Direct numerical simulation of the sedimentation of a particle pair in a shear-thinning fluid

By combining the momentum exchange method and a unified interpolation bounce-back scheme, a multiple-relaxation-time lattice Boltzmann method is developed to simulate the particle dynamics in a power-law fluid. With this method, the sedimentation of a particle pair in a shear-thinning power-law fluid for the generalized Archimedes number $\left(\mathrm{A}{\mathrm{r}}^{*}\right)$ varying from 100 to 400 is studied. Depending on the value of $\mathrm{A}{\mathrm{r}}^{*}$ and initial geometrical configuration, the particle pair is found to experience several different movement states, namely, the steady oblique doublet, periodic oscillation, period-doubling bifurcation, steady horizontal doublet, and chaos. Distinct from two groups of multiple stable states in the Newtonian system, three groups of multiple stable states are clearly identified in the present shear-thinning system: the steady oblique coexists with the periodic oscillation, the steady horizontal doublet coexists with the period-doubling bifurcation, and the steady horizontal doublet coexists with the chaos state. Moreover, the drafting, kissing, and tumbling (DKT) behavior of a particle pair observed in the Newtonian system is absent in the present shear-thinning system. It is attributed to the high-viscosity region between the particles, which can increase the viscous drag acting on the particle, thus preventing the particles from approaching each other and reducing particle aggregation. In addition, a critical value of the power-law index is obtained, below which the DKT state would not happen regardless of $\mathrm{A}{\mathrm{r}}^{*}$.

Figure 1

Schematic diagram of the unified interpolation bounce-back scheme and the refilling. The filled points represent the solid nodes within the particle and the hollow points represent the fluid nodes. The unknown distribution function at the boundary fluid node ${\mathit{x}}_{\mathrm{f}}$ is denoted by the red line with arrow. The locations of the moving particle at two neighboring time steps are shown as the blue dashed lines and the black solid lines, respectively. The red hollow circles represent the fluid nodes to be refilled, known as the fresh fluid nodes, and the black hollow circles are the fluid nodes used for evaluating the macroscopic variables of the fresh fluid node $P$.

Figure 2

Schematic diagram of the location of the imaginary particle that is tangential to the wall surface.

Figure 3

Time evolution of the $y$ component of particle velocity at three typical power-law indexes, where the results from Delouei et al. [39] are also shown for comparison.

Figure 4

Geometrical setup of the problem. Initially, the solid particles are of the same height, and are located on both sides of the center line of the channel with the horizontal distances of $X1$ and $X2$ away from the center line.

Figure 5

Terminal Reynolds number $R{\mathrm{e}}_{\mathrm{T}}$ versus Ar and the corresponding principal movement states for a pair of particles settling in Newtonian fluid. The present results are compared with those obtained from Verjus et al. [12] and Zhang et al. [13].

Figure 6

Snapshots of the particle pair settling in Newtonian fluid for $\mathrm{Ar}=600$, and the corresponding time points are given on the bottom or top of each box.

Figure 7

Phase portrait of $\mathrm{Ar}=900$, where the result from Zhang et al. [13] is also shown for comparison.

Figure 8

Phase portrait of $\mathrm{Ar}=980$ for a particle pair settling in Newtonian fluid.

Figure 9

Transversal displacements of the left particle versus time, which is normalized by $\sqrt{D/\phantom{Dg}g}$, for three different grid resolutions, $D/{\delta }_{\mathrm{x}}=16,32$, and 48. The simulations are conducted at $\mathrm{A}{\mathrm{r}}^{*}=140$ with horizontal locations of the particle pair initialized by run B.

Figure 10

Simulation results of $\mathrm{A}{\mathrm{r}}^{*}=130$: (a) sedimentation trajectories of the particles and (b) streamlines around the moving particles when they reach the steady motion. The rotation direction of each particle is shown by the white arrow.

Figure 11

Simulation results of $\mathrm{A}{\mathrm{r}}^{*}=140$: (a) sedimentation trajectories of the particles for two different initial configurations, (b) phase portrait, and (c) spectrogram obtained from the FFT for run B.

Figure 12

Simulation results of $\mathrm{A}{\mathrm{r}}^{*}=164$: (a) sedimentation trajectories of the particles for two different initial configurations, (b) phase portrait, and (c) spectrogram obtained from the FFT for run B.

Figure 13

Simulation results of $\mathrm{A}{\mathrm{r}}^{*}=166$: (a) sedimentation trajectories of the particles for two different initial configurations, (b) phase portrait, and (c) spectrogram obtained from the FFT for run B.

Figure 14

Simulation results of $\mathrm{A}{\mathrm{r}}^{*}=200$: (a) sedimentation trajectories of the particles for two different initial configurations, (b) streamlines around the moving particles for run A, (c) phase portrait, and (d) spectrogram obtained from the FFT for run B.

Figure 15

Simulation results of $\mathrm{A}{\mathrm{r}}^{*}=281$: (a) sedimentation trajectories of the particles for two different initial configurations, (b) phase portrait, and (c) spectrogram obtained from the FFT for run B.

Figure 16

Chaotic attractor (a) and the corresponding spectrogram (b) for $\mathrm{A}{\mathrm{r}}^{*}=400$.

Figure 17

Terminal Reynolds number ${\mathrm{Re}}_T$ versus the generalized Archimedes number $\mathrm{A}{\mathrm{r}}^{*}$ and the corresponding principal movement states for the sedimentation of a particle pair in a shear-thinning power-law fluid ($n=0.8$). Red squares and blue circles represent the oscillating state and steady doublet state, respectively. Three gray regions marked by I, II, and III represent the multiple stable states where two different states occur at an identical $\mathrm{A}{\mathrm{r}}^{*}$.

Figure 18

DKT and non-DKT states obtained at different values of $\mathrm{A}{\mathrm{r}}^{*}$ and $n$.

Figure 19

Viscosity fields at $\mathrm{A}{\mathrm{r}}^{*}=400$ for three different power-law indexes: (a) $n=0.8$, (b) $n=0.9$, and (c) $n=0.99$.

Figure 20

Time evolution of horizontal resistant force $F_x$ (normalized by the particle pure gravity $F_g$) of left particle for three different power-law indexes at $\mathrm{A}{\mathrm{r}}^{*}=400$.

Figure 21

Schematic diagram of a spherical particle settling in an enclosure filled with Newtonian fluid.

Figure 22

Time evolution of (a) the particle trajectory and (b) the particle velocity for a spherical particle settling in an enclosure, where the previous experimental and numerical results are also shown for comparison.