Nonisothermal, nonNewtonian Hele-Shaw flows, part II: asymptotics and existence of weak solutions
Nonlinear Analysis: Theory, Methods & Applications
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Polymer manufacturing processes such as compression molding, sheet forming and injection molding can be modeled by squeeze-flows. In this paper, we consider a squeeze-flow between two parallel plates, separated by a small gap. We treat both a Newtonian fluid and power-law fluid with a partial wall slip. We first derive a system of equations, using the lubrication theory approximation, for a non-isothermal, power-law fluid in a thin mould. We use the Hele-Shaw model with a Navier partial slip condition. Although we do not assume formation of a slip layer in our derivation, it still allows us to verify analytically that Navier's slip coefficient is a function of the slip layer thickness and the shear viscosity. We then prove the existence and regularity of weak solutions to the initial boundary value problems for a non-isothermal, Newtonian case in a more general setting. This is a coupled, non-linear problem as the pressure enters the heat equation in non-linearly. Our numerical experiments indicate that the degree of wall slip becomes more significant as the mold walls gets closer, which implies that wall slips should always be used in micro and nano processes.