Exact trajectory solutions of a spherical microswimmer under flow and external fields

The main objective of the present study is the derivation of exact analytical expressions for the orientation and trajectory of a spherical microswimmer submitted to general linear flows and to an external (gravitational or magnetic) force field, a problem known as gyrotaxis or magnetotaxis. We consider linear shear, hyperbolic, solid-rotation, and stagnation flows. The evolution equations of the swimmer orientation and its position are nonlinear and analytical results are the exception rather than the rule. Most available results for cell orientation and trajectories are obtained numerically. The solution for the swimmer orientation is inspired from a method due to Bretherton, initially developed for a different nonlinear equation. We show here that this method can be generalized to our evolution equation. We see that the swimmer under flow exhibits both run (a motion where the orientation angle is kept constant with time) and tumble (the orientation angle is cyclic with time) regimes, and a variety of cell trajectories are extracted analytically, such as parabolic, elliptic, and helical. This study offers a framework to generalize the results to other types of flows.

Figure 1

Motion of orientation $\mathit{p}$ in the tumbling case. $\tilde{\lambda }=0.5,\dot{\gamma }=1$, and $a=4$.

Figure 2

Example of a periodic trajectory in the absence of external torque [Eqs. (39, 40, 41)]. Parameters are ${\theta }_0=\pi /2,a_0=0.5,v_s=1$, and $\dot{\gamma }=2.$

Figure 3

Trajectory showing a cycloid in the absence of external torque [Eqs. (39, 40, 41)]. Parameters are ${\theta }_0=\pi /3,a_0=0.5,v_s=1$, and $\dot{\gamma }=2.$

Figure 4

Trajectory showing a helical shape. $\tilde{\lambda }=0.6, v_s=1, \dot{\gamma }=1, \mathrm{\Gamma }=0.8$.

Figure 5

Run trajectories when vorticity and external torques (gravity) are perpendicular. The fluid velocity is given by (45). Parameters are $\tilde{\lambda }=1.5,{\dot{\gamma }}_1=2$, and $v_s=0.5.$ For the parabola (dashed-dotted blue line) ${\dot{\gamma }}_2=0.5$ and for the ellipse (solid red line) ${\dot{\gamma }}_2=-0.5.$ For both trajectories the initial condition is $x\left(0\right)=y\left(0\right)=0.$

Figure 6

A tumble trajectory in the $x-y$ plane according to (59) with ${\mathbf{x}}_c^{\prime }\left(0\right)=0.$ Parameters are $\tilde{\lambda }=0.6, v_s=4, \dot{\gamma }=1,{\dot{\gamma }}_1=0.5,$ and $a=3.$

Figure 7

Trajectory showing a helical shape. $\tilde{\lambda }=0.6, v_s=1, \dot{\gamma }=1, \mathrm{\Gamma }=0.8$.

Figure 8

An example of trajectories showing that the microswimmer approaches a pseudoattractor. Parameters are $\lambda =1,{\tilde{\lambda }}_g=2,\dot{\gamma }=1,{\theta }_0=\pi /3,v_s=1$, and $d=4.$

Figure 9

Different $x-y$ trajectories of the microswimmer as a function of the initial position [$x\left(0\right),y\left(0\right)$] and the parameter $d$ or the initial orientation ${\varphi }_0$. Here, the parameters are $\lambda =1,{\tilde{\lambda }}_g=2,\dot{\gamma }=1,{\theta }_0=\pi /3,v_s=1$, and $d=4$ (dotted-dashed brown line), $d=8$ (solid black line), and $d=20$ (dashed blue line). Trajectories show that the microswimmer approaches a circular path in the $x-y$ plane.

Figure 10

Exact solutions to the ordinary differential equation, (95), with ${\dot{\gamma }}^2/v_0=1,v_s=1,\rho \left(0\right)=2$ and for different values of $w_3$ ($w_3^2=3$, solid brown line; $w_3^2=3.25$, dotted-dashed black line; and $w_3^2=3.5$, dotted blue line). $\rho$ reaches the fixed point $2v_0/\dot{\gamma }$ at finite time, $t^{★},$ and satisfies $\rho =2v_0/\dot{\gamma }$ for all $t>t^{★}.$

Figure 11

An example of trajectories, for large $t,$ of a gyrotactic swimmer in a fluid acceleration. Parameters are $\dot{\gamma }=1,\kappa =1,v_s=1,a_0=2$, and ${\theta }_0=\pi /3.$ The particle reaches the $z$ axis.

Figure 12

Projection of a particle trajectory in the $x-y$ plane according to Eq. (121). Parameters are $\delta =1,v_s=1$, and ${\mathit{j}}_i^{★}=1/\sqrt{3},i=1,2,3.$ The initial condition is $x\left(0\right)=0$ and $y\left(0\right)=1$ for the solid red curve, which is a part of the full hyperbolic trajectory, (121); the complementary trajectory is shown as the dotted blue line.