Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM. This is the accepted version of the article published in:

Three non-Newtonian constitutive models are employed to investigate how fluid rheological properties influence the development of laterally asymmetric flows past confined cylinders. First, simulations with the shear-thinning but inelastic Carreau–Yasuda model are compared against complementary flow velocimetry experiments on a semidilute xanthan gum solution, showing that shear-thinning alone is insufficient to cause flow asymmetry. Next, simulations with an elastic but non-shear-thinning finitely extensible non-linear elastic dumbbell model are compared with experiments on a constant viscosity solution of poly(ethylene oxide) (PEO) in an aqueous glycerol mixture. The simulations and the experiments reveal the development of an extended downstream wake due to elastic stresses generated at the stagnation point but show no significant lateral asymmetries of the flow around the sides of the cylinder. Finally, the elastic and shear-thinning linear Phan–Thien–Tanner (l-PTT) model is compared with experimental velocimetry on a rheologically similar solution of PEO in water. Here, at low flow rates, lateral symmetry is retained, while the growth of a downstream elastic wake is observed, in qualitative similarity to the non-shear-thinning elastic fluids. However, above a critical flow rate, the flow bifurcates to one of the two stable and steady laterally asymmetric states. Further parameter studies with the l-PTT model are performed by varying the degrees of shear-thinning and elasticity and also modifying the confinement of the cylinder. These tests confirm the importance of the coupling between shear-thinning and elasticity for the onset of asymmetric flows and also establish stability and bifurcation diagrams delineating the stable and unstable flow states.

Blood plasma has been considered a Newtonian fluid for decades. Recent experiments (Brust et al., Phys. Rev. Lett., 2013, 110) revealed that blood plasma has a pronounced viscoelastic behavior. This claim was based on purely elastic effects observed in the collapse of a thin plasma filament and the fast flow of plasma inside a contraction-expansion microchannel. However, due to the fact that plasma is a solution with very low viscosity, conventional rotational rheometers are not able to stretch the proteins effectively and thus, provide information about the viscoelastic properties of plasma. Using computational rheology and a molecular-based constitutive model, we predict accurately the rheological response of human blood plasma in strong extensional and constriction complex flows. The complete rheological characterization of plasma yields the first quantitative estimation of its viscoelastic properties in shear and extensional flows. We find that although plasma is characterized by a spectrum of ultra-short relaxation times (on the order of 10-3-10-5 s), its elastic nature dominates in flows that feature high shear and extensional rates, such as blood flow in microvessels. We show that plasma exhibits intense strain hardening when exposed to extensional deformations due to the stretch of the proteins in its bulk. In addition, using simple theoretical considerations we propose fibrinogen as the main candidate that attributes elasticity to plasma. These findings confirm that human blood plasma features bulk viscoelasticity and indicate that this non-Newtonian response should be seriously taken into consideration when examining whole blood flow.

To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are compared to experimental data and predictions of previously proposed TEVP models in simple rheometric flows, including steady and step-shear tests, flow reversal, intermittent step tests, small amplitude oscillatory shear (SAOS) and large amplitude oscillatory shear. In most cases, the proposed model reproduces more accurately these experimental data than the other models, highlighting its predictive capabilities. Moreover, SAOS illustrates that introducing viscoplasticity via the Saramito model necessarily reduces G″ to zero in the linear strain regime. This calls for model adjustments in the solid state. Finally, we examined the proposed model in uniaxial elongation and concluded that it is important to include this flow in the rheological characterization and modeling of such systems.

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