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Solving procedure for the Kelvin–Kirchhoff equations in case of non-stationary rotations of slim disc

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A Correction to this article was published on 13 March 2021

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Abstract

We have presented in this communication a new solving procedure for Kelvin–Kirchhoff equations, considering the dynamics of the falling or ascending the rigid disc in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the falling or ascending body, I1 = I2. Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of disc’s rotation has been explored in regard to the existence of an analytic way of presentation for the approximated solution (where we consider the case of laminar flow at slow regime of deceleration of the symmetric disc’s rotation). This scenario is associated with consideration of that the Stokes boundary layer phenomenon on the boundaries of the body has also been assumed at formulation of basic Kelvin-Kirchhoff equations. It allows us to use the results of Jeffery’s fundamental work, in which analytical solution for rotations of spheroid’s particle in Euler angles was obtained. The results of calculations for the components of angular velocity {Ωi} should then be used for solving momentum equation of Kelvin–Kirchhoff system.

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Acknowledgements

Sergey Ershkov is thankful to Dr. A.A.Kurkin with respect to review in scientific literature in his seminal article [6].

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Authors and Affiliations

  1. Plekhanov Russian University of Economics, Scopus number 60030998, Moscow, Russia

    Sergey V. Ershkov

  2. Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine

    Dmytro Leshchenko

  3. Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 24 Minina st., Nizhny Novgorod, Russia, 603155

    Ayrat R. Giniyatullin

Authors
  1. Sergey V. Ershkov
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  2. Dmytro Leshchenko
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  3. Ayrat R. Giniyatullin
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Contributions

In this research, Dr. SE is responsible for the general ansatz and the solving procedure, simple algebra manipulations, calculations, results of the article in Sections 1–5 and also is responsible for the search of approximate solutions. Dr. DL is responsible for theoretical investigations as well as for the deep survey in literature on the problem under consideration. Ayrat Giniyatullin is responsible for testing the initial conditions for the approximated solutions. All authors agreed with the results and conclusions of each other in Sections 1–7.

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Correspondence to Sergey V. Ershkov.

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Ershkov, S.V., Leshchenko, D. & Giniyatullin, A.R. Solving procedure for the Kelvin–Kirchhoff equations in case of non-stationary rotations of slim disc. Arch Appl Mech 91, 2921–2929 (2021). https://doi.org/10.1007/s00419-021-01890-9

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  • DOI: https://doi.org/10.1007/s00419-021-01890-9

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