Fluid transport and mixing by an unsteady microswimmer

We study the fluid drift due to a time-dependent dumbbell model of a microswimmer. The model captures important aspects of real microswimmers such as a time-dependent flagellar motion and a no-slip body. The model consists of a rigid sphere for the body and a time-dependent moving Stokeslet representing the flagella. We analyze the paths of idealized fluid particles displaced by the swimmer. The simplicity of the model allows some asymptotic calculations very near and far away from the swimmer. The displacements of particles near the swimmer diverge in a manner similar to an isolated no-slip sphere, but with a smaller coefficient due to the action of the flagellum. Far from the swimmer, the time dependence becomes negligible due to both being very fast and decaying with distance. Finally, we compute the probability distribution of particle displacements and find that our model has fatter tails than previous steady models, due to the presence of a no-slip surface that drags particles along.

Figure 1

(a) Diagram of Chlamydomonas reinhardtii and the swimming stroke of its flagella. (b) Time-dependent dumbbell swimmer model, swimming to the right.

Figure 2

Time-averaged velocity fields of a three-Stokeslet model similar to that of [60] for (a) the in-plane cross section [$xy$ plane containing flagella; compare to Fig. 2(a) in 61], (b) the out-of-plane cross section ($xz$ plane), and (c) the azimuthally averaged flow. (d) Time-averaged velocity field of the dumbbell model.

Figure 3

(a) Position of the body sphere center $A\left(t\right)$ and the flagellar Stokeslet $a\left(t\right)$, also showing the extent of the body. These are in the comoving frame during one full period ($\tau =0.5T$), plotted along the nondimensionalized axes using the scales in Table 2. (b) Instantaneous swimming speed for the dumbbell model with the parameters in Table 2 (solid line) compared to the measurements of Guasto et al. [61] (dashed line). The agreement is only qualitative.

Figure 4

Particle trajectories in the laboratory frame starting at $(x_0,y_0,0)$. The swimmer travels a net distance of $\lambda =40U\tau$. From top to bottom, $ln(y_0/R)=1,0,-1,-2,-3$. The trajectories are offset vertically for clarity. The initial position of the particles is marked by closed circles and the final position by open circles. The sharp spikes in the trajectories are due to the back-and-forth motion of the flagellum and body.

Figure 5

Plot of particle displacements ${\mathrm{\Delta }}_{\lambda }(x_0,y_0)/R$ as a function of initial particle position $(x_0,y_0,0)$ for a swimmer starting at $(A\left(0\right),0,0)$ and swimming for 20 periods (or a net distance of $\lambda \approx 9.8R$). The white disk is the initial position of the swimmer's body.

Figure 6

The swimmer starts centered at the origin and swims for 100 periods (a distance of about $49R$), passing through the initial square of fluid particles (dashed line) and deforming it (solid line). The time dependence creates characteristic lobe structures.

Figure 7

Averaged tangential speed (29), after dropping the lead coefficient. The dashed line is for an isolated rigid sphere ($\mathcal{W}\equiv 0$) moving at the same speed. The front of the swimmer is to the right.

Figure 8

The averaged streamline (35) (solid line) is closer to the swimmer's body than for an isolated no-slip sphere ($\mathcal{W}\equiv 0$, right) moving at the same speed. The streamline is closest to the body at the front of the swimmer (right).

Figure 9

Ratio of final to initial $y$ value, representing the jump in streamline due to the time dependence. The phase is expressed as a fraction of the period $\tau$. The dashed line is the time-averaged expression (36).

Figure 10

Net particle displacement ${\mathrm{\Delta }}_{\lambda }$ as a function of the the initial ($y_0$, dotted line) and final ($y_1$, solid line) distances from the swimming axis. The particle starts far ahead of the swimmer and ends far behind. The dashed line is the asymptotic form (40), which agrees with the displacement as a function of the final position $y_1$.

Figure 11

Particle paths (a) near the swimmer and (b) far from the swimmer. Paths caused by the full model (solid lines) are from Sec. 2b and the far-field approximation of the mean flow (dashed lines) is from Sec. 5a. Better agreement is seen for particle paths further from the swimmer.

Figure 12

Integrand of the enhanced diffusivity integral (55) with $\lambda \approx 9.8R$. The inset is the steady stresslet approximation, with ${\beta }_{\left(0\right)}\approx 5.2R^2$.

Figure 13

(a) Dimensionless values of the second moment of particle displacements and (b) effective diffusivity for varying path lengths of swimmers.

Figure 14

Probability distribution function of particle displacements for varying swimmer volume fractions for steady squirmers (dashed line) and the time-dependent model in this paper (solid line). The average stresslet strength $\beta \approx 5.2$ is the same in both models. The swimmers move a net distance of $\lambda /R\approx 3$, which is comparable to the experiments of Leptos et al. [5].