In this work, we implement models that are able to describe complex rheological behaviour (such as shear-banding and elastoviscoplasticity) in the HiGTree/HiGFlow system, which is a recently developed Computational Fluid Dynamics (CFD) software that can simulate Newtonian, Generalised-Newtonian and viscoelastic flows using finite differences in hierarchical grids. The system uses a moving least squares (MLS) meshless interpolation technique, allowing for more complex mesh configurations while still keeping the overall order of accuracy. The selected models are the Vasquez-Cook-McKinley (VCM) model for shear-banding micellar solutions and the Saramito model for viscoelastic fluids with yield stress. Development of solvers and numerical simulations of inertial flows of these models in 2D channels and planar-contraction 4:1 are carried out in the HiGTree/HiGFlow system. Our results are compared with those predicted by two other methodologies: the OpenFOAM-based software RheoTool that uses the Finite-Volume-Method and an in-house code that uses the Vorticity-Velocity-Formulation (VVF). We found an excellent agreement between the numerical results obtained by these three different methods. A mesh convergence analysis using uniform and refined meshes is also carried out, where we show that great convergence results in tree-based grids are obtained thanks to the finite difference method and the meshless interpolation scheme used by the HiGFlow software. More importantly, we show that our methodology implemented in the HiGTreee/HiGFlow system can successfully reproduce rheological behaviour of high interest by the rheology community, such as non-monotonic flow curves of micellar solutions and plug-flow velocity profiles of yield-stress viscoelastic fluids.

In this work, we present the implementation and verification of HiGTree-HiGFlow solver (see for numerical simulation of the KBKZ integral constitutive equation. The numerical method proposed herein is a finite difference technique using tree-based grids. The advantage of using hierarchical grids is that they allow us to achieve great accuracy in local mesh refinements. A moving least squares (MLS) interpolation technique is used to adapt the discretization stencil near the interfaces between grid elements of different sizes. The momentum and mass conservation equations are solved by an implicit method and the Chorin projection method is used for decoupling the velocity and pressure. The Finger tensor is calculated using the deformation fields method and a three-node quadrature formula is used to derive an expression for the integral tensor. The results of velocity and stress fields in channel and contraction-flow problems obtained in our simulations show good agreement with numerical and experimental results found in the literature.

This work presents a comprehensive numerical implementation of a viscoelastic thixotropic model known as the modified-Bautista–Manero (MBM) model (a model widely used to characterize the rheological behavior of wormlike micellar solutions). This implementation is integrated into the OpenFOAM computational fluid dynamics software, specifically using the rheoTool library. In contrast to previous studies, our approach includes simulations of inertial flows and we account for convective terms within all the governing equations. Moreover, we use two numerical techniques that have been proven to be efficient methods in computational simulations of viscoelastic fluids through a planar contraction: (1) the high-resolution scheme called convergent and universally bounded interpolation scheme for the treatment of advection to discretize the convective terms of the momentum, viscoelastic, and thixotropic equations; and (2) the logarithm-conformation reformulation to tackle the high-Weissenberg number problem. This expansion of the model's scope results in a more realistic representation of the fluid's behavior and, consequently, more accurate computational results. To validate the implementation of the MBM model, we performed numerical simulations in two different industrial-relevant geometries: the planar 4:1 contraction and the 4:1:4 contraction–expansion configurations. Two fluids with different structural responses are studied: one with quick structural recovery and the other with mildly slow recovery. The numerical method developed in this work allowed us to run stable simulations of these fluids up to Reynolds and Deborah number values of Re = 1.11 and De = 14.0, respectively, and the influence of inertial, viscoelastic, and thixotropic effects on various flow field variables is investigated. These variables include velocity, viscosity, normal stresses, and corner vortex size. Transient and steady solutions of corner vortexes across a range of Deborah and Reynolds numbers are also presented. In addition, the results obtained in this work are also directly compared with numerical simulations obtained using the non-thixotropic rubber network-based exponential Phan-Thien–Tanner model. For the planar 4:1 contraction simulations, the analysis of the results is based on three dimensionless numbers (the viscoelastic Mach, the elasticity, and the thixoelastic numbers), which allows to conclude that vortex enhancement is mainly seen when high elasticity is coupled with quick structural reformation and very low inertial effects. We further report from the planar 4:1:4 contraction–expansion results that an increase in inertia leads both to vortex inhibition in the upstream channel and to slight vortex enhancement in the downstream channel. Finally, we demonstrate the strong influence of fluid convection on fluidity profiles and on the size of the upstream/downstream corner vortex.