Geodesic
A refined model of interaction of spherical gas bubbles in a~liquid
Matematičeskoe modelirovanie, Tome 21 (2009) no. 9, pp. 89-98.

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A refined mathematical model of interaction of two spherical gas bubbles in a liquid is proposed. It is of the fifth order of accuracy relative to the ratio of the sum of the radii of the bubbles to the distance between their centers, which is by an order higher than the accuracy of the existing models. This allows one to study the interaction in the regimes with the less distant bubbles. The model is a system of the second order ordinary differential equations in the radii of the bubbles and coordinates of their centers. It approximately allows for the effects of the liquid viscosity and compressibility, the gas in the bubbles is assumed homobaric. 
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A. A. Aganin; A. I. Davletshin. A refined model of interaction of spherical gas bubbles in a~liquid. Matematičeskoe modelirovanie, Tome 21 (2009) no. 9, pp. 89-98. http://geodesic.mathdoc.fr/item/MM_2009_21_9_a8/

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