Direct numerical simulation of particle sedimentation in a Bingham fluid

The settling efficiency, and stability with respect to settling, of a dilute suspension of infinite circular cylinders in a quiescent viscoplastic fluid is examined by means of direct numerical simulations with varying solid volume fraction, $\varphi$, and yield number, $Y$. For $Y$ sufficiently large we find higher settling efficiency for increasing $\varphi$, similar to what is found in shear-thinning fluids and opposite to what is found in Newtonian fluids. The critical yield number at which the suspension is held stationary in the carrier fluid is found to increase monotonically with $\varphi$, while the transition to settling is found to be diffuse: in the same suspension, particle clusters may settle while more isolated particles remain arrested. In this regime, complex flow features are observed in the sedimenting suspension, including the mobilization of lone particles by nearby sedimentation clusters. Understanding this regime, and the transition to a fully arrested state, is relevant to many industrial and natural problems involving the sedimentation of viscoplastic suspensions under quiescent flow conditions.

Figure 1

Left: An overlapping grid consisting of two structured curvilinear component grids, $\mathit{x}={\mathcal{G}}_1\left(\mathit{r}\right)$ and $\mathit{x}={\mathcal{G}}_2\left(\mathit{r}\right)$. Middle and right: Component grids for the square and annular grids in the unit square parameter space $\mathit{r}$. Grid points are classified as discretization points, interpolation points, or unused points. Ghost points are used to apply boundary conditions.

Figure 2

Left: Close-up view of a representative computational grid used, showing the overlapping particle and background grids and near-surface grid refinement. Right: Mean sedimentation velocity at increasing yield numbers for $\varphi =(0,0.01,0.05)$, with error bars indicating the standard error of the mean particle sedimentation velocity.

Figure 3

Left: Average effective viscosity near the particle surface for $Y=0.043$, with longitudinal (dashed line) and lateral (dash-dot line) axes of symmetries indicated in the $\varphi =0$ panel. Right: Average effective viscosity along the longitudinal (top) and lateral (bottom) axes of symmetry for $Y=0.043$.

Figure 4

Color maps of the fluid velocity magnitude with particles overlaid in gray and yield surface indicated by the solid black lines for $Y=0.087$ with $\varphi =0.01$ (left) and $\varphi =0.05$ (right) for five different configurations.

Figure 5

Color maps of ${log}_{10}(\dot{\gamma }/\mathcal{G})$ for $\varphi =0.01$ at $Y=(0,0.022,0.065,0.087,0.13)$ (left panel) and $\varphi =0.05$ at $Y=(0,0.043,0.087,0.13,0.17)$ (right panel).

Figure 6

Proportion of settling particles in the suspension at increasing yield strengths with the critical yield number for a single particle indicated by the dashed line.

Figure 7

Examples of the complex flow features found in regime II. Left: Close-up view from a $\varphi =0.01$ suspension with $Y=0.087$ showing the mobilization of an isolated particle by a passing sedimentation cluster. Right: Close-up view from a $\varphi =0.05$ suspension with $Y=0.173$ showing lone particles swept up in the strong recirculating flow generated by a sedimentation cluster. A large unyielded region undergoing rigid body motion with embedded particles is visible. For each panel, the left plots show color maps of the fluid velocity magnitude, with particles overlaid in gray and their individual velocity vectors indicated by arrows. The right plots show color maps of the normalized second invariant of the velocity gradient tensor, with static unyielded regions masked off, yield surfaces indicated by the solid white lines, and particles overlaid in gray.

Figure 8

Convergence behavior of the functionals $a(\mathit{u},\mathit{u})$ (circles), $j\left(\mathit{u}\right)$ (squares), and $L\left(\mathit{u}\right)$ (diamonds), for a single particle (blue), $\varphi =0.01$ (yellow), and $\varphi =0.05$ (red). Symbols are computed, and lines are power-law fits with exponent $m$.

Figure 9

Convergence at large $B$ for three volume fractions. Symbols are computed, and lines are power-law fits with exponent $m$.

Figure 10

Normalized energy density per unit volume per unit $\mathrm{\Lambda }$ for a single disk (left), $\varphi =0.01$ (middle), and $\varphi =0.05$ (right).

Figure 11

Probability density functions of $\mathrm{\Lambda }$ with increasing $Y$ for a single disk (left), $\varphi =0.01$ (middle), and $\varphi =0.05$ (right).