Systems of nonlinear Hele-Shaw partial differential equations

Injection and compression molding methods are two widely used technologies for manufacturing polymers. In many applications of injection and compression molding methods, the molds are very thin relative to their length and width. In such cases, the Hele-Shaw approximation often simplifies the analysis.; Chapter 1 of this dissertation is an introduction to Hele-Shaw flows. We derive systems for the first order term (&thetas;0, p 0) and the second order term (&thetas;1, p1 ) for the unknown pressure p and temperature &thetas; in the presence of body forces. We discovered that body forces behave nonlinearly for non-Newtonian Hele-Shaw flows. We believe that the nonlinear effects of body forces in non-Newtonian Hele-Shaw flows will exhibit much richer behaviors than in the Newtonian case. We find that body forces have a nonlinear effect on the systems. We also derive a nonlinear “Darcy law”.; The second chapter is devoted to the existence of solutions to nonlinear systems arising from non-isothermal, non-Newtonian Hele-Shaw flows in the presence of body forces in injection molding.; In Chapter 3, we consider the system described in Chapter 2 as a free boundary problem by formulating it as a variational inequality. To the best of our knowledge, a nonlinear system of a partial differential equation coupled with a variational inequality seems to be “terrae incognitae”.; Since the pressures generated during a squeeze-flow are often quite large, it is necessary to include compressibility. In Chapter 4, we investigate a model where the flow is compressible. The difference between the compressible and the traditionally incompressible approaches is expressed as a correction term which relies on the equation of state. It is worthwhile to mention that compressibility has already been considered in injection molding.