Nonlinear systems arising from nonisothermal, non-Newtonian Hele-Shaw flows in the presence of body forces and sources

Abstract

In this paper, we give a formal derivation of several systems of equations for injection moulding. This is done starting from the basic equations for nonisothermal, non-Newtonian flows in a three-dimensional domain. We derive systems for both ( T 0, p 0) and ( T 1, p 1) in the presence of body forces and sources. We find that body forces and sources have a nonlinear effect on the systems. We also derive a nonlinear “Darcy law”. Our formulation includes not only the pressure gradient, but also body forces and sources, which play the role of a nonlinearity. Later, we prove the existence of weak solutions to certain boundary value problems and initial-boundary value problems associated with the resulting equations for ( T 0, p 0) but in a more general mathematical setting.