Modelling, Analysis and Computation in Viscoelasticity

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Viscoelastic materials with fading memory exhibit behavior that is intermediate between the nonlinear hyperbolic response of elastic materials and the strongly diffusive, parabolic response of viscous materials; they incorporate a subtle dissipative mechanism induced by the effects of fading memory. Understanding these effects is crucial to the proper design of computational methods for modeling both unsteady and steady viscoelastic flows. Analytical and computational techniques will be developed for unsteady problems by adapting methods from the study of hyperbolic conservation laws. Building upon the understanding of classical solutions and their breakdown, a study of weak solutions to the governing equations will be conducted using two approaches: the method of compensated compactness and the random choice method. The latter requires an understanding of the Riemann problem for viscoelastic flows; its solution is an essential ingredient for designing computational algorithms. These ideas will be used to broaden the scope of the established program of simulation of steady, two-dimensional viscoelastic flows, with the objective of resolving serious computational difficulties at high Weissenberg numbers. This research represents a program in scientific computation that unites methods from mathematical, computational, and engineering disciplines.