Base flow decomposition for complex moving objects in linear hydrodynamics: Application to helix-shaped flagellated microswimmers

The motion of microswimmers in complex flows is ruled by the interplay between swimmer propulsion and the dynamics induced by the fluid velocity field. Here we study the motion of a chiral microswimmer whose propulsion is provided by the spinning of a helical tail with respect to its body in a simple shear flow. Thanks to an efficient computational strategy that allowed us to simulate thousands of different trajectories, we show that the tail shape dramatically affects the swimmer's motion. In the shear dominated regime, the swimmers carrying an elliptical helical tail show several different Jeffery-like (tumbling) trajectories depending on their initial configuration. As the propulsion torque increases, a progressive regularization of the motion is observed until, in the propulsion dominated regime, the swimmers converge to the same final trajectory independently on the initial configuration. Overall, our results show that elliptical helix swimmer presents a much richer variety of trajectories with respect to the usually studied circular helix tails.

Figure 1

Sketch of microswimmer locomotion in a shear flow. The microswimmer body is a prolate ellipsoid of major axis $r_{h1}$ and minor axis $r_{h2}$. In (a) model I, the tail is a circular helix, i.e., a helix built on a circular cylinder of radius $r_{t1}$, whereas in (b) model II, the tail is an elliptical helix, i.e., a helix built on an elliptical cylinder of radii $r_{t1}$ and $r_{t2}=3r_{t1}$. (c) The shear flow is on the ${\mathbf{X}}_1{\mathbf{X}}_3$ plane of the global coordinate frame $\mathbf{O}{\mathbf{X}}_1{\mathbf{X}}_2{\mathbf{X}}_3$. A body coordinate frame ${\mathbf{O}}^{\prime }{\mathbf{e}}_1{\mathbf{e}}_2{\mathbf{e}}_3$ moves with the swimmer body. The polar $\theta$ azimuthal $\varphi$ and rotation $\psi$ angles are used to describe the orientation of the body frame with respect to the global frame. Moreover, we also define the angle between the swimmer axis $\mathit{p}={\mathbf{e}}_3$ and ${\mathbf{X}}_2$ as $\eta =arccos(sin\theta sin\varphi )$. The tail rotates with respect to the ${\mathbf{e}}_3$ so that each point of the rigid tail describes a circle on plane ${\mathbf{e}}_2{\mathbf{e}}_3$ with ${\psi }_t$ as the corresponding rotation angle. The motion of the microswimmer body is completely defined when the translational velocity $\mathit{U}$ of the body center, the body rotational velocity $\mathit{\Omega }$, and the tail spinning ${\omega }_t=\dot{{\psi }_t}$ are given.

Figure 2

Microswimmer motion in a simple shear flow. For each case, from ${10}^3$ to ${10}^4$ simulations with different initial conditions were run. Panel (a) reports the fraction of the circular helix swimmers that have a drift velocity oriented as ${\mathbf{X}}_2(U_2>0)$. The bars in panel (a) indicate the lateral velocity $U_2$ for the different initial conditions whereas the bars in panel (b) refer to the normalized average angle $\langle \eta /\pi \rangle$ between ${\mathbf{X}}_2$ and the microswimmer head orientation $\mathit{p}$. In the passive regime, all the trajectories converge to the same final state [${\mathrm{I}}_1$, panel (e)] where the swimmer is oriented as $-{\mathbf{X}}_2$ whereas its velocity is $U_2\cong {10}^{-4}$. In the active regime, the swimmer is again oriented as $-{\mathbf{X}}_2$, but $U_2<0$, see configuration (f) ${\mathrm{I}}_2$ and (g) ${\mathrm{I}}_3$. The dot-dashed line corresponds to the fraction of swimmers for which $U_2>0$. Panels (c) and (d) refer to velocity $U_2$ and orientation $\langle \eta /\pi \rangle$ for the elliptical helix tail. Here, a new intermediate regime appears between the active and passive regimes. In this intermediate regime, both positive and negative drift velocities $U_2$ are possible. Panels (e)–(k) report examples of the swimmer Jeffery-like tumbling motion (shear on the $X1X3$ plane). The solid lines on the spheres represent the direction of the swimmer axis $\mathit{p}$ along one period whereas the red and blue arrows refer to the direction of the average velocity along $X_2$.

Figure 3

Elliptical helix tail microswimmer. (a) Lateral velocity $U_2$, (b) normalized average angle $\langle \eta /\pi \rangle$, and (c) the absolute tail spin ${\dot{\psi }}^{\prime }=\dot{\psi }+{\dot{\psi }}_t$ as functions of ${\omega }_t$. Results refer to 450 trajectories for each ${\omega }_t$. The green color scale indicates the probability that one initial condition converges to the corresponding value on the vertical axis, for instance, in the passive case, almost 75% of the swimmers converge to the trajectory ${\mathrm{II}}_2$ that corresponds to $\langle \eta \rangle \cong 0.7\pi$ (dark green). The black solid line in panel (a) is the velocity of the same microswimmer in a fluid at rest. Panels (d) and (e) report a detailed view of the regions enclosed by the violet dotted boxes. The freezing tail phenomenon discussed in the text is sketched in panels (f) and (g) whereas the corresponding average ${\psi }^{\prime }$ is reported as a solid blue line between $0.8<{\omega }_t<3.2$ in panel (c).

Figure 4

Sketch of the method of fundamental solutions. (a) A solid body (domain ${\mathrm{\Omega }}_1$) moves in a bulk fluid (domain ${\mathrm{\Omega }}_2$). The solid blue line $\partial {\mathrm{\Omega }}_1$ is the boundary of the solid body. Boundary points ${\mathit{x}}_i^v$ (blue circles) are selected on $\partial {\mathrm{\Omega }}_1$ whereas source points ${\mathit{x}}_i^f$ (red squares) are placed inside the solid body. Panel (c) shows the discretization used for a quarter of an ellipse whereas the swimmer head is in panel (d). Panel (e) reports the tail centerline whereas panel (f) refers to the discretization of the swimmer tail. Each section of the tail is modeled as a circle where, again, red squares correspond to force sources and blue circles to the boundary. In panel (g) a short section of the swimmer tail is shown.

Figure 5

Sketch of the kinetic decoupling of the microswimmer in an external flow. (a) Active microswimmer motion $({\tilde{\mathit{U}}}_a,{\tilde{\mathit{\Omega }}}_a)$ in bulk fluid at rest. (b) Passive microswimmer translation ${\mathit{u}}_b$ in the external flow. (c) Passive microswimmer motion $({\tilde{\mathit{U}}}^E,{\tilde{\mathit{\Omega }}}^E)$ in the symmetric (deviatoric) part of the external flow. (d) Passive microswimmer rotation ${\mathit{\Omega }}_p^S$ in the antisymmetric part of the external flow.

Figure 6

Validation of the numerical method: Jeffery orbits. (a) Sketch of an ellipse orbit in a shear flow. The shear flow is on the ${\mathbf{X}}_1{\mathbf{X}}_3$ plane of the global coordinate frame $\mathbf{O}{\mathbf{X}}_1{\mathbf{X}}_2{\mathbf{X}}_3$. The polar $\theta$ and azimuthal $\varphi$ angles are used to describe the orientation of the body frame with respect to the global frame. The unit vector $\mathit{p}$ denotes the orientation of the ellipsoid. (b) and (c) Time evolution of angles $\theta$ and $\varphi$ for an ellipse with aspect ratio $r_{h1}/r_{h2}=3$ moving in a shear flow. The initial orientation of the ellipse is $(\theta =0.21\pi ,\varphi =0.23\pi )$. Orange points represent our numerical solution whereas the analytical solution [35] are reported as blue lines.