Hydrodynamical characteristics of a pair of elliptical squirmers in a channel flow of power-law fluids

Abstract  

The locomotion state and motion type of elliptical squirmers in a channel flow of power-law fluids are simulated numerically. Three locomotion states (independent, coupled, related) and three types of motions (upstream, intermediate, downstream) for pairs of squirmers are found and identified. The effect of height difference (0.5 ~ 10) between the initial positions of two squirmers, aspect ratio (0.3 ~ 1.0), particle Reynolds numbers (0.5 ~ 10), self-propelling strength of the squirmers (− 9 to 9), and power-law index (0.4 ~ 1.5) of the fluid on the locomotion state and motion type of a pair of squirmers are explored, and the corresponding hydrodynamical characteristics are analyzed in detail. Head-to-head coupled structures and body-to-body coupled structures are observed for a pair of pullers and a pair of pushers, respectively. It is found that coupled structures are easy to be broken for squirmers with larger aspect ratio or larger particle Reynolds number and self-propelling strength. The movement characteristics of squirmers are closely related to the initial positions of squirmers in strong shear-thinning fluid, but not to the initial positions in strong shear-thickening fluid. The dependence of viscosity on shear will also significantly affect the flow velocity, thus changing the motion type of squirmers.

Graphical Abstract

Appendix 1

Validation test and feasibility of stiffness parameter.

Validation of the power-law flow

The theoretical result for the power-law flow in a two-dimensional channel had been proposed by Bird et al. (Bird et al. 1987) as:

$$u\left(h\right)\text{=}{U}_{\mathrm{max}}\left[\text{1} - {\left(\frac{h}{0.5H}\right)}^{1+\frac{1}{n}}\right]$$

(34)

where u(h) and U_max are the velocity distribution and the maximum velocity, respectively; h and H are the vertical distance from the centerline and the height of the channel, respectively. The computational domain of this verified case is the same as that of mentioned in problem definition. As shown in Fig. 19, the present results are in good agreement with the theoretical ones.

Fig. 19

Comparison between present results and the available velocity profiles of the channel flow at different power-law indexes

Validation of a squirmer in a Newtonian channel flow

In this section, we used the power-law fluid model to reappear the cases in former work (Liu et al. 2022a)—the migration and rheotaxis of a single squirmer in a Newtonian channel flow with AR = b/a = 0.8, Re_p = U(2a)/ν = 0.5, β = 3, and κ = 0.17. In which a is the characteristic length and is set to 10. Correspondingly, in present simulation, we have m = ρU⁽²⁻ⁿ⁾(2a)ⁿ/Re_p, where n = 1, a = 10, Re_p = 0.5. As shown in Fig. 20, both trajectories and orientations of the squirmer can be successfully reappeared by the using of the power-law fluid model.

Fig. 20

Comparison of a trajectories and b orientations in the present study and that in former work for a single squirmer in a channel flow

Feasibility of parameters selection in collision model

The stiffness parameter ε for the repulsive force is suitable for a squirmer in both of shear-thinning and shear-thickening flows. We compare the trajectories of two squirmers with different ε in Fig. 21 a and b. As Δh = 1.0 in these cases, the repulsive force works at the beginning of two squirmers’ movement. Figure 21a and b show that the trajectories of squirmers in flows with n = 0.4 and n = 1.4 are independent of ε from 10⁻⁶ to 10⁻⁴. Thus, it is feasible to set ε = 10⁻⁵ in this paper.

On the other hand, in Fig. 21c and d, the trajectories in shear-thinning flows are dependent on ξ. As the ξ determines the shortest distance between two squirmers where repulsive force works. That is, for cases with Δh = 1.0, the repulsive force works at the beginning of the squirmers movement with ξ ≤ 1.0 while it does not work for the cases with ξ > 10. Thus, the presence of repulsive force makes a difference for the trajectories of squirmers with ξ = 0.5/1.0 and ξ = 1.5. However, the trajectories of ξ = 0.5 and ξ = 1.0 are coincident. On the other hand, trajectories of squirmers in shear-thickening flows are independent with ξ. Hence, it is feasible to set ξ = 1.0 in this paper.

Fig. 21

The trajectories of a pair of squirmers for a, c n = 0.4 and b, d n = 1.4 with a, b different ε and c, d different ξ. (β = 3, AR = 0.7, Re_p = 0.5)