Gyrotactic suppression and emergence of chaotic trajectories of swimming particles in three-dimensional flows

We study the effects of imposed three-dimensional flows on the trajectories and mixing of gyrotactic swimming microorganisms and identify phenomena not seen in flows restricted to two dimensions. Through numerical simulation of Taylor-Green and Arnold-Beltrami-Childress (ABC) flows, we explore the role that the flow and the cell shape play in determining the long-term configuration of the cells' trajectories, which often take the form of multiple sinuous and helical “plumelike” structures, even in the chaotic ABC flow. This gyrotactic suppression of Lagrangian chaos persists even in the presence of random noise. Analytical solutions for a number of cases reveal the how plumes form and the nature of the competition between torques acting on individual cells. Furthermore, studies of Lyapunov exponents reveal that, as the ratio of cell swimming speed relative to the flow speed increases from zero, the initial chaotic trajectories are first suppressed and then give way to a second unexpected window of chaotic trajectories at speeds greater than unity, before suppression of chaos at high relative swimming speeds.

Figure 1

Simulations of cells in the TGV flow with $V=0.1$ and $G=1$: (a) $\alpha =0$, (b) $\alpha =0.5$, and (c) $\alpha =1$.

Figure 2

Simulations of cells in the TGV flow with $V=3$ and $G=1$: (a) $\alpha =0$, (b) $\alpha =0.5$, and (c) $\alpha =1$.

Figure 3

Projections of cell trajectories in the TGV flow for $V=3$ and $G=1$.

Figure 4

Simulations of cells in the TGV flow with $V=9$ and $G=1$: (a) $\alpha =0$, (b) $\alpha =0.5$, and (c) $\alpha =1$.

Figure 5

Graph of the principal Lyapunov exponent when $G=1$ corresponding to the flows illustrated in Tables 3 and 4, showing the existence $\left(\lambda >0\right)$ and suppression $\left(\lambda <0\right)$ of Lagrangian chaos.

Figure 6

Graph of the principal Lyapunov exponent when $G=4$ corresponding to the flows illustrated in Table 5, showing the existence $\left(\lambda >0\right)$ and suppression $\left(\lambda <0\right)$ of Lagrangian chaos.

Figure 7

Simulations of cells in the ABC flow with $V=0.1$ and $G=1$: (a) $\alpha =0$, (b) $\alpha =0.5$ and (c) $\alpha =1$.

Figure 8

Graph of the principal Lyapunov exponent when $V=4$ corresponding to the flows illustrated in Table 6, showing the existence $\left(\lambda >0\right)$ and suppression $\left(\lambda <0\right)$ of Lagrangian chaos.

Figure 9

Simulations of cells in the ABC flow with $V=9$ and $G=1$: (a) $\alpha =0$, (b) $\alpha =0.5$, and (c) $\alpha =1$.

Figure 10

Time for cells to complete one circuit of a plume when $G=0.1$ and $\alpha =0$.

Figure 11

Simulations of cells in the ABC flow with $V=4$ and $G=4$: (a) $\alpha =0$, (b) $\alpha =2/3$, and (c) $\alpha =1$.

Figure 12

Projections of cell trajectories in the ABC flow with $\alpha =2/3,V=4$, and $G=4$.

Figure 13

Change in $p_x$ for one period from $t=300$ to 319.03 for the ABC flow with $V=4,G=4$, and $\alpha =2/3$.

Figure 14

Distance between cells and their nearest neighbor, averaged over all cells.

Figure 15

Poincaré sections in the $z=0$ plane with $V=G=4$ and $\alpha =2/3$, showing the effect of noise on cell trajectories when (a) $\eta =0$, (b) $\eta =0.0001$, (c) $\eta =0.001$, and (d) $\eta =0.01$.