A NUMERICAL SIMULATION OF THE FLOW OF VISCOELASTIC MATERIALS IN THE DIE-ENTRY REGION

A major obstacle in the prediction of stress profiles in the viscoelastic flow of polymers is a mysterious breakdown of the numerical procedure, which occurs at relatively small values of the Deborah number. Numerous papers have been devoted to analyzing the reason for this failure, but the exact cause remains unclear. Amazingly, all constitutive equations attempted and all kinds of different numerical procedures employed have run into the very same problem. We have investigated several currently popular constitutive models in a viscometric flow field and have found serious limitations in shear flows which may be the source of numerical problems. There is often a lack of appreciation for the computational uses of fluid models in the process of formulating constitutive equations. In the future, the use of the corrotational time derivative, which appears to create this trouble, may be prohibited. Alternatively, it may be possible to avoid the limitation by adding a proper retardation time in constitutive models which use the corrotational time derivative. The well known upper convected Maxwell model does not exhibit limitation but it is believed that a different source of numerical instability may be inherently present. We have adopted the cylindrical axisymmetric 4:1 contraction channel (the die-entry region) to simulate this fluid using the mixed finite element method. The worrisome infinite elongational viscosity predicted by this model in steady extensional flow, indeed, is responsible for the singular behavior of the stress solution field. The most difficult region for convergence, as might be expected, is found to be just after the reentrant corner