An extensive study of shear thinning flow around a spherical particle for power-law and Carreau fluids

We develop and implement a high-order finite element formulation to solve incompressible shear thinning flows using power-law and Carreau rheology models. We verify the implementation using the Method of Manufactured Solutions (MMS) and demonstrate that the implementation preserves the order of accuracy of the FEM scheme. We run high-order flow-past-a-sphere simulations for Re \(\in [0.1, 100]\) for Newtonian and shear thinning flows. Power-law simulations cover \(n \in [0.3, 0.9]\), Carreau simulations cover the same range of n and dimensionless relaxation times \(\Lambda \in [0.1, 100]\). We use 3D Cartesian grids, adopting a high-order sharp-interface immersed boundary method (IBM) to impose the no-slip boundary condition on the surface of the sphere. We use dynamic mesh adaptation with a Kelly error estimator to adaptively refine the grid. Newtonian drag coefficients are compared to Clift et al.’s model. Using the power-law model for shear thinning flow, results show that when decreasing n : the drag coefficient increases for Re ⪅ 5, it decreases for Re ⪆ 5, and that for higher Reynolds the wake region is longer and the separation point is moved upstream. These drag coefficient results are in adequation with previously established simulation results. We provide a new drag coefficient correlation for power-law fluids that is valid for a wide range of n and for Re \(\in [0.1, 100]\), with \(R^2 = 0.998\). The power-law model is singular in the zero-shear limit, and this limit is bound to occur in a flow-past-a-sphere geometry. For that matter, the Carreau model is well-posed. As the Carreau-modeled results show, the drag force on the sphere is considerably affected by the initial Newtonian plateau of the fluid. Drag coefficient results in the creeping flow regime are compared to experimental results, allowing validation of the numerical model. This work provides a better understanding of shear thinning flow past a sphere, for which the relationship between drag and flow regimes are highly nonlinear.

Uncontrolled Keywords

• computational fluid dynamics
• power-law
• Carreau
• shear thinning flow
• high-order methods
• drag force on sphere