Dispersion of model microorganisms swimming in a nonuniform suspension

Although diffusion properties of a suspension of swimming microorganisms in equilibrium have been studied intensively, those under nonequilibrium conditions remain unclear. In this study, we investigate the spreading of model microorganisms from high concentration to low by the Stokesian dynamics method. The results reveal that the spreading is neither purely diffusive nor ballistic. When the dipole component of the swimming velocity is small, the cells actively direct themselves towards lower concentrations. The concentration distribution shows stronger oscillations than would be expected for ballistic swimmers with constant orientations. The mechanism can be explained by the near-field hydrodynamic interactions between cells and the spatial gradient of the collision rate. Comparison of the numerical results with a simple continuum model and a Monte Carlo simulation shows that those conventional models can capture the basic features of the present results. These new findings pave the way towards a mathematical description of the dispersion of microorganisms in various environments.

Figure 1

Distribution of 900 squirmers per unit computational domain at $t=0$ and 20 $(c_0=0.1,\beta =1$, and $L=30$). Tops and bottoms of symmetry axes of squirmers are shown as white and black points, respectively. The computational cell is located at the center as indicated by the white lines.

Figure 2

Time change of concentration and orientation distributions during $t=0-30 (c_0=0.1,\beta =1$, and $L=30$): (a) dimension-free concentration distribution and (b) $x$ component of the orientation vector averaged over squirmers at position $x$.

Figure 3

The effect of swimming mode $\beta$, average concentration $c_0$, and the domain size $L$ on the time change of $I_{\mathrm{dif}}$, where the analytical results of the ballistic spreading are drawn for comparison. (a) Effect of $\beta$ ($c_0=0.1$ and $L=30$), (b) effect of $c_0$ and $L \left(\beta =0\right)$, and (c) effect of $L$, where the horizontal axis is scaled by $t/L^2$ ($c_0=0.1$ and $\beta =5$).

Figure 4

The effect of swimming mode $\beta$ on the cell orientation. (a) The $x$ component of the averaged orientation vector at $t=10$, where the ballistic spreading is drawn for comparison ($c_0=0.1$ and $L=30$). (b) Correlation between the nearest-neighbor distance $r_{\mathrm{near}}$ and the orientation change of the two squirmers $\Delta e_{\mathrm{near}}$. ABS stands for average of the absolute values.

Figure 5

Behavior of cells initially placed in the region $L/6<x<2L/6$ ($c_0=0.1$ and $L=30$): (a) trajectories of cells with their initial orientation $|e_{x,0}|<1/2 \left(t=0-8,\beta =0\right)$, where cells with $e_{x,0}>1/2$ are shown in blue (gray), (b) the collision rate for particles moving toward lower or higher concentration as a function of $c_0$ ($0.5\le c_0\le 0.15,-5\le \beta \le 5$ and $L=30$), and (c) the collision rate rescaled by $c_0(1+2dR/L)$, where all data are plotted ($0.5\le c_0\le 0.15,-5\le \beta \le 5$ and $30\le L\le 60$).

Figure 6

Comparison of results between the Stokesian dynamics simulation (SD simulation) and the diffusion-drift model (DD analysis): (a) effect of $L$ ($c_0=0.1,\beta =5$ and $\kappa =9$) and (b) effect of $\beta$ ($c_0=0.1,L=30,\kappa =9$ for $\beta =-5$, and $\kappa =6$ for $\left|\beta \right|=1$).

Figure 7

Comparison of results between the Stokesian dynamics simulation (SD simulation) and the Monte Carlo simulation (MC simulation). The MC simulation was performed with ${\kappa }_2=1.5$ and $\lambda =0$ as well as with ${\kappa }_2=0$ and $\lambda =0.003$. The results of the ballistic limit are plotted for comparison: (a) effect of $L$ on $I_{\mathrm{dif}}$ ($c_0=0.1,\beta =5$ and ${\kappa }_2=0.9$), (b) effect of $\beta$ on $I_{\mathrm{dif}}$ ($L=30$ and $c_0=0.1$), and (c) $x$ component of the averaged orientation vector at $t=10 (L=30$ and $c_0=0.1$).