Transformation of Irregular Solid Spherical Harmonics with Parallel Translation of the Coordinate System
Solid spherical harmonics and spherical functions are widely used for studying physical phenomena in spatial domains bounded by spherical or nearly-spherical surfaces. In this case, it is frequently needed to transform these functions with a parallel translation of the coordinate system. Specifically, this scenario arises in describing the hydrodynamic interaction of spherical or weakly-nonspherical gas bubbles in the unbounded volume of an incompressible fluid. In the two-dimensional (axisymmetric) case, when Legendre polynomials act as spherical functions, the transformation can be conducted with a well-known compact expression. In the three-dimensional case, similar well-known expressions are rather complex (for example, the Clebsch–Gordan coefficients are used in these expressions), which makes their use difficult. This paper describes a derivation of such an expression that naturally leads to a compact form of the respective coefficients. Actually, these coefficients are a generalization to the three-dimensional case of similar well-known coefficients in the two-dimensional (axisymmetric) case.
KeywordsSolid spherical harmonics parallel translation
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- 2.E. Fermi, Notes of Quantum Mechanics (Univ. Chicago Press, Chicago, 1995).Google Scholar
- 3.G. N. Duboshin, Handbook on Celestial Mechanics and Astrodynamics (Nauka, Moscow, 1976) [in Russian].Google Scholar
- 4.N. I. Idel’son, Potential Theory and its Applications to the Questions of Geophysics (GTTI, Moscow, 1932) [in Russian].Google Scholar
- 8.A. A. Aganin, A. I. Davletshin, and D. Yu. Toporkov, “Dynamics of a line of cavitation bubbles in an intense acoustic wave,” Vychisl. Tekhnol. 19 (1), 3–19 (2014).Google Scholar
- 9.A. I. Davletshin and T. F. Khalitova, “Equations of spatial hydrodynamic interaction of weakly nonspherical gas bubbles in liquid in an acoustic field,” J. Phys.: Conf. Ser. 669, 012008 (2016). doi 10.1088/1742-6596/669/1/012008Google Scholar
- 10.A. A. Aganin and A. I. Davletshin, “A refinedmodel of spatial interaction of spherical gas bubbles,” Izv. UNTs RAN, No. 4, 9–13 (2016).Google Scholar