A NUMERICAL STUDY OF NONLINEAR VISCOELASTIC FLOW AND NONISOTHERMAL POWER-LAW FLOW
Doctor of Philosophy
Using the finite difference technique, the steady flow of a viscoelastic fluid through a contraction/expansion and a modified Graetz heat transfer problem for power-law fluids between parallel plates are numerically studied. The constitutive model used for the viscoelastic fluid flow extends the Maxwell equation of linear viscoelasticity to the non-linear region by letting both relaxation time and elastic modulus depend upon the existing structure. Based on the calculations for a 3:1/1:3 contraction/expansion geometry with fully developed simple shearing flows at the entrance and exit, there exists a transition range for this fluid model, in terms of the Weissenberg number, beyond which elastic effects appear to become less dominant. The upstream vortex detachment length before the contraction is shown to grow with increasing Weissenberg number, and then reach a maximum value, until no converged solution can be obtained or possibly unsteady flow starts to develop. Heat transfer to polymer melts or solutions flowing in a parallel plate system is of great importance in polymer processing, as for example in extrusion through a large aspect ratio slit die. The present work was undertaken to solve the energy equation for power-law fluids under various circumstances, including a temperature-dependent viscosity, viscous dissipation and heat convection across streamlines induced by the abrupt change of boundary temperature and subsequent velocity field rearrangement. In addition to the numerical solutions given, an analytical approximation method for the Graetz-Nusselt problem is also presented for comparison. This method divides the domain into two sections for which two corresponding solutions, approximated by polynomials, are solved by a thermal boundary layer theory and an integral method respectively. These two approximate solutions are then matched at the intersection to give a continuous and consistent temperature profile. Although the analytical solutions application is limited to the case with a temperature-independent viscosity, this simple analysis is believed to be better than the previous methods available in the literature.