Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme

TL;DR

This paper presents a unified hyperbolic model and a semi-implicit numerical scheme for simulating non-Newtonian viscoplastic flows, effectively capturing solid-fluid transitions and comparing well with classical models on benchmark tests.

Contribution

It introduces a novel first-order hyperbolic model and a structure-preserving semi-implicit scheme for non-Newtonian flows, including yield-stress fluids, with demonstrated applicability on standard benchmarks.

Findings

Model accurately simulates non-Newtonian flows.

Numerical results agree with classical Navier-Stokes solutions.

Applicable to low-Mach number and complex flow scenarios.

Abstract

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.