Geodesic
Interaction of spherical bubbles with centers located on the same line
Matematičeskoe modelirovanie, Tome 25 (2013) no. 12, pp. 3-18.

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A mathematical model and a technique for solving problems of close-range interaction of two and more spherical gas bubbles in a liquid, centers of which are located on the same line, are proposed. In the model, the effects of the liquid viscosity and compressibility, heat transfer between the bubbles and liquid are taken into account. The bubble dynamics is described by a system of the second order ordinary differential equations for the radii of the bubbles and their spatial positions and the first order ones for the gas temperature in each bubble. The equations for the radii of the bubbles and their spatial positions contain coefficients of expansion of the fluid velocity potential in Legendre polynomials, which are determined from a system of linear algebraic equations. The order of accuracy of the proposed model is arbitrary relative to the maximum value, over all pairs of interacting bubbles, of the ratio of sum of the radii of the bubbles in a pair to the distance between their centers. Verification of the proposed model and method of solution is conducted. Some examples of application of the proposed model to describing interaction between the two and three bubbles are presented.
Keywords: interaction of bubbles, acoustic field, fluid velocity potential. 
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A. A. Aganin; A. I. Davletshin. Interaction of spherical bubbles with centers located on the same line. Matematičeskoe modelirovanie, Tome 25 (2013) no. 12, pp. 3-18. http://geodesic.mathdoc.fr/item/MM_2013_25_12_a0/

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