BUBBLE GROWTH FROM FIRST PRINCIPLES

We consider the simplest relevant problem in the foaming of molten plastics: the growth of a single bubble in a highly viscous Newtonian fluid without interference from other bubbles. This problem has defied accurate solution from first principles. Classical approaches from first principles have neglected the temperature rise in the surrounding fluid, and we find that this oversimplification greatly accelerates growth prediction. We use transport phenomena to analyze the growth of a solitary bubble, expanding under its own pressure. We consider a bubble of ideal gas growing without the accelerating contribution from mass transfer into the bubble. We find that bubble growth depends upon nucleus radius and nucleus pressure. We begin with a detailed examination of the classical approaches. Our failure to fit data with these approaches sets up the second part of our paper, a novel exploration of the essential decelerating role of viscous heating. We explore both isothermal and adiabatic expansions, and also the decelerating role of surface tension. The adiabatic analysis accounts for the slight deceleration due to the cooling of the expanding gas, which depends on gas polyatomicity. We explore the pressure profile, and the components of the extra stress tensor, in the surrounding fluid. These stresses can be frozen into foamed plastics. We find that our new theory compares well with measured size, when the nucleus radius, nucleus pressure, and melt viscosity are fitted. We include a detailed dimensional worked example to help process engineers with foam design.