KINETIC-THEORY FOR BUBBLY FLOW .1. COLLISIONLESS CASE

A kinetic theory for incompressible dilute bubbly flow is presented. T he 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 approx imation. A stability condition which depends on the variance of the bu bbles momenta and the void fraction is derived. If the variance is sma ll then the linearized initial-value problem is ill posed. If it is su fficiently large, then the initial-value problem is well posed and a p henomenon similar to Landau damping is observed. The ill-posedness is found to be the result of an unstable eigenvalue, whereas the Landau d amping arises from a resonance pole. Numerical simulations of the Vlas ov equation in one dimension are performed using a particle method. So me evidence of clustering is observed for initial data with small vari ance in momentum.