Space-Time Modeling and Numerical Simulations of Non-Newtonian Fluids Using Internal Variables

The modeling of fluids is an important field for mechanics of materials. In this work, we demonstrate that Hamilton's principle, which is well-known for the modeling of solids, can also be formulated to derive the Navier–Stokes equations, which paves the way for easy inclusion of complex material constraints. Furthermore, we expand Hamilton's principle to enable the introduction of “internal variables”, which describe the space- and time-dependent evolution of the material properties. Hereby, a novel strategy for the modeling of non-Newtonian fluids is given. Eventually, Hamilton's principle inherently enables a space-time formulation with the automatic derivation of the correct formal functional setting, which covers different scales of viscosity through the internal variable. The resulting system is a space-time multiscale model for fluid flow, which is based on an additional partial differential equation. The model constitutes thus a much more adaptive description of the complex processes in non-Newtonian fluid flow as possible for classical models based on algebraic constitutive laws. This also includes a spatially and temporally local evolution of the effective viscosity, depending on the local flow conditions rather than material parameters and resulting in both shear-thinning and shear-thickening behavior. Numerical examples substantiate our proposed setting by some studies from Newtonian flow to non-Newtonian regimes with fading or increasing viscosity.