On Maxwell models in viscoelasticity that are more computable

Maxwell models for the flow of viscoelastic fluids are not usually computable unless the dimensionless relaxation time is small. Problems with numerical instabilities arise. A new approach is presented that alleviates this problem. In this new approach, a viscous term is added which has the effect of regularizing the model. By an appeal to the kinetic theory of gases and turbulence, it is shown that this addition can be formally justified. The regularization comes from changing the constitutive equation type from purely hyperbolic to a mix that is largely parabolic and makes the model more computable. This can also alleviate the problem with specifying boundary conditions. The prospects for future research in viscoelastic fluids are thoroughly discussed.