Solving the behavior of individual bubbles would be computationally extensive and expensive. Instead, consider a phenomenological continuum model for predicting the behavior of the foaming process. More precisely, consider the model proposed by Arefmanesh [[1]] for predicting bubble growth. The Arefmanesh model assumes that the matrix behaves as a purely viscous material, and that the time scale of gas diffusion is longer than that of bubble expansion. As will be seen, the model is such that all physics is lumped in a single transport equation that dictates the bubble radius.

The model does not necessarily predict an equilibrium size of the bubbles: gas pressure decreases with increasing bubble radius but does not vanish. In actual cases, however, bubbles do not grow infinitely, since only a finite supply of gas is available for their growth, and bubbles attain equilibrium once the gas gets depleted in the polymer. Also, surface tension may possibly play a role if gas bubbles are small enough.

To avoid any misunderstanding or confusion, a few definitions are required as part of the introduction of the physical foaming model. As described below, the polymer matrix will denote the polymer without any blowing agent (gas bubbles). The dense polymer will refer to the polymer in which gas bubbles are already injected and which keep their initial radius . In this sense, the dense polymer is very close to the polymer matrix. The mixture, foam, or foamed polymer will denote the fluid system at any state between the initial and final foaming states.

In physical foaming, extrudate swelling is largely dictated by the growth of the gas bubbles, which plays a role that is more important than the rheology. This allows the focus to be on the bubble growth.

Reviewing the model proposed by Arefmanesh. you can first consider a single cell consisting of a polymer matrix surrounding a massless gas bubble of radius R. Its constant mass is , while its volume is given as the sum of the polymer volume V and the gas bubble volume. On the one hand, the density of the cell is given by

(20–1)

On the other hand, the density of the polymer matrix (without blowing agent, that is, with R=0) is given by . Let be the number of cells per volume unit of gas/polymer mixture. Then, the following can be calculated

(20–2)

This equation gives the density of the mixture as a function of the polymer matrix density , the number of cells N and the current gas bubble radius R. Also, while foaming progresses, the macroscopic viscosity of the mixture decreases since the volume increase results only from gas expansion. Hence, with the bubble growth, one can assume that the zero-shear viscosity of the mixture decreases on the same way as the density:

(20–3)

The density and the viscosity have been shown to be related to the bubble radius R. A constitutive equation is now needed for R. The growth of the bubble radius obeys the following simplified transport equation

(20–4)

where is the gas pressure and p is the matrix pressure. The parameter may play the role of surface tension that can slow down the growth of the bubble, and the quantity is a macroscopic viscosity of the matrix. With the assumption of isothermal foaming, the conservation of the gas quantity in a bubble gives

(20–5)

where the subscript i refers to the initial state characterized by an initial gas pressure within bubbles of non-vanishing initial radius . Thus, the previous equation now becomes:

(20–6)

with as initial condition. This equation involves three parameters that still need to be assigned: , and . The parameter can be set in equilibrium with the pressure at the inlet of the die. The parameter is usually set to 0 unless one wants to moderate the growth rate. The parameter corresponds to a viscosity. For not slowing down the bubble growth, you could select a value that more or less matches the macroscopic viscosity at the end of the foaming.

The foaming model involves several non-linearities. Next to the rheology and the moving boundaries, there is also the equation for the bubble growth and the dependence of density and viscosity with respect to the bubble radius R. Thus, you should progressively incorporate these non-linearities into the calculation. For this, the simulation should have the appropriate convergence strategies applied on rheology/slipping, on foaming, and on moving boundaries.