On the motion of bubbles in a periodic box

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Abstract
The motion of spherical bubbles in a box containing an incompressible and irrotational liquid with periodic boundary conditions is studied. Equations of motion are deduced using a variational principle. When the bubbles have approximately the same velocity their configuration is Lyapunov stable, provided they are arranged in such a way so as to minimize the effective conductivity of a composite material where the bubbles are treated as insulators. The minimizing configurations become asymptotically stable with the addition of gravity and liquid viscosity. This suggests that a randomly arranged configuration of bubbles, all with approximately the same velocity, cannot be stable. Explicit equations of motion for two bubbles are deduced using Rayleigh's method for solving Laplace's equation in a periodic domain. The results are extended for more than two bubbles by considering pairwise interactions. Numerical simulations with gravity and liquid viscosity ignored show the bubbles form clusters if initially they are randomly arranged with identical velocities. The clustering is inhibited, however, if the initial velocities are sufficiently different. The variance of the bubbles’ velocities is observed to act like a temperature; a lesser amount of clustering occurs with increasing temperature. Finally, in considering the effects of gravity and liquid viscosity, the long time behaviour is found to result in the formation of bubble clusters aggregated in a plane perpendicular to the direction of gravity.

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