3.1 General tractions

To determine the droplet trajectories, we start from the $\ell =1$ modes of active tractions (see appendix A) at a fixed time $t$ , which we write as

Here, $\unicode[STIX]{x1D735}_{s}$ denotes the angular part of the gradient operator, i.e. $\unicode[STIX]{x1D735}_{s}g(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})=r\unicode[STIX]{x1D735}g(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ . The $\ell =1$ part of the active traction field is determined by nine multipole moments $\unicode[STIX]{x1D6FC}_{m}$ , $\unicode[STIX]{x1D6FD}_{m}$ , $\unicode[STIX]{x1D6FE}_{m}$ , which correspond to nine real parameters. For any given traction field, these moments are obtained by projection of $\boldsymbol{t}^{\,act}$ onto the corresponding vector spherical harmonics. Higher- $\ell$ moments do not contribute to self-propulsion.

We solve the Stokes equations in the interior and exterior regions of the droplet and find for the velocity fields in the LF

The expansion coefficients are determined by the boundary conditions of (2.2), (2.3), as shown in detail in appendix A.2.

The traction forces are of two kinds. The first one corresponds to tractions with $\unicode[STIX]{x1D6FE}_{m}=0$ that give rise to a linear velocity of the droplet but no rotational velocity. The centre of mass velocity is determined by the flow field according to (2.6). Straightforward computation yields

with

Here, the ratio of interior to exterior viscosity has been denoted by $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D702}^{-}/\unicode[STIX]{x1D702}^{+}$ . Such tractions can, for example, be generated by Marangoni flow, which is represented by an inhomogeneous surface tension $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ . For this example, the tractions are given by

and hence cannot give rise to a rotational velocity of the droplet.

The second class of tractions has only $\unicode[STIX]{x1D6FE}_{m}\neq 0$ . The internal flow, as given in (3.3), corresponds to rigid body rotation $\boldsymbol{v}^{-}(\boldsymbol{r})=\unicode[STIX]{x1D74E}\times \boldsymbol{r}$ , with the rotational velocity given by

3.2 Force- and torque-free case

In this section, we discuss the special case where no external force and no external torque are present, as required for an autonomous swimmer. We first work out the consequences for self-propulsion by tractions and then compare our results with the squirmer model, in which an active velocity on the surface of a particle is described. The squirmer model has been discussed extensively in the literature, mainly for a solid particle.

Vanishing of the torque, $\boldsymbol{N}=\int _{\unicode[STIX]{x2202}K(t)}\,\text{d}^{2}S\boldsymbol{r}\times \boldsymbol{t}_{act}=8\unicode[STIX]{x03C0}a^{3}\unicode[STIX]{x1D702}^{+}\unicode[STIX]{x1D74E}=0$ , implies that all $\unicode[STIX]{x1D6FE}_{m}=0$ . In other words, tractions cannot generate rotational motion of an autonomous swimmer. Vanishing of the total force, $\boldsymbol{F}=\int _{\unicode[STIX]{x2202}K(t)}\,\text{d}^{2}S\boldsymbol{t}_{act}=0$ , implies a restriction on the multipole moments of the traction forces, namely $\unicode[STIX]{x1D6FC}_{m}=-2\unicode[STIX]{x1D6FD}_{m}$ . Consequently, the Stokeslet vanishes, i.e. $a_{m}^{+}=0$ , and the exterior flow field with $\ell =1$ decays as $1/r^{3}$ . The flow fields outside and inside the force-free droplet are explicitly given by

and the centre of mass velocity follows from equations (3.4) and (3.5) or directly from the constant part of the velocity field in the interior of the droplet,

Instead of considering a force-free droplet, one can ask what force is needed to keep the active droplet at rest in the LF. We now set $\boldsymbol{v}_{CM}=0$ and compute the stall force,

As one would expect, the stall force is proportional to $\boldsymbol{v}_{CM}$ ,

The mobility $\unicode[STIX]{x1D707}$ is seen to depend on both the interior and exterior viscosities. This reflects the fact that traction forces generate active surface velocities that depend on both viscosities. In the limit that the fluid in the interior becomes highly viscous or even solid-like ( $\lim \unicode[STIX]{x1D702}^{-}\rightarrow \infty$ ), the mobility reduces to the one for a solid particle, $\unicode[STIX]{x1D707}=1/(6\unicode[STIX]{x03C0}a\unicode[STIX]{x1D702}^{+})$ . In the opposite limit $\lim \unicode[STIX]{x1D702}^{-}\rightarrow 0$ , we recover the mobility of a gas bubble, $\unicode[STIX]{x1D707}=1/(4\unicode[STIX]{x03C0}a\unicode[STIX]{x1D702}^{+})$ . In between, the mobility increases monotonically.

We now compare our results with the squirmer model, which prescribes the active velocity field on the surface of the sphere (Blake Reference Blake1971),

There is no radial component to the active velocity, because we require the shape and size of the droplet to be preserved. The exterior and interior flow fields $\boldsymbol{v}^{\pm }(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ have to fulfil the following boundary condition in the LF:

The centre of mass velocity is yet unknown and has to be determined from the vanishing of the total force. Solving the Stokes equation for $\boldsymbol{v}^{\pm }$ with the above boundary condition and the requirement of vanishing total force, we reproduce the result of (3.9) for the centre of mass velocity, if we identify the coefficients of the active tractions with those of the active velocity according to

The rotational motion of a squirmer has been discussed by Pak & Lauga (Reference Pak and Lauga2014) and more recently by Pedley, Brumley & Goldstein (Reference Pedley, Brumley and Goldstein2016). Only the $l=1$ component of the active velocity contributes to the rotational velocity of the squirmer and is equivalent to a rigid rotation of the surface with angular velocity $\unicode[STIX]{x1D74E}$ , i.e. $\boldsymbol{v}_{act}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})=a\unicode[STIX]{x1D74E}\times \boldsymbol{e}_{r}$ . The active velocity gives rise to a rotation in the LF which exactly cancels the above rotational component of $\boldsymbol{v}_{act}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ , so that the $l=1$ components of the external and internal flow fields are not affected. Rotational components with $l>1$ in $\boldsymbol{v}_{act}$ do modify the flow fields inside and outside of the droplet, but do not contribute to  $\unicode[STIX]{x1D74E}$ .

3.3 Energy dissipation

Energy is dissipated by both the exterior and the interior flows,

where we have used that $\unicode[STIX]{x2202}_{k}\unicode[STIX]{x1D70E}_{ik}^{\pm }=0$ and the normal is always defined as pointing outward from the sphere. The second inequality in the above equation just says that the energy dissipated in the volumes $V^{\pm }$ is balanced by the energy inflow from the boundary. The latter is more easily computed, and we obtain for the force-free case

so that the ratio is independent of the driving amplitudes $\{\unicode[STIX]{x1D6FD}_{m}\}$ ,

In the stationary state, the total dissipation is balanced by the energy input due to surface tractions, $\unicode[STIX]{x2202}_{t}E^{+}+\unicode[STIX]{x2202}_{t}E^{-}=\unicode[STIX]{x2202}_{t}E^{act}=\int _{\unicode[STIX]{x2202}V}\,\text{d}^{2}xt_{i}^{act}v_{i}$ .

3.4 Sedimenting active droplets

Consider the simple case of a passive droplet in an external potential, which is assumed to be continuous across the interface. For example, if the fluids are not density matched, then buoyancy forces act on the droplet. These can be represented by a potential $U=\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g(z-a)$ for $z\leqslant a$ , where $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ denotes the difference in density between the inside and outside fluids. We assume that the sedimentation velocity, which is controlled by $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ , is sufficiently small, so that it is justified to still work in the limit of vanishingly small Reynolds number.

In the general case, the potential is expanded in spherical harmonics $U=\text{const.}+\sum _{m}U_{m}(r)Y_{1m}$ , keeping only $l=1$ . In the interior of the droplet, we then have to solve the Stokes equation

in the presence of an external force $\boldsymbol{f}_{\!ext}=\unicode[STIX]{x1D735}U$ . The force derived from a potential can be absorbed by defining an effective pressure $p_{eff}=p^{-}-U$ . The effective pressure has to fulfil Laplace’s equation and determines the inhomogeneous solution of the Stokes equation in the same way as does $p^{-}$ without external force. The force balance equation for the interface (2.3) involves only the pressure $p^{-}$ , and hence the potential $U$ is equivalently represented as a traction force according to

If there are no active tractions or body forces, we recover the velocity of the sedimenting passive droplet, $\boldsymbol{v}_{CM}=\unicode[STIX]{x1D707}\boldsymbol{f}_{\!ext}$ , with the mobility $\unicode[STIX]{x1D707}=(1+\unicode[STIX]{x1D706})/(2\unicode[STIX]{x03C0}a\unicode[STIX]{x1D702}^{+}(2+3\unicode[STIX]{x1D706}))$ . In the general case of an active droplet in an external field, the motion is a linear superposition of the velocity generated by the activity and the velocity caused by the external field,

so that sufficiently strong activity may turn a sedimenting droplet into a rising one.

3.5 Conclusion: active tractions

We have analysed general active tractions in the interface of the droplet, and related the linear and angular momentum of the droplet to the vector multipole moments of the traction. For an autonomous swimmer, such tractions can only generate translational motion, which is equivalently generated by an active velocity on the surface of the droplet. Energy is dissipated both inside and outside of the droplet, with the ratio being determined by the two viscosities. A sedimenting active droplet is discussed as a simple example for an active swimmer in an external force field.

4.1 A class of body forces

We start from the $\ell =1$ part (see appendix A) of a given force density $\boldsymbol{f}^{\,act}$ ,

which is the analogue of (3.1). In the droplet interior, the Stokes equation becomes inhomogeneous,

Incompressibility requires $\unicode[STIX]{x0394}p^{-}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{f}^{\,act}$ . The general solution is given as $p^{-}=p_{hom}^{-}+p_{inh}^{-}$ as the sum of the general solution, $p_{hom}^{-}$ , of the homogeneous equation and a particular solution, $p_{inh}^{-}$ , of the inhomogeneous equation. The former has already been constructed in § 3. Similarly, the solution of (4.2) is decomposed into the previously found flow field for vanishing body force, $\boldsymbol{v}_{0}^{-}=\boldsymbol{v}^{-}(\,\boldsymbol{f}^{a}=0)$ , and a particular solution, $\boldsymbol{w}$ , of the inhomogeneous equation,

To obtain explicit expressions for $p_{inh}$ and $\boldsymbol{w}$ , we treat a class of body forces, $\boldsymbol{f}^{\,act}$ , consisting of linear combinations of power laws $\boldsymbol{f}^{\,act}(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})=\sum _{\unicode[STIX]{x1D708}}(r/a)^{\unicode[STIX]{x1D708}}\boldsymbol{f}_{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ , which may be adapted to specific situations. Due to the linearity of the problem, we can restrict the discussion to a single power, so that $\unicode[STIX]{x1D6FC}_{m}(r)=(r/a)^{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D6FC}_{m}$ , $\unicode[STIX]{x1D6FD}_{m}(r)=(r/a)^{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D6FD}_{m}$ and $\unicode[STIX]{x1D6FE}_{m}(r)=(r/a)^{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D6FE}_{m}$ . More general solutions are then obtained by superposition.

For $\unicode[STIX]{x1D708}\neq 0$ , a special solution of the inhomogeneous equation for the pressure, $\unicode[STIX]{x0394}p_{inh}^{-}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{f}^{\,act}$ , is given by

We substitute this solution into (4.3) and solve for $\boldsymbol{w}$ ,

where the coefficients are given by

After a special solution $\boldsymbol{w}$ has been found, the coefficients appearing in the homogeneous solution, $\boldsymbol{v}^{+}$ and $\boldsymbol{v}_{0}^{-}$ , equations (3.2) and (3.3), are obtained by matching the boundary conditions at $r=a$ , requiring continuity of the velocity equation (2.2), $\boldsymbol{v}^{+}=\boldsymbol{v}_{0}^{-}+\boldsymbol{w}$ and balance of tractions (2.3). The resulting system of linear equations is solved in appendix A.2 and discussed in the following sections for an autonomous swimmer.

4.2 Force-free translations and torque-free rotations

For an autonomous swimmer, we require that the total force and total torque of the force density given by (4.1) vanish, $\boldsymbol{F}=\int \boldsymbol{f}^{\,act}(\boldsymbol{r})\,\text{d}V=0$ and $\boldsymbol{N}=\int \boldsymbol{r}\times \boldsymbol{f}^{\,act}(\boldsymbol{r})\,\text{d}V=0$ . If $\unicode[STIX]{x1D6FC}_{m}(r)=-2\unicode[STIX]{x1D6FD}_{m}(r)$ , the forces within each radial shell $[r,r+\text{d}r]$ are balanced. Let us specialise to this simple case, with $\boldsymbol{f}^{\,act}=(r/a)^{\unicode[STIX]{x1D708}}\boldsymbol{f}_{\unicode[STIX]{x1D708}}$ and $\unicode[STIX]{x1D6FC}_{m}=-2\unicode[STIX]{x1D6FD}_{m}$ . Consequently, the amplitude of the Stokeslet vanishes. The centre of mass velocity can be calculated from (2.6), which yields

A balance in each radial shell is not possible for the torque. Therefore, within our class of models, a force density with vanishing total force and torque requires at least a sum of two different powers, $\boldsymbol{f}=r^{\unicode[STIX]{x1D708}}\boldsymbol{f}_{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})+r^{\unicode[STIX]{x1D707}}\boldsymbol{f}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ . As a simple example of a rotating torque-free droplet, let us consider an active body force density $\boldsymbol{f}^{\,act}(\boldsymbol{r})=\sum _{m}\unicode[STIX]{x1D6FE}_{m}(r)\boldsymbol{e}_{r}\times \unicode[STIX]{x1D735}_{s}Y_{1m}$ with

The total torque $\boldsymbol{N}$ vanishes, if we require $\unicode[STIX]{x1D6FE}_{m,1}/(\unicode[STIX]{x1D708}_{1}+4)=-\unicode[STIX]{x1D6FE}_{m,2}/(\unicode[STIX]{x1D708}_{2}+4)=\tilde{\unicode[STIX]{x1D6FE}}_{m}$ . The flow field $\boldsymbol{v}^{-}=\boldsymbol{v}_{0}^{-}+\boldsymbol{w}$ inside the droplet is given by

The flow field $\boldsymbol{v}^{+}$ in the ambient fluid vanishes exactly, as it does for the case of a squirmer particle (Pak & Lauga Reference Pak and Lauga2014), but the internal flow gives rise to a non-zero angular momentum of the droplet which can be computed using (2.7),

with $c_{\unicode[STIX]{x1D6FE}}=\sum _{i}(-1)^{i}/(\unicode[STIX]{x1D708}_{i}+6)$ .

Figure 1 shows an illustrative example of a chiral self-propelling droplet with an $\ell =1$ component of body force density given by (here $a=1$ )

Coefficients are chosen such that both the total force and the total torque generated by this density vanish. In figure 1, the flow lines of the force density $\boldsymbol{f}$ and the flow fields $\boldsymbol{v}^{\pm }$ are shown in the $x{-}z$ plane. This simple example already shows that it is not straightforward to guess the internal flow field for a given force density field.

Figure 1.

(a) Flow lines of the force density, and (b) internal and (c) external flow fields in the $xz$ plane with parameters $\unicode[STIX]{x1D6FC}_{0}=1$ , $\unicode[STIX]{x1D6FC}_{1}=-\unicode[STIX]{x1D6FC}_{-1}^{\ast }=1.7+0.5\,\text{i}$ , $\unicode[STIX]{x1D6FE}_{0}=1$ , $\unicode[STIX]{x1D6FE}_{1}=-\unicode[STIX]{x1D6FE}_{-1}^{\ast }=1.2+1.7\,\text{i}$ . It should be noted that it is not straightforward to guess the internal flow from the force field and the external flow. Magnitudes of vectors are coded in spectral sequence (black–violet–blue–green–yellow–red) online.

4.3 Conclusions: active body forces

In this section, we have related the linear and rotational motion of the droplet to the vector multipole moments of a given field of active body force densities and calculated the corresponding internal and external flow. Force-free translational motion can be achieved by a general power law dependence of the force density, $\boldsymbol{f}^{\,act}=(r/a)^{\unicode[STIX]{x1D708}}\boldsymbol{f}_{\unicode[STIX]{x1D708}}$ , such that forces are balanced in each radial shell inside the droplet. The translational velocity of the droplet is determined by the amplitude and power $\unicode[STIX]{x1D708}$ of the force as well as the two viscosities of the inside and outside fluid. Torque-free rotational motion requires at least two different power laws for the force density and cannot be fulfilled for each radial shell separately. The rotational flow inside the droplet depends in a non-trivial way on the assumed force density, whereas the rotational flow outside vanishes, as for a squirmer (Pak & Lauga Reference Pak and Lauga2014). Consequently, the rotational velocity of the droplet is a function of the interior viscosity only.

6.1 Rigid active matter and droplet velocities

In the previous sections, we have calculated the flow fields of a droplet in the LF at a single fixed time. Now, we want to construct time-dependent flows of the fluids and trajectories of the droplet as a continuum of such single time snapshots. A necessary condition for the validity of this quasisteady-state approximation is that – in the limit of small Reynolds number $Re$ – the product of Reynolds and Strouhal numbers is also small, i.e. $Re\cdot S=\unicode[STIX]{x1D6FA}L^{2}/\unicode[STIX]{x1D702}^{-}\ll 1$ (Kim & Karrila Reference Kim and Karrila2005). Here, $\unicode[STIX]{x1D6FA}^{-1}$ is a characteristic time scale for changes of the force density and $L$ is a characteristic length scale, which, for internal flow, should be smaller than  $a$ . A typical time scale for dissipative momentum transport over $a=1~\unicode[STIX]{x03BC}\text{m}$ in aqueous solution ( $\unicode[STIX]{x1D702}=10^{-6}~\text{m}~\text{s}^{-2}$ ) is $a^{2}/\unicode[STIX]{x1D702}=10^{-6}~\text{s}$ , so that the quasi-static approximation should be valid for $\unicode[STIX]{x1D6FA}\ll 10^{6}$  Hz if the viscosity contrast $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D702}^{-}/\unicode[STIX]{x1D702}^{+}$ stays $O(1)$ . Even if this condition is met, the induced acceleration of the droplet may lead to additional forces, which may change the trajectories. These forces were first found by Stokes (Reference Stokes1851), Boussinesq (Reference Boussinesq1885) and Basset (Reference Basset1888). We will postpone a study of these effects to future work and neglect them in the following. A critical study of the validity of this assumption for passive particles can be found in Feng & Joseph (Reference Feng and Joseph1995).

The time dependences of active force densities and tractions in the LF have not yet been specified. These densities are generated by active material, which shares the droplet interior with the Newtonian fluid, and which follows its own evolution equations. The material in the interior of micro-organisms has been modelled in many ways, for example as mixtures of simple (Dembo & Harlow Reference Dembo and Harlow1986) and complex (Kruse et al. Reference Kruse, Joanny, Jülicher, Prost and Sekimoto2005; Joanny et al. Reference Joanny, Jülicher, Kruse and Prost2007) active and passive fluids, or as a biphasic poro-elastic medium (Biot Reference Biot1955; deGennes Reference deGennes1976; Moeendarbary et al. Reference Moeendarbary, Valon, Fritzsche, Harris, Moulding, Thrasher, Stride, Mahadevan and Charras2013) with a distribution of active force densities or stress tensors, produced by molecular motors and (dis)-assembly of semi-flexible polymers (Loisel et al. Reference Loisel, Boujemaa, Pantaloni and Carlier1999). In all cases, the resulting evolution equations are quite complex. In the present subsection, we discuss a model of rigid active matter, which consists of a sparse rigid statistically isotropic structure, which carries the active forces and is immersed in a Newtonian fluid. This model, although highly simplified, is not unreasonable as an approximate continuum description on time scales that do not exceed the reorganisation time of the cytoskeleton. Its main advantage is that it allows for simple closed equations of motion of the droplet. A schematic view of our model is shown in figure 3.

Figure 3.

Schematic view of our model and the frames of reference used in the main text. The droplet is cut open to show its interior. Inside and outside flow is created by active force densities $\boldsymbol{f}^{\,act}(\boldsymbol{r},t)$ in the interior and/or by active tractions $\boldsymbol{t}^{\,act}(\boldsymbol{r},t)$ in the interface. The droplet interior contains a sparse rigid statistically isotropic structure, with its centre fixed to the centre of the droplet. It carries the active elements, which produce $\boldsymbol{f}^{\,act}$ and $\boldsymbol{t}^{\,act}$ , and it still can rotate relative to the interior fluid. The motion of the droplet is observed from the LF. The co-moving frame (CMF) is fixed to the rigid structure and the momentary material frame (MMF) is an LF, which coincides with CMF at one instant of time. Calculations of flow fields and translational and rotational velocities of the droplet have been performed in the MMF, and thus the polar axis of the spherical coordinates is the $z^{\prime }$ axis.

As in every continuum description, let us label an element of the active material by $X$ . At an initial time $t=0$ , we assume that the element is at rest in the LF at position $\boldsymbol{X}_{0}$ in a quiet fluid, and in the course of time it follows a trajectory $\boldsymbol{X}(t)=\boldsymbol{X}(\boldsymbol{X}_{0},t)$ with velocity $\boldsymbol{v}_{m}(X,t)=\text{d}\boldsymbol{X}(t)/\text{d}t$ . If, for example, the material elements do not interact and move convectively in the Newtonian fluid, the material velocity equals the fluid velocity $\boldsymbol{v}_{m}(X,t)=\boldsymbol{v}^{-}(\boldsymbol{X}(t),t)$ . In the following we will restrict the discussion to a particularly simple case. We suppose that the active material is tied to parts of a polymeric (cyto)-skeleton, which can be modelled as a rigid structure. Thus, it can only undergo rigid body motions, composed of rotations $\hat{\boldsymbol{R}}(t)$ with respect to some material point and translations $\boldsymbol{b}(t)$ , i.e. $\boldsymbol{X}(t)=\hat{\boldsymbol{R}}(t)\boldsymbol{X}(0)+\boldsymbol{b}(t)$ , and thus the corresponding material velocities are given by $\boldsymbol{v}_{m}(\boldsymbol{X},t)=\boldsymbol{v}_{0}+\unicode[STIX]{x1D74E}_{m}(t)\times [\boldsymbol{X}(t)-\boldsymbol{b}(t)]$ , with $\boldsymbol{v}_{0}=\text{d}\boldsymbol{b}/\text{d}t$ .

As an additional constraint, we require that the translational motion of the skeleton and the translational motion of the droplet coincide, i.e. $\boldsymbol{v}_{0}=\boldsymbol{v}_{CM}$ . This constraint can be realised in various ways, for example by anchoring at least four material points of the skeleton (not all in the same plane) within the interface. This fixes the centre of the sphere rigidly to the skeleton, reminiscent of the anchoring at the microtubule-organising centre. The motion of the skeleton is thus restricted to rotations within the droplet. Without such a constraint, the skeleton could perform translational motions relative to the centre of the sphere – like, for example, an active particle propelling in the interior of the droplet – which we do not want to discuss here.

In the following, we will refer to $\unicode[STIX]{x1D74E}_{m}$ as the angular velocity of the skeleton. To compute this angular velocity from the flow field $\boldsymbol{v}^{-}$ , we consider the dissipative forces exerted by the flow on the skeleton in the MMF as proportional to the relative velocity between skeleton and fluid, i.e. proportional to $\int _{K}\unicode[STIX]{x1D70C}_{p}(\boldsymbol{v}_{m}-\boldsymbol{v}^{-})\,\text{d}V$ . A part of these forces is compensated by the rigid constraints; the other part is the only remaining force on the skeleton, which thus has to vanish. A convenient way to calculate the uncompensated generalised force on the rigid skeleton is to start from the dissipation function (Landau & Lifshitz Reference Landau and Lifshitz1976) $D(\unicode[STIX]{x1D74E}_{m})=(\unicode[STIX]{x1D705}/2)\int _{K}\unicode[STIX]{x1D70C}_{p}(\boldsymbol{r})(\boldsymbol{v}^{m}(\boldsymbol{r})-\boldsymbol{v}^{-}(\boldsymbol{r}))^{2}\,\text{d}V$ . The friction coefficient $\unicode[STIX]{x1D705}>0$ is unimportant in the following. The derivative of $D$ with respect to $\unicode[STIX]{x1D74E}_{m}$ is the generalised force, which has to vanish. The resulting relation $\int _{K}[\boldsymbol{r}\times (\boldsymbol{v}_{CM}+\unicode[STIX]{x1D74E}_{m}\times \boldsymbol{r}-\boldsymbol{v}^{-})]\unicode[STIX]{x1D70C}_{p}\,\text{d}V=0$ can be written in the form

Here, $\boldsymbol{I}_{ij}=\int \unicode[STIX]{x1D70C}_{p}(r^{2}\unicode[STIX]{x1D6FF}_{ij}-x_{i}x_{j})$ . It should be noted that $\unicode[STIX]{x1D74E}_{m}$ depends on the structure of the skeleton and in general differs from the angular velocity $\unicode[STIX]{x1D74E}$ defined in (2.7). If the skeleton density is spherically symmetric, $\boldsymbol{I}_{ij}=I\unicode[STIX]{x1D6FF}_{ij}$ , with $I=(8\unicode[STIX]{x03C0}/3)\int _{0}^{a}\unicode[STIX]{x1D70C}_{p}(r)r^{4}$ , we obtain

For a homogeneous $\unicode[STIX]{x1D70C}_{p}(r)=\unicode[STIX]{x1D70C}_{p}$ , the constant $I$ becomes $I=(8\unicode[STIX]{x03C0}a^{5}/15)\unicode[STIX]{x1D70C}_{p}$ . In this case, $\unicode[STIX]{x1D74E}_{m}$ coincides with $\unicode[STIX]{x1D74E}$ from (2.7).

For other structures, there will be a relative rotational motion of the skeleton and the internal fluid. If, for example, the skeleton mass is essentially concentrated in a spherical nucleus with radius $a_{nuc}<a$ , the motion is reminiscent of the observed rotation of a cell nucleus in the interior of a cell (Kumar et al. Reference Kumar, Maitra, Sumit, Ramaswamy and Shivashankar2014). On the other hand, if the skeleton were concentrated in a thin shell at the interface, i.e. $\unicode[STIX]{x1D70C}_{p}(r)=\unicode[STIX]{x1D70C}_{p}\unicode[STIX]{x1D6FF}(r-a)$ , then $I=(8\unicode[STIX]{x03C0}a^{4}/3)\unicode[STIX]{x1D70C}_{p}$ , and the rigid shell would rotate relative to the internal fluid.

Three frames of reference, which are used in the following, are defined with respect to the skeleton (see figure 3). At time $t=0$ , when each $X$ is placed at its corresponding $\boldsymbol{X}_{0}$ , we choose a Cartesian system, with its origin at the centre of the sphere, $\boldsymbol{r}_{CM}$ , and unit vectors $\boldsymbol{e}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}=1,2,3$ . This is our LF. The frame at any other time, with its origin at $\boldsymbol{X}(\boldsymbol{X}_{0}=\mathbf{0},t)$ and rotated unit vectors $\hat{\boldsymbol{R}}(t)\boldsymbol{e}_{\unicode[STIX]{x1D6FC}}$ , is called the CMF. At $t=0$ , the CMF and the LF coincide, i.e. $\hat{\boldsymbol{R}}(0)=\hat{\mathbf{1}}$ and $\boldsymbol{b}(0)=\mathbf{0}$ . Per definition, the skeleton is at rest in the CMF. Its density $\unicode[STIX]{x1D70C}_{p}(X)$ carries the active force density $\boldsymbol{f}^{\,act}(X,t)$ , which may depend upon time. Another LF, with origin and axes coinciding with those of the CMF at a fixed instant $t$ will be called the MMF in the following. The calculations of §§ 3–5 have been performed in this frame.

The active matter in the droplet interior will in general also modify the internal flow field. For a droplet filled with a rigid gel and a viscous fluid, there is an additional permeation force (Milner Reference Milner1993; Callan & Jülicher Reference Callan and Jülicher2011), which enters the force balance. This leads to a modified Stokes equation of the form $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{-}=\unicode[STIX]{x1D735}p-\unicode[STIX]{x1D701}(\unicode[STIX]{x1D70C}_{p})(\boldsymbol{v}-\boldsymbol{v}^{m})-\boldsymbol{f}^{\,act}-\boldsymbol{f}^{ext}$ . Here, $\unicode[STIX]{x1D701}(\unicode[STIX]{x1D70C}_{p})$ is a permeation coefficient, which depends upon the detailed structure of the gel. In the following, we will assume that the gel fraction is small, and $\unicode[STIX]{x1D701}$ , which vanishes with $\unicode[STIX]{x1D70C}_{P}$ , is so small that we can neglect the permeation force. Then, $\boldsymbol{v}^{-}$ is entirely due to the active and external force densities, and can be calculated using (2.1).

The time dependence of the force density $\boldsymbol{f}^{\,act}(\boldsymbol{r},t)$ and the skeleton density $\unicode[STIX]{x1D70C}_{p}(\boldsymbol{r},t)$ produced by active material at position $\boldsymbol{r}$ in the LF are now easily obtained. In the CMF, the position and orientation of each material element are time-independent, and only the explicit time dependence of the force density remains, i.e. $\unicode[STIX]{x1D70C}_{p}(\boldsymbol{r},t)=\unicode[STIX]{x1D70C}_{p}(\boldsymbol{X}_{0}=\boldsymbol{r})$ and $\boldsymbol{f}^{\,act}(\boldsymbol{r},t)=\boldsymbol{f}^{\,act}(\boldsymbol{X}_{0}=\boldsymbol{r},t)$ . In the LF, these quantities are co-translated and co-rotated along the trajectory of the active material element, so that

and correspondingly $\text{d}\unicode[STIX]{x1D70C}_{p}(\boldsymbol{r},t)/\text{d}t=(\boldsymbol{v}_{CM}+\unicode[STIX]{x1D74E}_{m}\times \boldsymbol{r})\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}_{p}(t)$ .

6.2 Trajectories

To determine the trajectories of the droplet in the LF, we can use (2.6) and (6.1), (6.2). Due to our assumption that the active material is tied to the skeleton, the dynamics of this skeleton in the LF is fully characterised by the transformation of the MMF, spanned by coordinate unit vectors $\boldsymbol{e}_{i}$ , $i=1,2,3$ , and rooted at the centre of the sphere, $\boldsymbol{r}_{CM}(t)$ , into the Cartesian LF spanned by $\boldsymbol{a}_{i}$ , $i=1,2,3$ . The equations of motion for the droplet skeleton thus become those of a rigid body,

with translational and rotational velocities, which are linearly related to the multipole moments of the internal force distributions via the flow fields $\boldsymbol{v}^{\pm }$ . In the previous sections, we have calculated these relations explicitly. In this section, we discuss how features of the trajectories are linked to features of the body force distributions of self-propelled motion by considering a few illustrating cases. We restrict the examples to a droplet with a homogeneous skeleton, i.e. $\unicode[STIX]{x1D74E}_{m}=\unicode[STIX]{x1D74E}$ , and choose a radius $a=1~\unicode[STIX]{x03BC}\text{m}$ to produce representative numerical values, for which we measure lengths in $\unicode[STIX]{x03BC}\text{m}$ , time in s and forces in pN.

Force multipoles and velocities are related by (4.7), (4.11), which can be written in the form

and

with $C=\sqrt{(2\unicode[STIX]{x03C0}/3)}3(\unicode[STIX]{x1D708}+5)(2+3\unicode[STIX]{x1D706})(\unicode[STIX]{x1D702}^{+}/a^{2})$ and $D=\sqrt{(2\unicode[STIX]{x03C0}/3)}(\unicode[STIX]{x1D702}^{-}/ac_{\unicode[STIX]{x1D6FE}})$ . For our illustrations, we choose $\unicode[STIX]{x1D706}=5$ and $\unicode[STIX]{x1D708}=1$ in $C$ and $\unicode[STIX]{x1D708}_{1}=1$ and $\unicode[STIX]{x1D708}_{2}=2$ (see (4.11)) in $c_{\unicode[STIX]{x1D6FE}}$ . Then, the constant $C=c_{v}\times 10^{9}~\text{s}^{-1}\approx 0.44\times 10^{9}~\text{s}^{-1}$ and $D=c_{\unicode[STIX]{x1D714}}\times 10^{9}~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}\approx 0.41\times 10^{9}~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}$ .

The force $\boldsymbol{f}^{\,act}$ and the corresponding multipole moments $\unicode[STIX]{x1D6FC}_{m}$ , $\unicode[STIX]{x1D6FE}_{m}$ are force densities per mass density with units $\unicode[STIX]{x03BC}\text{m}~\text{s}^{-2}$ . We prefer to relate velocities to force densities $\unicode[STIX]{x1D746}_{f}$ per volume, with corresponding multipoles $\widehat{\unicode[STIX]{x1D6FC}}_{m}$ , $\widehat{\unicode[STIX]{x1D6FE}}_{m}$ given in $\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}=10^{-12}~\text{N}~\unicode[STIX]{x03BC}\text{m}^{-3}$ , to facilitate the comparison with forces generated by molecular mechanisms. For aqueous solutions, the volume force density is related to $\boldsymbol{f}^{\,act}$ by $\boldsymbol{f}^{\,act}[\unicode[STIX]{x03BC}\text{m}~\text{s}^{-2}]=\unicode[STIX]{x1D746}_{f}/\unicode[STIX]{x1D70C}_{0}=10^{9}\unicode[STIX]{x1D746}_{f}[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ , so that

When comparing with molecular mechanisms of force generation, one should remember that $\widehat{\unicode[STIX]{x1D6FC}}_{a}$ and $\widehat{\unicode[STIX]{x1D6FE}}_{a}$ only contain the $\ell =1$ part of the total force density.

Figures 4–7 depict trajectories of the centre of mass (scales in  $\unicode[STIX]{x03BC}\text{m}$ ) in (a) and illustrate the rotational motion in (b). For the latter, we follow a marked point on the interface in time. This point has position vector $\boldsymbol{y}(t)$ in the LF and develops in time according to $\boldsymbol{y}(t)=\hat{\boldsymbol{R}}(t)\boldsymbol{y}(0)+\boldsymbol{r}_{CM}(t)$ . We plot $\unicode[STIX]{x0394}\boldsymbol{y}(t)=\boldsymbol{y}(t)-\boldsymbol{r}_{CM}(t)=\hat{\boldsymbol{R}}(t)\boldsymbol{y}(0)=\sum _{i}b_{i}\boldsymbol{e}_{i}(t)$ and have chosen $b_{i}=1$ , $i=1,2,3$ .

Case 1. For the simple case of force densities, which are time-independent in the CMF, the resulting trajectories from (6.4), (6.5) are helices, as shown in figure 4. The axis of the helix is parallel to $\unicode[STIX]{x1D74E}$ , which we have chosen to be the $z$ -axis. If the chiral parts of the force densities vanish (i.e. $\unicode[STIX]{x1D6FE}_{m}=0$ ), the trajectories become straight lines.

Figure 4.

For time-independent force densities, the trajectories are helices (or straight lines, if $\unicode[STIX]{x1D74E}=0$ ). The example shows a trajectory generated by force density multipoles $\widehat{\unicode[STIX]{x1D6FE}_{0}}=10\,\sqrt{2}c_{\unicode[STIX]{x1D714}}$ , $\tilde{\unicode[STIX]{x1D6FE}}_{1}=0$ and $\widehat{\unicode[STIX]{x1D6FC}}_{0}=10\,\sqrt{2}c_{v}$ , $\widehat{\unicode[STIX]{x1D6FC}}_{1}=c_{v}(-3+2\,\text{i})$ , given in $[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ . Panel (a) depicts $\boldsymbol{r}_{CM}(t)$ and panel (b) shows the rotational motion of $\unicode[STIX]{x0394}\boldsymbol{y}(t)$ , as defined in the main text.

Case 2. Force densities, which oscillate with a single frequency and vanish when averaged over a period, can create unbounded translational motion, as shown in figure 5. The unbounded translation requires non-vanishing chiral components of the force densities.

Figure 5.

Periodic force densities with vanishing averages over one period may nevertheless lead to unbounded translations. Panel (a) shows the trajectory of $\boldsymbol{r}_{CM}$ and panel (b) depicts the rotational motion of $\unicode[STIX]{x0394}\boldsymbol{y}(t)$ (see the main text) for force multipoles (in $[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ )  $\widehat{\unicode[STIX]{x1D6FE}}_{0}=0.1\,\sqrt{2}\,c_{\unicode[STIX]{x1D714}}\sin (t)$ and $\widehat{\unicode[STIX]{x1D6FE}}_{1}=0$ , $\widehat{\unicode[STIX]{x1D6FC}}_{0}=0$ , $\widehat{\unicode[STIX]{x1D6FC}}_{1}=0.1\,c_{v}(-\text{sin}(t)+\text{i}\cos (t))$ .

Case 3. On the other hand, it is also possible to get helical motion without any chiral components of the forces. In figure 6(a), the chiral components and thus the angular velocity $\unicode[STIX]{x1D74E}$ vanish. The force density oscillates with a single frequency but has a non-vanishing average over one period. If a constant chiral force density is added to the oscillating force, rosette-shaped trajectories appear, as shown in figure 6(b). These may also be superimposed with a translational motion.

Figure 6.

(a) Helical motion arising from periodic non-chiral force densities, with multipoles (in $[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ )  $\widehat{\unicode[STIX]{x1D6FC}_{0}}=0$ and $\widehat{\unicode[STIX]{x1D6FC}_{1}}=c_{v}(-(0.1+0.6\cos (t))+\text{i}(0.1+0.6\sin (t)))$ . (b) A chiral force density with $\widehat{\unicode[STIX]{x1D6FE}}_{0}=0.2\,\sqrt{2}c_{\unicode[STIX]{x1D714}}$ , $\widehat{\unicode[STIX]{x1D6FE}}_{1}=0$ is added to the system of (a), resulting in rosette-shaped trajectories. The depicted rosette is simple and closed, because the ratio of the frequencies in $\boldsymbol{v}_{CM}$ (which is 1) and $\unicode[STIX]{x1D74E}$ is a rational number ( $1/5$ ).

Case 4. Finally, if the forces contain more than one frequency, the zoology of the trajectories gets more and more involved and will be discussed elsewhere. As an example, figure 7 shows the translational (figure 7 a) and rotational (figure 7 b) motion of a three-dimensional trajectory resulting from five different non-vanishing frequencies and a constant component of forces. Although this trajectory is smooth, it superficially resembles run-and-tumble type trajectories when observed from a fixed viewing angle.

Figure 7.

This example shows a complex trajectory of $\boldsymbol{r}_{CM}$ and $\boldsymbol{b}$ (see the main text), resulting from a chiral force density $\widehat{\unicode[STIX]{x1D6FE}}_{0}=\sqrt{2}c_{\unicode[STIX]{x1D714}}[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ , $\widehat{\unicode[STIX]{x1D6FE}}_{1}=c_{\unicode[STIX]{x1D714}}(-0.5\sin (t)+0.3\,\text{i}\,\sin (\sqrt{2}t))[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ and $\widehat{\unicode[STIX]{x1D6FC}}_{0}=0.4\,\sqrt{2}c_{v}\sin (0.3t)[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ , $\widehat{\unicode[STIX]{x1D6FE}}_{1}=c_{v}(-0.6\sin (1.9t)+0.8\,\text{i}\,\sin (1.1t))[\text{pN}~\unicode[STIX]{x03BC}\text{m}^{-3}]$ .

We have presented these few cases to illustrate that simple changes of the time dependences of active processes can lead to qualitatively different trajectories of self-propelled motion. Such changes in time dependences thus offer rich possibilities for steering self-propelled motion.

Simple models of self-propelled particles have been studied extensively over recent years, mainly to understand the collective behaviour of active matter. In these approaches, a swimmer is modelled as an anisotropic particle with constant linear and angular velocity, and is subject to noise due to the surrounding fluid. Here, we derive these velocities from the internal flow, generated by active structural elements within the swimmer. Constant velocities are shown to result from a time-independent forcing machinery, whereas oscillating forces give rise to time-dependent velocities and complex trajectories. The trajectories that we observe bear striking similarities with those of self-propelled particles (Bechinger et al. Reference Bechinger, Leonardo, Löwen, Reichhardt, Volpe and Volpe2016). In the latter case, complex behaviour results from the interplay of activity and (persistent) noise, whereas in our case, it is the result of the inherent machinery of a single swimmer. Important differences concern the small scales of the trajectories, which have features of a random walk for self-propelled particles, but are completely smooth in our case.