Transformation of Irregular Solid Spherical Harmonics at Parallel Translation of the Coordinate System

A.A. Aganin^*, A.I. Davletshin^**

Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, 420111 Russia

E-mail: *aganin@kfti.knc.ru, **anas.davletshin@gmail.com

Received December 28, 2016

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Abstract

When studying physical phenomena in spatial regions bounded by spherical or slightly non-spherical surfaces, spherical functions and solid spherical harmonics are widely used. A problem of transformation of those functions and harmonics with translation of the coordinate system frequently arises. Such a situation occurs, in particular, when the hydrodynamic interaction of spherical or slightly non-spherical gas bubbles in an unbounded volume of incompressible fluid is described. In the two-dimensional (axisymmetric) case, when the role of spherical functions is played by the Legendre polynomials, such a transformation can be performed using a well-known compact expression. Similar known expressions in the three-dimensional case are rather complex (they, for example, include the Clebsch–Gordan coefficients), which makes their application more complicated. The present paper contains derivation of such an expression, naturally leading to a compact form of its coefficients. Those coefficients are, in fact, a generalization to the three-dimensional case of the analogous known coefficients in the two-dimensional (axisymmetric) case.

Keywords: solid spherical harmonics, parallel translation

Acknowledgments. This study was supported by the Russian Science Foundation (project no. 17-11-01135).

Figure Captions

Fig. 1. Agreed notations.

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For citation: Aganin A.A., Davletshin A.I. Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 5–12. (In Russian)

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