  We discuss the applicability of a unified hyperbolic model for continuum
fluid and solid mechanics to modeling non-Newtonian flows and in particular to
modeling the stress-driven solid-fluid transformations in flows of viscoplastic
fluids, also called yield-stress fluids. In contrast to the conventional
approaches relying on the non-linear viscosity concept of the Navier-Stokes
theory and representation of the solid state as an infinitely rigid
non-deformable solid, the solid state in our theory is deformable and the fluid
state is considered rather as a "melted" solid via a certain procedure of
relaxation of tangential stresses similar to Maxwell's visco-elasticity theory.
The model is formulated as a system of first-order hyperbolic partial
differential equations with possibly stiff non-linear relaxation source terms.
The computational strategy is based on a staggered semi-implicit scheme which
can be applied in particular to low-Mach number flows as usually required for
flows of non-Newtonian fluids. The applicability of the model and numerical
scheme is demonstrated on a few standard benchmark test cases such as Couette,
Hagen-Poiseuille, and lid-driven cavity flows. The numerical solution is
compared with analytical or numerical solutions of the Navier-Stokes theory
with the Herschel-Bulkley constitutive model for nonlinear viscosity.
