Analysis of diffusion-induced bubble growth in polymeric liquids

The growth and collapse of gas bubble in liquids is encountered in numerous chemical and material processing technologies. Diffusion-induced bubble growth and collapse in viscous liquids, which is prevalent in polymer foam processing, is complicated by momentum transfer in the liquid surrounding the bubble. In the present study, a rigorously-formulated transport model for diffusion-induced bubble growth and collapse in viscous liquids is formulated focusing on the coupling of mass and momentum transfer processes. Numerical solutions to this non-linear moving-boundary problem are obtained over a wider range of conditions which are representative of conditions encountered in a typical foaming process.; Despite extensive research previously conducted on the subject of bubble growth, several important questions remain unanswered concerning the development of appropriate transport models. Specifically, the use of several simplifying assumptions has not been justified, and the influence of fluid rheology has not been examined. Results from the rigorous model formulated in the present study are used to address these unresolved questions.; Thin boundary layer approximations, which greatly simplify bubble growth modeling, have been introduced in all previous studies using different mathematical approaches. In addition, a more subtle approximation, referred to as the infinitely dilute solution approximation, has also been used extensively in previous studies on bubble growth. The validity of both the thin boundary layer (TBL) and infinitely dilute solute (IDS) approximations are evaluated in the present study by comparing predictions for bubble growth rate from rigorous model with those from simplified models derived using these approximations. Results from this analysis show that use of the TBL approximation is valid only at relatively high growth rates. However, the IDS approximation is shown using a similar analysis to be valid only at relatively low growth rates. Hence, the use of both the TBL and IDS approximations which is required in one commonly used mathematical approach, severely limits the range of applicability for model based on these approximations.; In the diffusion-controlled limit where momentum transfer can be ignored, an analysis was carried out to examine the influence of concentration dependent diffusivity on bubble growth and collapse. This analysis showed that growth rate was decreased and collapse rate was increased when a diffusivity with a concentration dependence typical of polymer-solvent systems was used in the transport model. Numerical results over a wide range of conditions were correlated and for the case of bubble growth were analyzed using the TBL approximation.; The influence of fluid rheology on bubble growth was examined using several widely studied rheological constitutive equations. The effect of fluid elasticity was examined using an upper-converted Maxwell model. It was found that fluid elasticity accelerated bubble growth up the diffusion-controlled limit. The effect of non-linear (shear thinning) rheological behavior on bubble growth was examined using the Phan-Thien Tanner model. Shear thinning was shown to have only minor effects on bubble growth rate at early times. This can be explained by the fact that even at high growth rates, the diffusion controlled limit is rapidly approached such that the Newtonian limit is also rapidly approached.