2.1. Flow field produced by a micro-swimmer

By changing their shape, swimmers exert forces and torques on the fluid they move in. We model these as a distribution of point forces, $\boldsymbol{f}({\bf\xi})$ , summed over the swimmer’s surface $S$ located at position $\boldsymbol{R}+{\bf\xi}$ . The force distribution produces flows around the swimmer

where $\unicode[STIX]{x1D611}_{ij}(\boldsymbol{r})$ is the appropriate Green’s function of the Stokes equations. For an unbounded fluid, this Green’s function is the Oseen tensor $\unicode[STIX]{x1D60E}_{ij}(\boldsymbol{r})=({\it\delta}_{ij}/r+r_{i}r_{j}/r^{3})/(8{\rm\pi}{\it\mu})$ where ${\it\delta}_{ij}$ is the Kronecker delta and $r=|\boldsymbol{r}|$ . When a no-slip boundary is present, the Green’s function is the Blake tensor (Blake Reference Blake1971a )

where $H=\boldsymbol{R}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{x}>0$ and where the derivatives $\partial _{j}^{\ast }=\unicode[STIX]{x1D614}_{jl}\partial _{l}$ and ${\rm\nabla}^{\ast 2}={\rm\nabla}^{2}$ . For a free-slip interface the Green’s function is $\unicode[STIX]{x1D60D}_{ij}(\boldsymbol{r},\boldsymbol{r}^{\ast })=\unicode[STIX]{x1D60E}_{ij}(\boldsymbol{r})+\unicode[STIX]{x1D614}_{jk}\unicode[STIX]{x1D60E}_{ik}(\boldsymbol{r}^{\ast })$ .

We expand (2.1) in a multipole expansion to write the fluid flow generated by a micro-swimmer moving in the $z$ direction as

which contains Stokes dipole $\boldsymbol{u}^{D}\sim O(r^{-2})$ , quadrupole $\boldsymbol{u}^{Q}\sim O(r^{-3})$ , octupole $\boldsymbol{u}^{O}\sim O(r^{-4})$ and higher-order terms. A detailed explanation of the multipole expansion can be found in the work by Jeffrey & Onishi (Reference Jeffrey and Onishi1984) or Kim & Karilla (Reference Kim and Karilla1991, p. 27), or more explicitly in the context of micro-swimmers in the work by Yeomans, Pushkin & Shum (Reference Yeomans, Pushkin and Shum2014).

Many micro-swimmers have, to a good approximation, a shape that is cylindrically symmetric about the swimming direction. This helpfully restricts the number of non-zero components of the swimmer multipole moments. In the supplementary data (available at http://dx.doi.org/10.1017/jfm.2015.269) we compute the residual flow fields in terms of a substantially reduced number of coefficients, the dipole moment ${\it\kappa}$ , the quadrupolar moments $Q_{\Vert ,\bot ,r}$ , etc., which are experimental observables for a particular individual organism. In the following paragraphs we list the flow fields up to octupolar order, and describe their physical significance.

An autonomous swimmer cannot rely on external forces to move forwards. Therefore there are no Stokeslet terms in (2.3), assuming that the effects of gravity are negligible. This assumption is true for swimmers that are neutrally buoyant, or if their sedimentation velocity is negligible compared to their swimming speed (Ishikawa et al. Reference Ishikawa, Simmonds and Pedley2006). Therefore the leading-order term in (2.3) is the dipolar flow field

where $u_{i}^{S}(\boldsymbol{r},\hat{\boldsymbol{e}}_{j})=\hat{\boldsymbol{e}}_{j}\boldsymbol{\cdot }\unicode[STIX]{x1D611}_{ij}(\boldsymbol{r})$ is the Stokeslet, the flow generated by a point force along $\hat{\boldsymbol{e}}_{j}$ , and $\partial _{z}=(\hat{\boldsymbol{e}}_{z}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})$ is the derivative with respect to the relative position $\boldsymbol{r}$ in the swimming direction. The dipolar term describes the balancing forces of propulsion and drag. If ${\it\kappa}>0$ the swimmer is known as a pusher, with an extensile flow field (e.g. a sperm cell or Escherichia coli bacterium), where the organism is propelled from behind. If ${\it\kappa}<0$ the swimmer is termed a puller, with a contractile flow field (e.g. the Chlamydomonas reinhardtii alga, which performs a breast stroke). The dipole moment ${\it\kappa}$ has been measured in experiment and time-averaged for the microphytes Volvox carteri and C. reinhardtii (Drescher et al. Reference Drescher, Goldstein, Michel, Polin and Tuval2010) and for E. coli (Drescher et al. Reference Drescher, Dunkel, Cisneros, Ganguly and Goldstein2011).

The quadrupolar flow field of a cylindrically symmetric swimmer can be written as

where ${\rm\nabla}_{\Vert }^{2}=(\hat{\boldsymbol{e}}_{z}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})^{2}$ and ${\rm\nabla}_{\bot }^{2}={\rm\nabla}^{2}-{\rm\nabla}_{\Vert }^{2}$ are the Laplace operators parallel to, and in the plane perpendicular to, the swimming direction $\hat{\boldsymbol{e}}_{z}$ . The source doublet flow field, a particular combination of the quadrupole terms with $Q_{\Vert }=Q_{\bot }$ , allows us to describe the flow around the swimmer’s body, which can help the tracer particles avoid encountering the singularity centre and hence account for the finite size of the cell. This will be discussed further in § 2.2. Another combination, $Q_{\Vert }=-Q_{\bot }$ , describes fore–aft asymmetries of the Stokeslet density along the swimmer’s surface, such as the size difference of the flagella compared to the smaller head. Finally, the term with coefficient $Q_{r}$ is a rotlet doublet that represents the rotation of the flagella and counter-rotation of the head. If $Q_{r}>0$ , looking from tail to head, the flagella (resp. head) rotate in the clockwise (resp. anticlockwise) direction. There are currently no experimental results in the literature for quadrupolar or higher-order multipoles.

The octupolar velocity field for a cylindrically symmetric swimmer is

These terms are fore–aft symmetric, and hence describe more detailed near-field features of the swimmer in the radial and azimuthal directions. The hexadecapolar flow field also has three coefficients, $H_{\Vert }$ , $H_{\bot }$ and $H_{r}$ , and may be obtained by acting on the expression (2.6) with $(\hat{\boldsymbol{e}}_{z}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})$ . These terms are fore–aft anti-symmetric and hence describe more detailed near-field features of the swimmer in the longitudinal direction. Higher-order flow fields may be derived in the same fashion. Figure S2 in the supplementary data gives a pictorial summary of flow fields up to octupolar order.

In the presence of a surface, additional boundary conditions have to be satisfied so that the flow fields given by (2.4)–(2.6) are transformed by replacing the Stokeslets $\boldsymbol{u}^{S}(\boldsymbol{r},\hat{\boldsymbol{e}}_{j})$ with the effective Stokeslets $\overline{\boldsymbol{u}}^{S}(\boldsymbol{r},\hat{\boldsymbol{e}}_{j})=\hat{\boldsymbol{e}}_{j}\unicode[STIX]{x1D609}_{ij}$ or $\hat{\boldsymbol{e}}_{j}\unicode[STIX]{x1D60D}_{ij}$ for a no-slip or free-slip boundary respectively. More details are given in § S2 of the supplementary data.

In general, the multipole moments are time-dependent quantities that relate to details of the swimming stroke. If the stroke period ${\it\tau}_{s}$ is small compared to a typical swimming time scale, say the time taken to swim one body length, ${\it\tau}_{a}$ , then the swimming parameters may be approximated by constants obtained through time-averaging over the swimming stroke. A non-zero stroke period, however, may give rise to additional average drift of tracer particles, analogous to Stokes drift. To first order, the magnitude of this effect is proportional to ${\it\tau}_{s}/{\it\tau}_{a}$ . For E. coli, for example, we have $V\sim 20~{\rm\mu}\text{m}~\text{s}^{-1}$ and body length $l_{sw}\sim 5~{\rm\mu}\text{m}$ , so that ${\it\tau}_{a}^{-1}\sim 4~\text{Hz}$ compared to the stroke frequency ${\it\tau}_{s}^{-1}\sim 100~\text{Hz}$ (Turner, Ryu & Berg Reference Turner, Ryu and Berg2000), which gives a good separation of time scales with corrections of $\mathit{O}(10^{-2})$ . For C. reinhardtii we have ${\it\tau}_{a}^{-1}\sim 100/8=13~\text{Hz}$ and ${\it\tau}_{s}^{-1}\sim 50~\text{Hz}$ (Kamiya & Hasegawa Reference Kamiya and Hasegawa1987).

Figure 2.

Flow fields corresponding to the given multipoles in the rest frame of a micro-swimmer moving with velocity $V=20~{\rm\mu}\text{m}~\text{s}^{-1}$ . Inclusion of quadrupole terms results in formation of a closed surface of streamlines separating the flow into the internal and external regions: see (b–d). The external region is singularity-free and can model the flow field of a finite-size swimmer. The separatrix shape can be adjusted by tuning the quadrupole parameters. It can, for example, produce elongated shapes characteristic of the head of a mammalian spermatozoon: see (d). Here colours portray the velocity magnitude, ranging from zero (dark blue) to $2V$ (light blue), and white denotes the singularity centre. Arrows indicate the velocity direction in the $(z,{\it\rho})$ plane. The thick orange line is the separatrix. (a) Pure dipole: ${\it\kappa}=30~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ , $Q_{\Vert }=0~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , $Q_{\bot }=0~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ . (b) Source doublet: ${\it\kappa}=0$ , $Q_{\Vert }=30~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , $Q_{\bot }=30~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ . (c) Combined: ${\it\kappa}=30~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ , $Q_{\Vert }=30~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , $Q_{\bot }=30~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ . (d) Combined: ${\it\kappa}=5~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ , $Q_{\Vert }=10~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , $Q_{\bot }=30~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ .

2.2. Modelling the swimmer’s near-field

Including quadrupole terms in the expansion of the swimmer velocity field allows us to model the swimmer body and hence avoid the singularities of the multipole expansion at the origin (Stokes Reference Stokes1846). Consider the flow field in the rest frame of the swimmer, $\boldsymbol{u}^{rest}=\boldsymbol{u}^{lab}-V\hat{\boldsymbol{e}}_{z}$ , where the laboratory frame flow fields are listed in § 2.1. The flow field of a purely dipolar pusher, given by (2.4) with ${\it\kappa}>0$ , is shown in the swimmer rest frame in figure 2(a). Notice a stagnation point a distance $z^{\ast }$ directly in front of the swimmer. This point corresponds to $u^{D,rest}(z^{\ast })=0$ , which can be inverted to give $z^{\ast }=\sqrt{2{\it\kappa}/V}$ . The incoming fluid moves around the stagnation point and the streamlines join again at the singularity. However, the near-field behaviour is unphysical because of the singularity – as expected for a far-field expansion.

Next we consider the source doublet flow given by (2.5) for $Q_{sd}\equiv Q_{\Vert }=Q_{\bot }$ and $Q_{r}=0$ , and shown in the swimmer rest frame in figure 2(b). For this multipole there is a surface of streamlines connecting two stagnation points in front and behind the swimmer. There is no flow crossing this surface, the separatrix, and hence the singularity region is separated from the rest of the flow. Therefore we can argue that, even though the swimmer is described by point singularities, it has an effective finite size. For a pure source doublet the position of the separatrix can be calculated exactly. Details of this calculations are given in § S3 of the supplementary data. The result is a spherical shell with radius

Figure 2(c) shows the case when a dipole term is added to the flow field, in addition to the source doublet. If $|{\it\kappa}|\ll |Q_{sd}|/a_{sw}$ the separatrix remains approximately spherical with radius $a_{sw}$ given by (2.7). If $|{\it\kappa}|$ takes larger values, the separatrix changes shape. Indeed, the shape can be adjusted by tuning the dipole and quadrupole parameters. For instance, figure 2(d) shows a separatrix with an elongated shape, approximately $4~{\rm\mu}\text{m}$ long and $2~{\rm\mu}\text{m}$ wide, characteristic of rod-shaped bacteria or the head of a mammalian spermatozoon (Katz et al. Reference Katz, Overstreet, Samuels, Niswander, Bloom and Lewis1986). In § S3 of the supplementary data we show that the separatrix exists for all finite values of ${\it\kappa}$ provided $Q_{\bot }>0$ and $-1<Q_{\Vert }/Q_{\bot }<2$ , and we derive an explicit expression for its position.

We shall use this result throughout the paper when numerically integrating tracer displacements, as it provides a useful and physically reasonable way of accounting for the near-field structure of flows created by micro-organisms.

2.3. Expansion of the tracer equation of motion

We aim to generate an expression for the tracer displacement as a power series in $1/r$ . The tracer particle is advected directly by the fluid, and its equation of motion is

which is an implicit equation subject to the initial condition $\boldsymbol{r}_{T}^{int}$ , and where we explicitly write $\boldsymbol{u}^{im}$ , the velocity field of the image system, needed when a boundary is present. Equation (2.8) can either be solved numerically or one can proceed analytically if the tracer displacement is small compared to the distance to the swimmer, $|{\rm\Delta}\boldsymbol{r}_{T}|\ll |\boldsymbol{r}|~\forall t$ . If the swimming motion is with a constant speed $V$ , then taking the time-derivative of this condition yields the equivalent condition $\left|\partial _{t}\boldsymbol{r}_{T}(t)\right|=\left|\boldsymbol{u}^{sw}\right|\ll V~\forall t$ .

We change variables in (2.8) to write $(\text{d}\boldsymbol{r}_{T}/\text{d}t)=(\text{d}{\rm\Delta}\boldsymbol{r}_{T})/(\text{d}t)=\boldsymbol{u}^{sw}\left(\boldsymbol{r}+{\rm\Delta}\boldsymbol{r}_{T}\right)+\boldsymbol{u}^{im}\left(\boldsymbol{r}^{\ast }+{\rm\Delta}\boldsymbol{r}_{T}\right)$ and expand for small values of ${\rm\Delta}\boldsymbol{r}_{T}$ , to give

To solve for ${\rm\Delta}\boldsymbol{r}_{T}$ as a power series in $1/r$ we write

where the terms ${\rm\Delta}\boldsymbol{r}_{T}^{(n)}$ scale as $r^{-n}$ . Since the swimmer moves with a constant speed $V$ along $z$ we may write $(\text{d}/\text{d}t)=V(\text{d}/\text{d}R_{z})=-V(\text{d}/\text{d}r_{z})=-V(\partial r/\partial z)(\text{d}/\text{d}r)$ . Therefore, the time derivatives $\partial _{t}{\rm\Delta}\boldsymbol{r}_{T}^{(n)}$ scale as $r^{-n-1}$ . Hence, substituting (2.10) into (2.9), using (2.3) and equating successive powers of $1/r$ yields

where we have used the shorthand notation $\boldsymbol{u}^{D}=\boldsymbol{u}^{D}(\boldsymbol{r}(t),~\hat{\boldsymbol{e}}_{z})$ and similarly for $\boldsymbol{u}^{Q}$ , $\boldsymbol{u}^{O}$ etc., into which the image system flow fields are incorporated.

The tracer trajectory can be found by integrating equations (2.11)–(2.14) successively, and substituting the resulting expressions into the higher-order equations of motion. In § S4.1 of the supplementary data we describe this procedure for a swimmer in an unbounded fluid, and in § S4.2 and § S4.3 we repeat the calculations for a swimmer near a no-slip and free-slip boundary respectively.

3.1. Tracer loops in the $({\it\rho},z)$ plane

Figure 3 shows the tracer displacements. The values of the swimming speed and the dipole strength were chosen to correspond to an E. coli bacterium (Drescher et al. Reference Drescher, Dunkel, Cisneros, Ganguly and Goldstein2011). As the run length $2L$ between two tumbling events of E. coli varies from 5 to $30~{\rm\mu}\text{m}$ (Berg & Brown Reference Berg and Brown1972), we have chosen $2L=10~{\rm\mu}\text{m}$ to emphasise the effects of a finite-length swimming path. Different initial tracer positions are depicted by planes of small black dots at $z_{T}^{int}=-8,-6,\dots ,6,8~{\rm\mu}\text{m}$ . Coloured dots show the final tracer positions after the swimmer has completed its run. In several cases trajectories are also shown, running from the open circles to the black dots. These results were obtained from a numerical solution of (2.8).

Figure 3.

Tracer trajectories and final displacements due to a micro-swimmer in an unbounded fluid. The swimming speed and the dipole strength characteristic for the E. coli bacterium have been used: ${\it\kappa}=30~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ and $V=20~{\rm\mu}\text{m}~\text{s}^{-1}$ . Quadrupole strengths $Q_{\Vert }=Q_{\bot }=5~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ were chosen in (b–d) to represent a swimmer with a characteristic size $a_{sw}\approx 1~{\rm\mu}\text{m}$ . All higher-order multipole coefficients set to zero. The trajectories are computed using numerical integration of (2.8). Figures (a–c) are in the laboratory frame, where the swimmer moves along a finite line from $-5~{\rm\mu}\text{m}$ to $L=5~{\rm\mu}\text{m}$ . Initially, tracers are located in planar sheets perpendicular to the $z$ -axis at $z_{T}^{int}=-8,-6,\dots ,6,8~{\rm\mu}\text{m}$ , uniformly distributed between ${\it\rho}=0$ and ${\it\rho}=3~{\rm\mu}\text{m}$ , and shown by solid black dots. The final positions of the tracer sheets are depicted by coloured (online) dots. Several trajectories of individual tracers are also shown as thin black lines, running from starting points denoted by open circles to ending points denoted by solid dots. (a) Purely dipolar pusher: ${\it\kappa}=30$ , $Q_{sd}=0$ . (b) Regularised pusher with finite size: ${\it\kappa}=30$ , $Q_{sd}=5$ . (c) The equivalent puller with finite size: ${\it\kappa}=-30$ , $Q_{sd}=5$ . Figure (d) is equivalent to (b) in the rest frame of the micro-swimmer, which itself is located at $z=5~{\rm\mu}\text{m}$ ; tracers were initially located $z_{T}^{int}=2,4,\dots ,16,18~{\rm\mu}\text{m}$ , and final tracer positions are shown after a time ${\rm\Delta}t=2L/V$ has passed. Full tracer trajectories are shown for the planar sheet $z_{T}^{int}=10~{\rm\mu}\text{m}$ .

Results for a purely dipolar pusher ( ${\it\kappa}>0$ ) are shown in figure 3(a). The tracer trajectories form loops in the region adjacent to the swimming path, where $|z_{T}^{int}|\ll L$ , as previously shown by Dunkel et al. (Reference Dunkel, Putz, Zaid and Yeomans2010). The shape of the trajectory is triangular for a dipolar swimmer and is a smaller three-loop structure for a purely quadrupolar swimmer. Notice that the tracer trajectories almost close on themselves, but there is a finite opening of the loop that defines the final tracer displacement ${\rm\Delta}\boldsymbol{r}_{T}$ .

Far away from the swimming path, where $|\boldsymbol{r}_{T}^{int}|\gg L$ , the final displacement scales as $1/r$ and reflects the dipolar flow field because the swimming path approximates a point source, $\lim _{r\gg L}{\rm\Delta}\boldsymbol{r}_{T}=\boldsymbol{u}^{sw}{\rm\Delta}t$ , where ${\rm\Delta}t=2L/V$ . Additional quadrupole moments do not affect the far-field displacement, as shown in figure 3(b), since these terms decay as $1/r^{2}$ . Tracers in front of a pusher are moved forwards, while those behind the swimmer are moved backwards. Displacement along the radial direction is towards the swimming path for $|z_{T}^{int}|<L$ and away from the swimming path for $|z_{T}^{int}|>L$ . For the equivalent puller ( ${\it\kappa}\rightarrow -{\it\kappa}$ ) the tracers move over the loops in the opposite direction and the displacement in the far-field reverses sign, as shown in figure 3(c).

Closer to the swimming path the tracer displacement does not simply reflect the flow field as the higher-order advective terms in (2.9) become significant. In other words, there is interference between the tracer displacements caused by different multipole components of the swimmer’s velocity field, as seen in the second term of (2.13). We observe that, predominantly, tracer trajectories are larger as the initial distance to the swimming path ${\it\rho}_{T}^{int}$ decreases. For tracers that start adjacent to the swimming path, where $|\boldsymbol{r}_{T}^{int}|\ll L$ , the final tracer displacement vanishes along the ${\it\rho}$ direction. Along the $z$ direction it becomes increasingly negative as ${\it\rho}$ decreases, for purely dipolar pushers and pullers alike (Pushkin et al. Reference Pushkin, Shum and Yeomans2013). However, as shown in figure 3(a), it becomes unphysical to use the pure dipolar swimmer model for tracers close to the swimmer where we expect entrainment in the forward direction. Furthermore, the pure dipole model neglects the effects of finite body size important for tracers close to the swimmer, as discussed in § 2.2.

Therefore, we present results in figure 3(b,c) for tracer trajectories near a finite-size swimmer obtained by taking $Q_{\Vert }=Q_{\bot }=5~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , corresponding to a swimmer with a characteristic size $a_{sw}\approx 1~{\rm\mu}\text{m}$ . The effect of the finite swimmer body size is clearly seen in the transition from backward displacement to forward entrainment for tracers with an initial position in the middle of the swimming path, at $z_{T}^{int}=0$ , around ${\it\rho}_{T}^{int}\approx 0.4~{\rm\mu}\text{m}$ . This transition can be calculated analytically for particles initially located close to the swimming path, i.e. in the limit $|\boldsymbol{r}_{T}^{int}|\ll L$ . In § 3.3 we present a detailed calculation and discussion of entrainment effects due to quadrupolar and higher-order terms.

The finite swimmer body size is also responsible for the shape of the tracer trajectories around $z_{T}^{int}=5$ . This becomes apparent in figure 3(d), where the trajectories are plotted in the rest frame of the swimmer. Here the tracer trajectories are explicitly shown for $z_{T}^{int,rest}=10~{\rm\mu}\text{m}$ and final displacements are shown for $z_{T}^{int,rest}=2,4,\dots ,18~{\rm\mu}\text{m}=z_{T}^{int,lab}+10~{\rm\mu}\text{m}$ compared to the laboratory frame. The tracers are advected by the flow, following a trajectory around the swimmer located at $z=5~{\rm\mu}\text{m}$ , and ending up in the same positions as the tracers in figure 3(b). The trajectories do not cross the separatrix around the swimmer, as discussed in § 2.2.

3.2. Azimuthal tracer displacements

Swimmers that rotate as they swim, such as numerous biological cells that propel by rotating their flagella, will also displace tracers in the azimuthal direction. Hence, a tracer trajectory will spiral about the direction of the swimmer. Since swimmers are torque-free as well as force-free, their heads and tails rotate in the opposite directions and may cause the tracer spiral trajectories to change chirality as the swimmer passes by. The resulting net azimuthal tracer displacement is far from obvious and will depend on the distance between the tracer and the swimmer path. We describe it using analytical results for tracers far from the swimmer path and numerical results for tracers closer to the swimmer path.

Far from the swimming path an analytic expression showing the contributions of the different multipoles to the tracer displacements follows from equations (S4.3)–(S4.8) of the supplementary data. Let us consider swimmers with a finite path from $-L$ to $L$ and tracers located initially in the mid-plane $z_{T}^{int}=0$ . To the lowest orders in ${\it\rho}^{-1}$ , where ${\it\rho}\gg L$ and noting that we use the shorthand notation ${\it\rho}\equiv {\it\rho}_{T}^{int}$ , the tracer displacement is

where $r_{L}^{2}=L^{2}+{\it\rho}^{2}$ . The first, dipolar, term confirms our conclusion that for a pusher (puller) the tracer is attracted to (repelled from) the swimming path. The second, quadrupolar, term corresponds to displacement along the swimming direction. The third term in (3.1), leading to azimuthal tracer displacement, is the octupolar rotlet. It is notable that the lower-order quadrupolar rotlet term does not appear here because, due to the forward–backward symmetry of the quadrupole rotlet velocity field component, the tracer is rotated in one direction by the head but then in the opposite direction by the tail through an equal angle. This angle, which may be relevant to fluid stirring, is

using equation (S4.8) of the supplementary data. For long swimming path lengths ${\rm\Delta}{\it\phi}_{T}$ tends to a finite value for a given value of ${\it\rho}$ .

Close to the swimmer path the expression (3.1) is invalid as the higher-order terms become important. Physically, these terms correspond to interference between the tracer displacements. In particular, the tracer displacement due to the dipole and the quadrupole rotlet components interfere in (2.14). Hence, the cancellation of the net azimuthal tracer displacement due to the forward–backward symmetry of the quadrupole rotlet velocity field does not occur even for tracers initially placed in the mid-plane. Therefore we should expect enhanced fluid stirring in the azimuthal direction near the swimmer. In the limit $|\boldsymbol{r}_{T}^{int}|\ll L$ this effect can be quantified analytically and we will present these results in § 3.3. Here we analyse the enhancement of azimuthal stirring close to the swimmer for a finite swimming path using numerical calculations.

Figure 4 shows the net tracer displacement in the azimuthal direction for a finite-size swimmer having the quadrupole rotlet strength $Q_{r}=10~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ . The enhancement of azimuthal stirring due to interference of the dipole and quadrupole rotlet terms is highlighted in figure 4(a). When ${\it\kappa}=0$ , tracers starting at $z_{T}^{int}=0$ have the net azimuthal displacement ${\rm\Delta}{\it\phi}_{T}=0$ . However, addition of a positive (pusher) dipole term leads to a significant increase of the displacement. A negative (puller) dipole term would also result in a significant displacement, though of a lesser magnitude than for a pusher of the same absolute dipole strength. At the two extrema $z_{T}^{int}=\pm L$ tracers are only rotated in one direction by the head (or tail). Inserting the parameters of figure 4(a) and $L\rightarrow 2L$ into (3.2) gives ${\rm\Delta}{\it\phi}_{T}=\mp 7.1^{\circ }$ at $z_{T}=\pm 5~{\rm\mu}\text{m}$ , in good agreement with the numerical results. At larger $z$ the interference effect becomes negligible and the ${\rm\Delta}{\it\phi}_{T}$ corresponding to different ${\it\kappa}$ values converge on the same curve.

Figure 4.

Azimuthal tracer displacement due to a rotating micro-swimmer moving along a finite path, from $-5$ to $5~{\rm\mu}\text{m}$ . Parameters used are ${\it\kappa}$ variable, $Q_{r}=10~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , $Q_{\Vert }=Q_{\bot }=1~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ , $V=20~{\rm\mu}\text{m}~\text{s}^{-1}$ , and all other multipole coefficients are set to zero. (a) Dependence of the azimuthal displacement on ${\it\kappa}$ . Initially, tracers are located along a line with ${\it\rho}=2~{\rm\mu}\text{m}$ and $z_{T}^{int}\in [-10,10]~{\rm\mu}\text{m}$ . Final tracer positions are found numerically for ${\it\kappa}=0$ (solid), ${\it\kappa}=+30$ (dotted), and ${\it\kappa}=-30~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ (dashed). (b) Displacements projected on the $(x,y)$ plane, with $\hat{\boldsymbol{e}}_{z}$ pointing into the paper; ${\it\kappa}=30~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ . Tracers are initially located in the plane $z_{T}^{int}=0$ , distributed on a grid, shown in light grey, with ${\it\rho}_{T}^{int}\in [0,3]~{\rm\mu}\text{m}$ with spacing $h_{{\it\rho}}=1/3~{\rm\mu}\text{m}$ and ${\it\phi}_{T}^{int}\in [-{\rm\pi},{\rm\pi}]$ with spacing $h_{{\it\phi}}=2{\rm\pi}/36=10^{\circ }$ . The final tracer displacement is shown by means of the transformed grid. Examples of tracer trajectories, initially located at ${\it\rho}_{T}^{int}=1,4/3,\dots ,3~{\rm\mu}\text{m}$ , run from open circles to black dots.

Figure 4(a) shows that overall pushers are more efficient at stirring fluid in the azimuthal direction than pullers of the same absolute dipole strength. This observation has a simple explanation: in (3.1) we have shown for a finite straight swimming path that tracers are attracted to the path if the swimmer is a pusher and repelled from it if the swimmer is a puller. Consequently, tracers spend more time closer to the path of a pusher and the interference effect is stronger. As the attraction/repulsion effect decays with the swimming path length $L$ , this difference between pushers and pullers wanes accordingly. Indeed, we will show in § 3.3 that for an infinite swimmer path the azimuthal tracer displacement is dominated by a term proportional to ${\it\kappa}Q_{r}/V^{2}$ that has the same magnitude for pushers and pullers of equal absolute dipole strength.

In figure 4(b) the tracer displacement is shown in the $(x,y)$ plane for tracers initially located at $z_{T}^{int}=0$ for ${\it\kappa}>0$ . Far away from the swimming path, whilst being attracted to the swimming path, the tracers are first rotated in the negative (anticlockwise) ${\it\phi}$ direction as the swimmer moves from $-L$ to 0, then back in the positive ${\it\phi}$ direction as the swimmer moves from 0 to $L$ . For example, the tracer trajectories starting between ${\it\rho}_{T}^{int}=2~{\rm\mu}\text{m}$ and $3~{\rm\mu}\text{m}$ show this well. The magnitude of the rotation is again predicted by (3.2). Close to the swimming path there is a strong rotation of tracer particles in the positive (clockwise) ${\it\phi}$ direction. This positive angular displacement corresponds to the dotted line in figure 4(a). In particular, the tracer trajectory starting at ${\it\rho}_{T}^{int}=1~{\rm\mu}\text{m}$ illustrates that a pusher first attracts the tracer particle towards the swimming path, rotates it several times about the $z$ -axis, and then repels it back to a ${\it\rho}$ value close to the initial value ${\it\rho}_{T}^{int}$ .

3.3. Tracer motion in the limit $|\boldsymbol{r}_{T}^{int}|\ll L$

We now consider tracer particles which are located near a long swimming path, so that $|\boldsymbol{r}_{T}^{int}|\ll L$ . For $L\rightarrow \infty$ the lowest-order expression for the net tracer displacement (3.1) vanishes. Physically this means that tracer trajectories become, generally, almost closed loops. Indeed, numerical results for the tracer displacement as a function of the swimming path length $L$ , presented in figure 5, show that the ${\it\rho}$  component of the displacement decays as ${\rm\Delta}\boldsymbol{r}_{T}^{{\it\rho}}\sim L^{-2}$ . This dependence follows directly from the first term in (3.1). The $z$ and ${\it\phi}$  components, however, remain finite for large swimming path lengths. In order to describe this behaviour analytically we need to take into account higher-order terms in the expansion for the tracer displacement (2.10).

Figure 5.

Dependence of the tracer displacement ${\rm\Delta}\boldsymbol{r}_{T}$ on the swimming path length  $2L$ . Initially, the tracer is located at $z_{T}^{int}={\it\phi}_{T}^{int}=0$ and ${\it\rho}_{T}^{int}=2~{\rm\mu}\text{m}$ . Final displacements, obtained numerically, in the ${\it\rho},{\it\phi},z$ directions are denoted by the solid, dotted and dashed lines respectively. Parameters used are $V=20~{\rm\mu}\text{m}~\text{s}^{-1}$ , ${\it\kappa}=30~{\rm\mu}\text{m}^{3}~\text{s}^{-1}$ , $Q_{\Vert }=Q_{\bot }=5~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ and $Q_{r}=1~{\rm\mu}\text{m}^{4}~\text{s}^{-1}$ . All other multipole coefficients are set to zero.

In § S4.1 of the supplementary data we compute contributions up to the sixth order inclusive, which gives us enough information to study both a short swimming path (3.1) and also the two leading-order contributions in powers of $1/{\it\rho}$ along each of the directions $z$ , ${\it\rho}$ and ${\it\phi}$ in the limit $L\rightarrow \infty$ . Hence, writing terms up to $O(1/{\it\rho}^{6})$ , we obtain the final tracer displacement

First, consider the final tracer displacement in the swimming direction, $\hat{\boldsymbol{e}}_{z}$ . Far from the swimmer, for large ${\it\rho}$ values, the ${\it\kappa}^{2}$ dipole term dictates a tracer displacement along the negative $z$ direction that scales as ${\rm\Delta}r_{z}\propto {\it\rho}^{-3}$ . This far-field result is the same for both pushers and pullers, since it is proportional to ${\it\kappa}^{2}$ . Closer to the swimming path, the negative displacement is countered by a forward entrainment due to the quadrupole term, which is positive for all values of $Q_{\Vert }$ and $Q_{\bot }$ . We have already seen this behaviour for tracers that start close to the middle of a finite swimming path in figure 3(b). In principle the octupole and higher-order terms, which will depend on individual swimmer details, could contribute to a backward displacement, although eventually a forward entrainment is expected very close to the swimmer due to the no-slip boundary condition at its surface. So, in general, although the expression (3.3) is only guaranteed to be exact in the limit of large ${\it\rho}$ , it gives a qualitatively correct picture of tracer displacements even for tracers initially located close to the swimmer. In particular, there is a characteristic critical radius ${\it\rho}={\it\rho}_{c}$ where the displacement in the swimmer direction changes sign:

For ${\it\rho}\gg {\it\rho}_{c}$ the far-field dipole approximation holds, whereas for ${\it\rho}\ll {\it\rho}_{c}$ the near-field dominates the tracer displacement. Notice that substituting the parameters of figure 3(b) into (3.4) gives ${\it\rho}_{c}=0.5~{\rm\mu}\text{m}$ , in reasonable agreement with the numerical results, given the small value $L=5$ chosen to emphasise the effects of a finite swimming path.

In the angular direction, $\hat{\boldsymbol{e}}_{{\it\phi}}$ , (3.3) shows that the displacement of tracer particles about the $z$ -axis scales as ${\rm\Delta}r^{{\it\phi}}\propto {\it\kappa}Q_{r}{\it\rho}^{-4}$ far from the swimming path. Therefore a micro-swimmer with a head (tail) rotating in the (anti-) clockwise direction will stir particles in the anticlockwise direction in the far-field. Dividing by the trajectory circumference $2{\rm\pi}{\it\rho}$ , the number of revolutions made by a tracer scales as ${\it\rho}^{-5}$ . Inserting the parameters of figure 4(b) into (3.3) gives ${\sim}2.7$ revolutions for a tracer particle with ${\it\rho}_{T}^{int}=1~{\rm\mu}\text{m}$ , which is in reasonable quantitative agreement with the figure where the particle is rotated ${\sim}3.2$ times around the swimming path.

Along the radial direction, $\hat{\boldsymbol{e}}_{{\it\rho}}$ , our calculations show that all contributions to the final tracer displacement vanish up to order $O({\it\rho}^{-7})$ and numerical integration of the implicit differential equation (2.8) also shows no resultant displacement to within machine precision. Although for $L=\infty$ the tracer does move along $\hat{\boldsymbol{e}}_{{\it\rho}}$ , it returns to its original distance from the swimming path ${\it\rho}_{int}$ once the swimmer has moved past. Physically, this is reasonable as we are assuming the fluid to be incompressible. That is, if ${\rm\Delta}r_{T}^{{\it\rho}}$ had a finite negative (positive) value, constant for all initial tracer positions $z_{T}^{int}$ along the infinite swimming path, then this would lead to a singular accumulation (depletion) of particles at the $z$ -axis, which would violate incompressibility. The final tracer displacement must be divergence-free and hence the radial displacement vanishes for a long swimming path.