Dynamics of a droplet driven by an internal active device

A liquid droplet, immersed into a Newtonian fluid, can be propelled solely by internal flow. In a simple model, this flow is generated by a collection of point forces, which represent externally actuated devices or model autonomous swimmers. We work out the general framework to compute the self-propulsion of the droplet as a function of the actuating forces and their positions within the droplet. A single point force, $\mathbf{F}$, with general orientation and position, ${\mathbf{r}}_0$, gives rise to both translational and rotational motion of the droplet. We show that the translational mobility is anisotropic and the rotational mobility can be nonmonotonic as a function of $|{\mathbf{r}}_0|$, depending on the viscosity contrast. Due to the linearity of the Stokes equation, superposition can be used to discuss more complex arrays of point forces. We analyze force dipoles, such as a stresslet, a simple model of a biflagellate swimmer and a rotlet, representing a helical swimmer, driven by an external magnetic field. For a general force distribution with arbitrary high multipole moments the propulsion properties of the droplet depend only on a few low order multipoles: up to the quadrupole for translational and up to a special octopole for rotational motion. The coupled motion of droplet and device is discussed for a few exemplary cases. We show in particular that a biflagellate swimmer, modeled as a stresslet, achieves a steady comoving state, where the position of the device relative to the droplet remains fixed. There are two fixed points, symmetric with respect to the center of the droplet. A tiny external force selects one of them and allows one to switch between forward and backward motion.

Figure 1

Decomposition of the interior of the droplet into an inner sphere $r\le r_0$ and an outer shell $r_0\le r\le R$, used for the construction of the flow field generated by a point force $\mathbf{F}\delta (\mathbf{r}-{\mathbf{r}}_0)$ in the interior.

Figure 2

(a) Upper half of the droplet with examples of devices of point forces at ${\mathbf{r}}_0$. (b) Stresslet. (c) Rotlet. (d) Quadrupolar device. The point forces are indicated by vectors.

Figure 3

Biflagellate swimmer, modeled by a small sphere with an attached stresslet and subject to an external force ${\mathbf{F}}^{\text{ext}}$, which will be tuned to the swimming activity to achieve a stable comoving state of droplet and device. All length scales are assumed to be small as compared to the droplet's radius.

Figure 4

Device velocity ${\mathit{U}}_{\text{device}}$ (solid line) and droplet velocity (dashed line) generated by the device described in the main text with point forces $F$ of 1 pN at a distance $d=2\mu \text{m}$ from a bead with radius $1\mu \text{m}$, in a droplet of radius $50\mu \text{m}$. Top: Without steering force $F^{\text{ext}}=0$. Stable fixed points are marked by circles; unstable fixed points are marked by squares. Bottom: With $F^{\text{ext}}=0.001\text{pN}$. The viscosity contrast is $\lambda =5$ and contributions up to $l=50$ are taken into account. No visible changes appear if contributions up to $l=100$ are added.