Symmetric-hyperbolic conservation laws modelling viscoelastic flows of Maxwell fluids

Many equations have been proposed to model viscoelastic flows.

Seminal equations have been proposed by Maxwell in 1867 for fluids with stress relaxation.

The famous Upper-Convected Maxwell equations are useful for one-dimensional flows in particular.

But the usability of Maxwell fluids for multi-dimensional viscoelastic flows has remained limited, as shown by numerous numerical simulations that do not converge when the discretization parameters are refined beyond a critical value for the relaxation-time of the stress.

As a remedy, we propose to consider a system of conservation laws with algebraic source terms (balance laws) to model viscoelastic flows.

Our system is symmetric-hyperbolic, similar to the famous K-BKZ integral viscoelastic models, but more versatile (purely differential).

It rigorously unifies fluid models with elastodynamics for compressible solids, and it can be manipulated for various applications of the viscoelastic flow concept in environmental hydraulics (shallow-water flows) or materials engineering (non-isothermal flows).