Each bubble experiences forces that cause it to move through the melted polymeric material. The internal temperature gradient that results from surface heating of the sphere causes gradients in surface tension and in viscosity, both of which cause the bubble to move toward the heated surface. The bubble is also influenced by the flow fields surrounding neighboring bubbles, including radial fields due to bubble growth as well as convective flow fields. On approach to the surface of the melted thermoplastic sample, deformation of the surface and the bubble occur, and the bubble slows. When bubbles are in close proximity to each other or the sample surface, short range forces such as van der Waals and double-layer forces act to keep bubbles separated while thin-film drainage takes place, resulting in eventual coalescence or surface bursting. The timescale over which coalescence occurs is strongly affected by the presence or absence of surface-acting agents (surfactants) in the melt. Finally, the g-jitter present in a real microgravity environment, such as the NASA Reduced-Gravity Aircraft (±0:01g) or the Space Shuttle, results in a randomly-oriented body force proportional to the difference between bubble and melt mass densities.
To simplify this model, flow fields from nearby bubbles and the gravitational force due to g-jitter are neglected. The bubbles are assumed to be far enough apart that each individual bubble can be moved in space as if it is the only bubble in the melt. The only forces that are taken into account, therefore, are those that are caused by viscosity and surface tension gradients, modified by the slowing of velocity as a bubble approaches the surface. The velocity of each bubble, therefore, is in the radial direction only.
The Reynolds number for bubble motion in the polymeric melt, Re = ρU(2R) / µp, is assumed to be much smaller than one, so that the Stokes flow approximation applies. For Stokes flow, the drag force on a bubble of radius R moving in a fluid of infinite extent with terminal velocity U is
|
(20) |
where the ratio of gas to liquid viscosity is κ = µg
/ µp 
1 for a
bubble in a polymer melt. Multiple
forces on a bubble may be linearly superposed. The total bubble velocity is
therefore calculated from
the summation of individual velocities due to viscosity gradient and surface
tension gradient forces.
The viscosity of the molten thermoplastic material depends strongly on temperature and molecular weight, and therefore varies considerably in space and time. The decreasing resistance to movement toward the hot surface of the polymeric sample causes an expanding bubble to move outwards with terminal velocity
 |
(21) |
where R is the bubble radius,
its growth rate,
d lnpµ / dT describes the variation of
viscosity with
temperature (note that the value in parentheses is positive), and ∂T
/ ∂z is the temperature gradient [17].
This result is based on the assumption that the bubble growth rate is much
larger than bubble translation.
The temperature gradient also sets up a non-uniform tangential stress on the surface of a non-contaminated bubble which must be balanced by a flow. This Marangoni, or surface-tension induced, flow is also in the direction of increasing temperature [42]:
 |
(22) |
where d ln σ / dT is the slope of the natural log of surface tension with temperature.
The approach of an axisymmetric deformable particle, either drop or bubble, toward an initially flat interface has been studied by Chi and Leal [43]. The liquids are immiscible, incompressible, and Newtonian. The interface separates the fluid of the dense medium through which the liquid particle moves from the less dense fluid into which the particle will eventually merge. Given the case with like fluids for the particle and outer medium, computations provide a set of curves describing nondimensional velocity as a function of nondimensional distance from the center of mass of the particle to the undeformed flat interface. For a bubble in a polymer melt, the particle-to-fluid viscosity ratio λ = µg / µp is low and the capillary number Ca = µpU∞/ω, where U∞ is the particle velocity far from the surface and ω is the surface tension, is high. An empirical fit for the curve most closely representing the case of interest was determined as:
| ƒ(x) = 0.45879 − 0.070324x + 0.50747ln(0.5 + x) for − 0.1 < x < 6 , |
where x is the distance between bubble center and undeformed surface nondimensionalized by the radius of the bubble. The velocity of each bubble in the model whose center is closer than six radii from the surface is then decreased through multiplication by this factor. The total radial velocity for a bubble, therefore, is given by
 |
(24) |
where ƒ(x) is given by equation (23) for x = (rS − rB)/RB) between −0.1 and 6, by one for x > 6, and by zero for x < −0.1. Also of interest from the paper by Chi and Leal [43] is the calculated shape of the film between particle and interface as the particle comes very close to the surface. For bubbles with low λ and high Ca, the film is thinnest directly above the center. This is a geometry that results in rapid bursting of the bubble, as compared to the "dimpled" mode in which the rim is thinnest in a circle some distance from the center.