z. Dimitrowova, et al (editors), Lisbon, Portugal, 9-12 September 2013
ABSTRACT: The focus of the paper is on the numerical analysis of natural vibrations and stability of cylindrical shells with arbitrary cross-sections, completely or partially filled with a quiescent or flowing fluid. A three-dimensional problem is considered and solved by the finite element method. The motion of the shell is described by applying the variational principle of virtual displacements, which incorporates the linearized Bernoulli equation for evaluating the hydrodynamic pressure. A compressible inviscid fluid is considered in the framework of the potential theory. The governing equations are converted using the Bubnov–Galerkin method. To demonstrate the possibilities of three-dimensional numerical simulation two problems on the natural vibrations and stability of elliptical cylindrical shells are investigated.
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