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
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.
References
1. Lamb H. Hydrodynamics. Cambridge, Cambridge Univ. Press, 1916. 728 p.
2. Fermi E. Notes on Quantum Mechanics: A Course Given by Enrico Fermi at the University of Chicago. Chicago, Univ. of Chicago Press, 1961. 171 p.
3. Duboshin G.N. Handbook for Celestial Mechanics and Astrodynamics. Moscow, Nauka, 1976. 864 p. (In Russian)
4. Idelson N.I. Potential Theory and Its Application to Geophysical Problems. Moscow, GTTI, 1932. 350 p. (In Russian)
5. Aganin A.A., Davletshin A.I. Simulation of interaction of gas bubbles in a liquid with allowing for their small asphericity. Mat. Model., 2009, vol. 21, no. 9, pp. 89–98. (In Russian)
6. Aganin A.A., Guseva T.S. Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields. Fluid Dyn., 2010, vol. 45, no. 3, pp. 343–354. doi: 10.1134/S0015462810030014.
7. Aganin A.A., Davletshin A.I. Interaction of spherical bubbles with centers located on the same line. Mat. Model., 2013, vol. 25, no. 12, pp. 3–18. (In Russian)
8. Aganin A.A., Davletshin A.I., Toporkov D.Yu. Dynamics of a line of cavitation bubbles in an intense acoustic wave. Vychisl. Tekhnol., 2014, vol. 19, no. 1, pp. 3–19. (In Russian)
9. Davletshin A.I., Khalitova T.F. Equations of spatial hydrodynamic interaction of weakly nonspherical gas bubbles in liquid in an acoustic field. J. Phys.: Conf. Ser., 2016, vol. 669, artic. 012008, pp. 1–4. doi: 10.1088/1742-6596/669/1/012008.
10. Aganin A.A., Davletshin A.I. A refined model of spatial interaction of spherical gas bubbles. Izv. Ufim. Nauchn. Tsentra Ross. Akad. Nauk, 2016, no. 4, pp. 9–13. (In Russian)
11. Doinikov A.A. Mathematical model for. collective bubble dynamics in strong ultrasound fields. J. Acoust. Soc. Am., 2004, vol. 116, no. 2, pp. 821–827. doi: 10.1121/1.1768255.
12. Takahira H., Akamatsu T., Fujikawa S. Dynamic.s of a cluster of bubbles in a liquid (theoretical analysis). JSME Int. J., Ser. B., 1994, vol. 37, no. 2, pp. 297–305.
13. Hobson E.W. The Theory of Spherical and Ellipsoidal Harmonics. Cambridge, Cambridge Univ. Press, 1931. 500 p. (In Russian)
14. Steinborn E.O., Ruedenberg K. Rotation and translation of regular and irregular solid spherical harmonics. Adv. Quantum Chem., 1973, vol. 7, pp. 1–81. doi: 10.1016/S0065-3276(08)60558-4.
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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