Liquid Sloshing in Circular Toroidal and Coaxial Cylindrical Shells

Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


Free liquid vibrations in circular toroidal and coaxial cylindrical shells are considered. The liquid is supposed to be an ideal and incompressible one, and its flow inside the reservoirs is irrotational. In these assumptions, there exists a velocity potential that satisfies the Laplace equation. The mixed boundary value problem to determine this potential and liquid pressure are formulated for the Laplace equation and further reduced to solving the system of one-dimensional singular integral equations. For its numerical implementation, the boundary element method is used taking into account the ring shape of the free surface. The effective numerical procedures are proposed to accurate calculations of singular integrals containing elliptical integrals in their kernels. Numerical simulations are provided for both circular toroidal and coaxial cylindrical shells for different filling levels and various widths of gaps. The analytical solution is received for coaxial cylindrical shells, including a limit case of the infinitesimal gap. This solution can be considered as a benchmark test and allows us to validate the proposed numerical method.


Free liquid vibrations Circular toroidal and coaxial cylindrical shells Boundary element method 



Financial support for Joint Ukraine-Indian Republic R&D Projects is gratefully acknowledged. The authors would also like to thank our foreign collaborator, Professor Alexander Cheng, University of Mississippi, USA, for his constant support and interest to our research.


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Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.V.N. Karazin, Kharkiv National UniversityKharkivUkraine
  2. 2.A. Pidgorny Institute of Mechanical Engineering Problems of the National Academy of Sciences of UkraineKharkivUkraine

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