Two-phase flow equations for a dilute dispersion of gas bubbles in liquid

Abstract

Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected). Next, by applying the same technique, the momentum-flux tensor of the entire mixture is obtained. An equation expressing the fact that the average force on a massless bubble is zero leads to a third relation. Complemented with mass-conservation equations for liquid and gas, these equations appear to constitute a completely hyperbolic system, unlike the systems with complex characteristics found previously. The characteristic speeds are calculated and shown to be related to the propagation speeds of acoustic waves and concentration waves.

A kinetic theory for particulate systems with bimodal and anisotropic velocity fluctuations

“…Since we are expressing P 12 1 in terms of T 12 1 , the momentum equation for V 1 1 is reduced to a first-order differential equation. Therefore, no boundary conditions for this equation need to be specified at the channel walls.…”

“…Finally, a boundary condition for T 1 1 , and using the limiting expression for thermal conductivity for dilute particulate systems ͓k = ͑225 ͱ / 576͒T 1/2 ͔, we obtain a relatively simple approximate boundary condition given by…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14] In particular, the special case of flows in which the hydrodynamic interactions among bubbles can be determined using the potential flow approximation has been extensively studied. [8][9][10]13,15 The potential flow approximation is expected to be valid when the Reynolds number based on the bubble radius and characteristic velocity is large compared with unity but the Weber number, the ratio of inertial to surface tension forces, is small enough so that the bubbles are approximately spherical, and the liquid is free of surface-active impurity.…”

Behaviour of a cloud of bubbles filled with vapour and a small amount of noncondensable gas (A theoretical and numerical study)

“…In the data available the percentages of gas are unknown, yet important according to the present model! Conservation of momentum (gas bubbles) BIESHEUYEL et al [3] gives :…”

The equations of motion of a one-velocity, one-temperature bubbly fluid (revisited)

“…The equation of motion of this new variable leads to a non-dilute form of the Rayleigh-Plesset equation. Our [5,6] or cell models [7,8] (5) and (7), and the RayleighPlesset equation (1) [1][2][3][4] for the rotating motion Vg = n x r, and it may also be compared to the internal momentum appearing in the motion of the bubble with a velocity w relative to the liquid. In fact the study of the motion of a rigid particle in a perfect fluid has already shown that the equations of motion are most elegantly rewritten using an overall internal momentum or angular momentum [ 16,17].…”