Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor

Abstract

We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are reformulated as a system for the triple pressure-velocity-apparent viscosity, where the apparent viscosity is given by a scalar implicit equation. We prove that the proposed numerical scheme has---on the discrete level---a solution, and using the proposed scheme we numerically solve several flow problems.

References in corpus (3)

• Shear Banding of Complex Fluids
• Thermodynamics of viscoelastic rate-type fluids with stress diffusion
• Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics

Cited by in corpus (6)

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• Thermodynamical extension of a symplectic numerical scheme with half space and time shift demonstrated on rheological waves in solids
• Four spacetime dimensional simulation of rheological waves in solids and the merits of thermodynamics
• A Hybrid High-Order method for incompressible flows of non-Newtonian fluids with power-like convective behaviour
• Implicit type constitutive relations for elastic solids and their use in the development of mathematical models for viscoelastic fluids
• On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary