Kinetic Theory for Bubbly Flow I: Collisionless case

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Abstract
A kinetic theory for incompressible dilute bubbly flow is presented. The Hamiltonian formulation for a collection of bubbles is outlined. A Vlasov equation is derived for the one-particle distribution function with a self-consistent field starting with the Liouville equation for the N-particle distribution function and using the point-bubble approximation. A stability condition which depends on the variance of the bubbles momenta and the void fraction is derived. If the variance is small then the linearized initial-value problem is ill posed. If it is sufficiently large, then the initial-value problem is well posed and a phenomenon similar to Landau damping is observed. The ill-posedness is found to be the result of an unstable eigenvalue, whereas the Landau damping arises from a resonance pole. Numerical simulations of the Vlasov equation in one dimension are performed using a particle method. Some evidence of clustering is observed for initial data with small variance in momentum.

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