A continuum model for foam-generating processes

This study develops a continuum model for foam-generating processes. The exact equations of continuity and momentum for the liquid and the gas phase are integrated over the foam mixture. The Gauss divergence theorem and Leibnitz integration rule are applied to develop “average” gradients and transients over the mixture. Three new field variables: the liquid volume fraction &phis;_L, the bubble radius R and the bubble expansion speed $u$_GR represent the foam structure in the continuum model. The drag force arising from slip between the bubbles and the liquid phase is represented by a fourth new variable, t _G. The four new variables require four additional equations for closure. The bubble expansion speed $u$_GR is exactly identical to the material derivative of R. Enforcing this identity for the volume averaged quantities in each phase yields two equations. A third equation is obtained from requiring continuity of normal traction at the bubble interface and averaging this constraint over the foam mixture. The fourth equation is a specified constitutive relation between the drag force and the slip speed. The new system of equations of foam flow in averaged variables reduce to the original Navier-Stokes equations for the gas-only or liquid-only limit cases. The foam material properties are represented in three material constants (the liquid shear viscosity μ _L, the inter-phase drag viscosity μ_LG, and the surface tension σ_L. The model predicts the bubble-count density as a response to the motion of the mixture. A solution to the inverse problem is presented to show the new foam model is determinate and self-consistent.