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Solutions of the Laplace Equation Satisfying the Condition of Impermeability on a Spherical Segment

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We construct a set of harmonic functions satisfying the condition of impermeability on a spherical segment. These functions can be used as a functional basis for the construction of approximate solutions of the boundary-value problems of liquid sloshing. The indicated harmonic functions are obtained as a result of the Kelvin inversion of the auxiliary functions satisfying the corresponding boundary condition on an interval of the horizontal line. The constructed system of functions is applied to the determination of natural sloshing frequencies in a spherical tank.

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References

  1. M. Ya. Barnyak and I. A. Lukovskii, “On the solution of problems of the dynamics of ideal incompressible liquid in a spherical cavity and in a horizontal circular channel,” Mat. Fiz. Nelin. Mekh., Issue 4 (38), 66–71 (1985).

  2. M. Ya. Barnyak, D. S. Zhavrots’kyi, and I. O. Lukovs’kyi, “Approximate method for the construction of solutions of boundary-value problems of the hydrodynamics of ideal liquids in spherical cavities,” in: Problems of the Dynamics and Stability of Multidimensional Systems [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2005), pp. 64–73.

  3. M. Ya. Barnyak, “Construction of the solutions of boundary-value problems for the Laplace equation in domains of revolution with ribbed boundaries,” Ukr. Mat. Zh., 61, No. 5, 579–595 (2009); English translation: Ukr. Math. J., 61, No. 5, 695–715 (2009).

  4. R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. 2, Partial Differential Equations, Wiley-Interscience, New York (1962).

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivs’ka Str., 3, 01601, Kiev, Ukraine

    M. Ya. Barnyak

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  1. M. Ya. Barnyak
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Correspondence to M. Ya. Barnyak.

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Translated from Neliniini Kolyvannya, Vol. 18, No. 3, pp. 313–322, July–September, 2015.

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Barnyak, M.Y. Solutions of the Laplace Equation Satisfying the Condition of Impermeability on a Spherical Segment. J Math Sci 220, 254–264 (2017). https://doi.org/10.1007/s10958-016-3182-6

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  • DOI: https://doi.org/10.1007/s10958-016-3182-6

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