Statistical description of a cloud of compressible bubbles

Abstract

We use the methods of statistical mechanics to describe the interaction of N compressible gas bubbles in an incompressible, inviscid and irrotational liquid. The governing equations for bubble positions, radii and corresponding momenta form a Hamiltonian system depending on the virtual mass matrix. An explicit expression of the virtual mass matrix is presented, which is calculated with accuracy (b/d)³, where b and d are respectively the mean bubble radius and the mean inter-bubble distance. We study two limit cases: the limit of moving rigid spheres and the limit of immobile oscillating bubbles. In each case, we construct a canonical ensemble partition function. In the limit of rigid spheres, we improve results by Yurkovetsky and Brady (phys Fluids 8(4): 881–895, 1996). In particular, we derive an analytic expression for the “attractive” potential which may be responsible for the clustering effect, and show why the accuracy (b/d)³ is not sufficient to characterize the “repulsive potential” . In the limit of immobile oscillating bubbles, we prove the existence of a long range repulsive potential.