2.1 Liquid-film stokeslet flow

The fluid is bounded by two parallel planar surfaces, a solid wall and an interface, at which no-slip and no-shear conditions must respectively be satisfied, in addition to a no-penetration condition (figure 1). We aim to solve the Stokes equations

Here, the fluid velocity $\boldsymbol{u}(\boldsymbol{x},t)$ and pressure $P(\boldsymbol{x},t)$ fields at location $\boldsymbol{x}=(x_{1},x_{2},x_{3})$ and time $t$ are due to a point force $\boldsymbol{f}\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{y})$ (stokeslet) that acts at position $\boldsymbol{y}=(y_{1},y_{2},y_{3}=h)$ . The boundary conditions for a film are the no-slip condition $\boldsymbol{u}(\boldsymbol{x},t)=0$ at the solid wall $x_{3}=0$ and both impermeability $u_{3}(\boldsymbol{x},t)=0$ and no-shear condition $\unicode[STIX]{x2202}_{3}\boldsymbol{u}(\boldsymbol{x},t)=0$ at the interface $x_{3}=H$ . In an unbounded fluid, the Green’s function is the Oseen tensor (Kim & Karilla Reference Kim and Karilla1991)

where $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{y}$ ,  $r=|\boldsymbol{r}|$ and $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta. The flow field is then given by $u_{i}^{S}(\boldsymbol{x},\boldsymbol{y},\boldsymbol{f})={\mathcal{J}}_{ij}\,f_{j}$ , where repeated indices are summed over. The corresponding pressure is $P(\boldsymbol{x},\boldsymbol{y},\boldsymbol{f})={\mathcal{P}}_{j}\,f_{j}$ with ${\mathcal{P}}_{j}=r_{j}/4\unicode[STIX]{x03C0}r^{3}$ .

If the film height is taken to infinity ( $H\rightarrow \infty$ ), then only the single no-slip boundary remains at $x_{3}=0$ and the stokeslet flow field $\boldsymbol{u}^{S}$ is altered by the addition of an auxiliary flow field, which can be written in terms of a system of images. This image flow field is given by the Blake tensor (Blake Reference Blake1971b ), which is centred at the position $\boldsymbol{Y}^{(0)}=(y_{1},y_{2},-y_{3})=\unicode[STIX]{x1D648}\boldsymbol{\cdot }\boldsymbol{y}$ , where the reflection matrix is $\unicode[STIX]{x1D648}=\text{diag}(1,1,-1)$ . The Blake tensor can be written in terms of the Oseen tensor (see Mathijssen, Pushkin & Yeomans Reference Mathijssen, Pushkin and Yeomans2015) as

which is a function of $\boldsymbol{x}$ and $\boldsymbol{y}$ , where the derivatives $\tilde{\unicode[STIX]{x2202}}_{j}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y_{j}=\unicode[STIX]{x1D614}_{jl}\,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}Y_{l}^{(0)}$ and $\tilde{\unicode[STIX]{x1D6FB}}^{2}=\tilde{\unicode[STIX]{x2202}}_{l}\tilde{\unicode[STIX]{x2202}}_{l}$ are with respect to the force position $\boldsymbol{y}$ . The tensor ${\mathcal{B}}_{ij}(\boldsymbol{x},\boldsymbol{Y}^{(0)})$ is recorded in the first row of table 1. The resulting flow field at $\boldsymbol{x}$ due to a point force at $\boldsymbol{y}=\boldsymbol{y}^{(0)}$ in the vicinity of a no-slip wall is then $u_{i}^{B}=[{\mathcal{J}}_{ij}(\boldsymbol{x},\boldsymbol{y}^{(0)})+{\mathcal{B}}_{ij}(\boldsymbol{x},\boldsymbol{Y}^{(0)})]\,f_{j}$ .

Similarly, if only a shear-free interface is present at $x_{3}=H$ , the auxiliary flow field of the stokeslet flow is a direct reflection centred at the position $\boldsymbol{Y}^{(-1)}=(y_{1},y_{2},2H-y_{3})$ , with the corresponding free-slip boundary tensor

This result is recorded in row two of table 1.

Table 1.

Recursion relations for the successive image systems of a stokeslet in a liquid film. The first image system of the Oseen tensor (2.3) from reflection in the bottom wall is the Blake tensor (2.4), and the second image from reflection in the top interface is the mirrored Oseen tensor (2.5). Subsequent image systems are obtained from further reflection operations, with $\text{B}$ denoting the ‘bottom’ (no-slip wall) and $\text{T}$ the ‘top’ (no-shear interface), which operate linearly on all the Oseen tensor terms ${\mathcal{J}}_{ij}$ of the image system tensor ${\mathcal{G}}_{ij}$ .

When both boundaries are present, the image system at $\boldsymbol{Y}^{(0)}$ corrects the boundary conditions at the solid no-slip wall ( $x_{3}=0$ ) but disturbs the boundary conditions at the film interface ( $x_{3}=H$ ), and vice versa for the image at $\boldsymbol{Y}^{(-1)}$ . This difficulty can be overcome by using an infinite series of images to find the flow that satisfies the film Stokes equations. That is, the image system at $\boldsymbol{Y}^{(0)}$ (or $\boldsymbol{Y}^{(-1)}$ ) can be reflected in the interface (or wall) to form a secondary image system at position $\boldsymbol{y}^{(-1)}$ (or $\boldsymbol{y}^{(1)}$ ), and so on. Hence, the positions of the image systems are

where $m=0,\pm 1,\pm 2,\ldots \,$ .

Figure 1 schematically shows the series of images. As the number of images goes to infinity, the boundary conditions at both surfaces are satisfied. Table 1 lists the procedure to find the image system tensors ${\mathcal{G}}_{ij}$ , and hence the velocity fields, of the first few image systems of a stokeslet in a film. Table 2 gives the resulting expressions of these tensors explicitly. The tensor of a given image system can be obtained by replacing all the Oseen tensors ${\mathcal{J}}_{ij}$ in the tensor of the previous image system by the appropriate Blake tensor ${\mathcal{B}}_{ij}$ or free-slip boundary tensor ${\mathcal{T}}_{ij}$ given by (2.4) and (2.5), respectively. It is key that all the resulting expressions are still in terms of Oseen tensors and their derivatives, which can again be replaced at the next reflection operation. Hence, by successively repeating the reflection operations (denoted by B or T for a ‘bottom’ or ‘top’ surface, operating from right to left), the image system tensor and thus the velocity field of the image systems are found via the recursion relations

where $m\geqslant 1$ , and the reflection operators $\text{B}$ and $\text{T}$ act linearly on all the Oseen tensors ${\mathcal{J}}_{ij}$ present in the image system tensor ${\mathcal{G}}_{ij}$ , as defined in table 1. The foundations of the recursion relations are

From these rules we obtain the Green’s function in a film from the infinite series

giving the film stokeslet flow $u_{i}^{F}(\boldsymbol{x},\boldsymbol{y},\boldsymbol{f})={\mathcal{F}}_{ij}\,f_{j}$ .

Table 2.

Explicit expressions of the image system tensors ${\mathcal{G}}_{ij}$ of the first few image systems of a stokeslet in a liquid film. The indices $i,j,k,l,o,p\in \{1,2,3\}$ and repeated indices are summed over. Added together, these tensors yield the Green’s function in a film (2.14).

This successive reflection method can also be used to construct the flow fields in more general confinement geometries, such as the flow bounded by two no-slip plates or the flow between a no-slip and a fluid–fluid (partial-slip) interface, by using the appropriate reflection operations, instead of those in (2.4) and (2.5). Furthermore, Staben et al. (Reference Staben, Zinchenko and Davis2003) showed that the flow field generated by a stokeslet between two no-slip plates can be written as two Blake images and a rapidly decaying integral term. We anticipate that the same could be achieved for the stokeslet flow in a liquid film.

2.2 Multipole expansion

In this section, we summarise how the flow field generated by a micro-swimmer is related to the stokeslet in a film. The micro-organism is modelled as a prolate spheroid with semi-major and semi-minor axes $a$ and $b$ , respectively, and aspect ratio $\unicode[STIX]{x1D6FE}=a/b$ . Unless otherwise stated, we use $a$ as the characteristic swimmer size throughout the text. The organism is located at position $\boldsymbol{y}$ with orientation $\boldsymbol{p}$ , swimming in a film of height $H$ (figure 1). In addition to its own motility (propulsion velocity $\boldsymbol{v}^{SW}=v^{SW}\boldsymbol{p}$ ), the motion of a swimming cell is affected by steric and hydrodynamic interactions with the bounding planes, plus any background flow.

As a micro-swimmer moves, it generates a flow $\boldsymbol{u}$ . This swimmer-generated flow field can be written in terms of a multipole expansion of the Stokes flow solution in the film (2.14). Because neutrally buoyant micro-swimmers do not subject their surrounding fluid to a net force or torque, we exclude the stokeslet and Stokes rotlet terms from the expansion. Similarly, the assumption that the swimmer is cylindrically symmetric about the swimming direction allows us to exclude the non-symmetric terms (see Mathijssen et al. Reference Mathijssen, Pushkin and Yeomans2015). Hence, an axisymmetric force-free and torque-free micro-swimmer generates a velocity field

where $\boldsymbol{u}^{D}$ is the Stokes dipole, $\boldsymbol{u}^{Q}$ the quadrupole, $\boldsymbol{u}^{SD}$ the source doublet and $\boldsymbol{u}^{RD}$ the rotlet doublet (Spagnolie & Lauga Reference Spagnolie and Lauga2012; Mathijssen et al. Reference Mathijssen, Pushkin and Yeomans2015). Though including more than four terms in the multipole expansion might more accurately describe the near-field flow due to swimmer-specific details, these few terms satisfactorily account for the universal attributes of a generic micro-swimmer.

Figure 2.

Flow fields produced by a micro-swimmer at the centre of a film of height $H$ , obtained using the recursive series method with $n=9$ images: shown are (a) dipolar $\boldsymbol{u}^{D}$ (2.16), (b) quadrupolar $\boldsymbol{u}^{Q}$ (2.17), (c) source doublet $\boldsymbol{u}^{SD}$ (2.18), and (d) rotlet doublet $\boldsymbol{u}^{RD}$ (2.19). The flow fields shown correspond to planar cuts close to the swimmer, $x_{1}=-a$ , $x_{2}=a$ and $x_{3}=h-a$ , where $h=H/2$ , $a=H/100$ and all lengths are scaled with respect to $H$ . The schematic swimmer points in the swimming direction $\boldsymbol{p}=\hat{\boldsymbol{e}}_{1}$ . Colour maps show the velocity magnitude normalised by its maximum on a logarithmic scale, ranging from $10^{-2}$ (blue) to $1$ (red), and are superimposed by streamlines (black lines).

Each contribution to the multipole expansion can be written in terms of derivatives of the Green’s function in the film (2.14). Specifically,

where the derivatives act on the swimmer position $\boldsymbol{y}$ . The multipole coefficients $\unicode[STIX]{x1D705}$ and ( $\unicode[STIX]{x1D708},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}$ ) have dimensions of ( $\text{velocity}\times \text{length}^{2}$ ) and ( $\text{velocity}\times \text{length}^{3}$ ), respectively.

These multipole flow fields (2.16)–(2.19) are shown in figure 2. The dipole arises from the opposing propulsion and drag forces exerted by the swimmer ( $\boldsymbol{u}^{D}$ ; figure 2 a). Pusher-type swimmers such as E. coli have a positive force dipole with $\unicode[STIX]{x1D705}>0$ (shown in figure 2 a). Pushers drive fluid out along the swimming direction and draw fluid in from the sides. Puller-type swimmers, on the other hand, have $\unicode[STIX]{x1D705}<0$ and draw fluid in along their swimming axis (Lauga & Powers Reference Lauga and Powers2009). The quadrupole flow field represents the fore–aft asymmetry of the micro-organism ( $\boldsymbol{u}^{Q}$ ; figure 2 b). For example, the quadrupole describes the weighting of propulsion forces towards the posterior of flagellated bacteria, in which case one expects $\unicode[STIX]{x1D708}>0$ . The source doublet represents the finite size of the swimmer ( $\boldsymbol{u}^{SD}$ ; figure 2 c). For ciliated organisms with a slip velocity at their surface, $\unicode[STIX]{x1D70E}>0$ ; whereas for non-ciliated swimmers, one would expect $\unicode[STIX]{x1D70E}<0$ because this corresponds to the Faxén correction to the stokeslet flow for a finite-sized solid sphere. Finally, the rotlet doublet represents the opposing rotation of the swimmer’s head and tail ( $\boldsymbol{u}^{RD}$ ; figure 2 d) (Spagnolie & Lauga Reference Spagnolie and Lauga2012; Mathijssen et al. Reference Mathijssen, Pushkin and Yeomans2015).

Figure 3.

The flow field generated by a micro-swimmer is modified in thin films: (a–c) flow fields for decreasing values of the film height, $H=10H_{0}$ , $H=3H_{0}$ and $H=H_{0}$ , respectively. The micro-swimmer is located at the middle of the film, at $h=H/2$ , and oriented in the direction $\boldsymbol{p}=\hat{\boldsymbol{e}}_{1}$ . Although the flow is purely dipolar far away from the boundaries (a), this lower-order multipole is screened with increasing confinement and a recirculating flow pattern appears (b,c). The number of images, planar cuts and colour legends are defined as in figure 2.

In a film of sufficiently large thickness, the flow field is relatively unaffected by the boundaries (figure 3 a). Upon decreasing the height of the film, the flow profiles in the $x_{1}$ – $x_{3}$ and $x_{2}$ – $x_{3}$ planes remain unaffected, except near the surfaces, where the boundary conditions must be satisfied. However, as the thickness of the film is reduced, the flow field in the $x_{1}$ – $x_{2}$ plane is modified and recirculating flow patterns appear close to the swimming cell (figure 3 b). Such patterns are reminiscent of those seen for a stokeslet between two parallel plates (Liron & Mochon Reference Liron and Mochon1976). The recirculating regions are enhanced in size as the film thickness is further reduced (figure 3 c) compared to the size of the swimmer.

The effect of reducing the film thickness on altering the flow structure is understood by considering the minimisation of the energy dissipation by a self-propelled organism in Stokes flow. As the thickness of the film is reduced, the propulsion energy is more effectively dissipated by the boundaries. This introduces a length scale of order ${\sim}H$ beyond which the primary contributions to the flow field are screened. As a result of the emergence of this screening length, the flow field is suppressed in the plane normal to the film height and recirculating flow patterns are formed.

To understand the case of strong confinement, we derive the stokeslet and swimmer-generated flow fields in the thin-film limit (appendix B). In this limit, the flows in the $i$ direction due to a swimmer pointing in the $j$ direction decay exponentially with the lateral distance if either $i$ or $j$ or both are equal to 3 (i.e. directed perpendicular to the film). Only the parallel components of the flow do not decay exponentially, and those have a half-parabolic profile along $x_{3}$ . The stokeslet has a recirculating pattern of two loops in the $x_{1}$ – $x_{2}$ plane, the dipole has four loops, the quadrupole has six loops, and the source doublet maintains its bulk-flow structure with two loops (appendix B; figure 10). The solution for higher-order multipoles between two no-slip surfaces, which we have not found elsewhere in the literature, are the same, but with a parabolic profile along $x_{3}$ . These thin-film limit expressions could be used, for example, to model swimmer–swimmer interactions in films of thickness comparable to the swimmer size.

Figure 4.

Comparison of flow fields produced by a stokeslet in a liquid film using (a,c) the solution obtained with the recursive series method (§ 2.1), where $n=9$ images have been used, and (b,d) the exact solution (appendix A). The film has height $H$ , the stokeslet is located at $h=0.45H$ just beneath the centre, an arbitrary non-trivial point, and it is oriented in the $x_{1}$ direction (a,b) or in the $x_{3}$ direction (c,d). The planar cuts and colour legends are defined as in figure 2. The white dashed lines mark the region $r_{1}/H,r_{2}/H<1$ , inside which the series solution is accurate.

Figure 5.

Convergence of the flow fields generated by a micro-swimmer in a film as a function of the number of images $n$ , normalised with respect to the value if only the swimmer itself is present ( $n=0$ ), on a logarithmic scale. Only odd values of $n$ are shown, which corresponds to pairs of image systems being added on both sides of the film. The swimmer is located just underneath the middle of the film $\boldsymbol{y}=(0,0,0.45H)$ , and the flow is sampled at arbitrary points on the two surfaces $\boldsymbol{x}=(0,H/4,0)$ and $\boldsymbol{x}=(0,H/4,H)$ . In (a–d) the swimmer is oriented in the $\hat{\boldsymbol{e}}_{1}$ direction, and in (e–h) the swimmer is oriented in the $\hat{\boldsymbol{e}}_{3}$ direction. The convergence of the boundary conditions is shown at the bottom wall (a,c,e,g), and top interface (b,d,f,h) for the normal and parallel flow components. Note that, if no marker is shown, the value is exactly zero.

2.3 Comparison to the exact solution

To establish the length scales for which the recursive series solution (§ 2.1) is applicable, we compare it to the exact solution (appendix A). There are four important length scales in the system of a micro-swimmer in a liquid film: the organism size $a$ , the film height $H$ , the distance to the nearest surface ${\hat{h}}=\min (h,H-h)$ , and the size of the flow region of interest, measured by the distance from the swimmer to where the flow is evaluated $r=|\boldsymbol{x}-\boldsymbol{y}|$ .

Figure 4 shows the flow fields generated by a stokeslet in a liquid film, using the recursive image series (a,c) and the exact solution (b,d). The stokeslet is located at a distance $h=0.45H$ from the bottom wall and is oriented in the direction parallel (a,b) and perpendicular (c,d) to the film surfaces. Close to the point force, where $a\sim r\ll {\hat{h}}<H$ , the flow in a bulk fluid given by the Oseen result (2.3) is recovered. If the film height is increased, with $h$ kept constant such that $a\sim r\sim h\ll H$ , then the Blake result (2.4) is recovered (see details in appendix C, figure 11). Similarly, if the stokeslet is located close to the top interface, the local flows can also be described with a single image system (2.5). However, for a stokeslet in the middle of the film ( ${\hat{h}}\sim H$ ), the image systems above and below the film contribute with equal significance. If one is interested in distant flows ( $a<{\hat{h}}\sim H\ll r$ ; figure 4, outside the white dashed lines), then all images are approximately equidistant from the point $\boldsymbol{x}$ , so that many terms are required in the series to eliminate differences between the two methods. This regime is equivalent to the thin-film limit (appendix B). If one is interested in local flows ( $a<r<{\hat{h}}\sim H$ ; figure 4, inside the white dashed lines), then there is good agreement between the recursive image series (a,c) and the exact solution (b,d), even with a limited number of images ( $n=9$ ). This can be understood because in this domain the series converges and can be truncated, as discussed quantitatively below. Since this work is concerned with the effect of hydrodynamic interactions with film surfaces on the dynamics of micro-swimmers themselves, the recursive image series method is seen to be accurate.

Convergence of the swimmer-generated flow fields at the surfaces is shown in figure 5 as a function of the number of images used in the recursive series. The values of the flow (or shear rate) at the boundaries are shown, and these must approach zero to accurately calculate the flows. Here $n=0$ means no images are present, and only the flow generated by the stokeslet in the bulk is considered. As $n$ increases, more image reflections are included, where the image numberings are defined in table 1. The swimmer location is chosen near the centre of the film, as a worst-case scenario, and we show the flows at the surfaces where the boundary effects are strongest and the convergence is the slowest, with ${\hat{h}}\sim H\sim r$ . Both at the bottom wall (figure 5 a,c,e,g) and the top interface (figure 5 b,d,f,h), however, the correction to the flow field is small after only a few images. Furthermore, with $n=1,5,9,\ldots ,$ image systems included, the boundary conditions at the bottom wall are satisfied exactly; and similarly, with $n=3,7,11,\ldots ,$ the top interface boundary conditions are fully satisfied. This feature of the recursive series method can be leveraged if an exact boundary condition is required at one of the two surfaces. For thin films compared to the swimmer size, or distances much greater than the film height, more images will be required for a given accuracy. The convergence can be justified by noting that each term of each image decays as ${\sim}(-1)^{m}|2mH|^{-1}$ . Since at every image reflection point ( $\boldsymbol{y}^{(m)},\boldsymbol{Y}^{(m)}$ ) the leading term is a stokeslet pointing in the direction opposite to the one at the previous reflection point, this infinite series of alternatingly opposing stokeslets converges and can be expressed as converging integral expressions (§ A.2). For a dipole swimmer, the leading terms decay as ${\sim}(-1)^{m}|2mH|^{-2}$ , so the infinite series of alternatingly opposing dipoles converges more rapidly.

In short, the recursive series can compute flows accurately in the region $r<H$ for any $h$ and $x_{3}$ (figure 4, inside the white dashed lines). This is a region of particular interest since it is, by construction, where the swimmer resides and where perturbations to the flow fields are most significant. Secondly, one requires $a\ll {\hat{h}}$ to evaluate hydrodynamic interactions with surfaces using the multipole expansion, but it is noteworthy that Spagnolie & Lauga (Reference Spagnolie and Lauga2012) argue that far-field hydrodynamic interactions give surprisingly accurate results even for small swimmer–wall separations $a\sim {\hat{h}}$ . When these two conditions are satisfied, we find that $n=9$ images are sufficient to describe micro-swimmer flows with an error less than ${\sim}1\,\%$ . Therefore, in this work, we utilise $n=9$ images in all presented figures.

The advantage of the traditional Fourier-transform solution is that it is exact and provides access to all regions of the film. In particular, it converges rapidly in regions far from the swimmer, $r\gg H$ , and therefore a tractable expression for the flows in the far-field limit can be extracted (appendix B). On the other hand, this exact solution can be more tedious to handle, especially when taking derivatives, as is necessary for a multipole expansion of a swimming microbe. In comparison, the image series in (2.14) is constructed purely from the Oseen tensor and its derivatives, and can be manipulated with ease, both analytically and computationally. The best choice of method therefore depends on the purpose in mind. In the far field the Fourier-transform method excels, and in the near field the recursive series method is more convenient. Since we are interested in local hydrodynamics, we work with the latter in the remainder of this paper.

3.1 Dipolar hydrodynamic interactions

The primary contribution to the flow field generated by a force-free micro-swimmer at low Reynolds number is the Stokes dipole. A flagellated swimmer with its propulsion mechanism located at the rear of its body is a pusher swimmer with dipole coefficient $\unicode[STIX]{x1D705}>0$ , whereas an organism propelling itself with flagella at the front of its body is modelled as a puller with $\unicode[STIX]{x1D705}<0$ (Lauga & Powers Reference Lauga and Powers2009).

To determine the dipolar hydrodynamic interactions with the film surfaces, we insert the auxiliary dipolar flow field (2.16), $\boldsymbol{u}^{D\ast }=\boldsymbol{u}^{D}-\boldsymbol{u}^{D\infty }$ , into (3.3) and (3.4). The first two image systems give

The complete list of dipolar hydrodynamic interactions due to the first 10 image systems is given in table 3 for a small swimmer, with $a\ll H$ in (3.3), in terms of the dimensionless position $\unicode[STIX]{x1D701}=h/H$ .

Table 3.

Dipolar hydrodynamic interactions of a micro-swimmer with the surfaces of a film. Given are the boundary-induced translational and rotational velocities due to the first image systems (tables 1 and 2) as a function of non-dimensionalised swimmer position $\unicode[STIX]{x1D701}=y_{3}/H$ and orientation $\boldsymbol{p}=(p_{1},p_{2},p_{3})$ with respect to the no-slip wall at $\unicode[STIX]{x1D701}=0$ and free-slip interface at $\unicode[STIX]{x1D701}=1$ . In each case, the functional term listed is to be multiplied by the stated prefactor. For simplicitly, we consider small swimmers with respect to the film height, so the higher-order Faxén term proportional to $a^{2}$ in (3.3) is omitted here.

A pusher-type swimmer is attracted towards both surfaces and tends to align parallel to them (figure 6 a) (Berke et al. Reference Berke, Turner, Berg and Lauga2008; Spagnolie & Lauga Reference Spagnolie and Lauga2012). The $h$ – $\unicode[STIX]{x1D719}$ phase space in figure 6(a) shows stable fixed points at $h=b,H-b$ , $\unicode[STIX]{x1D719}=0,\pm \unicode[STIX]{x03C0}$ , demonstrating that the specific trajectories shown (red, blue and green) are representative of the general behaviour of pusher swimmers in films.

Figure 6.

The effect of dipolar hydrodynamic interactions on the dynamics of a micro-swimmer in a film. Swimmer trajectories in real $x_{1}$ – $x_{3}$ space are shown in the left-hand panels, and the right-hand panels show the same dynamics in $h$ – $\unicode[STIX]{x1D719}$ phase space. For simplicity, we consider spherical swimmer bodies. The swimmer size is small compared to the film height, $a=b=H/100$ . The background colours in the phase-space plots indicate the sign of the velocity in the $x_{3}$ direction, where yellow is upward swimming and light blue is downward swimming. (a) A pusher swimmer with $\unicode[STIX]{x1D705}=(1/20)[H^{2}v^{SW}]$ . (b) The equivalent puller with $\unicode[STIX]{x1D705}=-(1/20)[H^{2}v^{SW}]$ . All other multipole moments are set to zero. The schematic black micro-organisms indicate the swimming direction.

A puller, on the other hand, tends to orient perpendicular to the planar boundaries (figure 6 b). A puller’s stable fixed points are found at $h=a,H-a$ , $\unicode[STIX]{x1D719}=\pm \unicode[STIX]{x03C0}/2$ . For both pusher and puller types, the hydrodynamic interactions are a factor of $3/2$ stronger at the solid, no-slip bottom wall than at the no-shear free interface. All dipolar swimmers initially oriented parallel to the surfaces and located below a critical height $h^{\ast }$ will accumulate at the bottom wall. The blue trajectories in figure 6(a,b) correspond to $h=H/2<h^{\ast }$ and demonstrate that cells that began swimming parallel to the surfaces along the centreline accumulate at the bottom wall. An analytic estimate for this critical height can be obtained using the first two image systems (3.5)–(3.7). This gives $h^{\ast }\approx (3-\sqrt{6})H$ compared to the numerically determined value $h^{\ast }=0.565H$ , which is different by $2.6\,\%$ . In brief, hydrodynamic interactions attract both pushers and pullers more towards the no-slip wall than towards the film interface, when averaged over all initial swimmer positions and orientations.

3.2 Source doublet hydrodynamic interactions

For ciliated micro-organisms, such as Paramecium and Volvox, that rely on local surface deformations as propulsion mechanism, or squirmer swimmers (Blake Reference Blake1971a ; Lauga & Powers Reference Lauga and Powers2009) with a slip velocity at their surface, such as active colloids (Paxton et al. Reference Paxton, Kistler, Olmeda, Sen, St Angelo, Cao, Mallouk, Lammert and Crespi2004; Howse et al. Reference Howse, Jones, Ryan, Gough, Vafabakhsh and Golestanian2007; Valadares et al. Reference Valadares, Tao, Zacharia, Kitaev, Galembeck, Kapral and Ozin2010), it is expected that the source dipolar coefficient $\unicode[STIX]{x1D70E}>0$ . For non-ciliated but flagellated organisms, one expects the coefficient $\unicode[STIX]{x1D70E}<0$ . Table 4 lists the hydrodynamic interactions of a small swimmer with the surfaces due to its source doublet flow field. The first two image systems give

Table 4.

Same as table 3, but for source doublet hydrodynamic interactions.

While the force dipole is the most significant contribution to the hydrodynamic interactions attracting swimmers to the film surfaces, we find that the higher-order terms can account for a boundary repulsion if $\unicode[STIX]{x1D70E}>0$ (figure 7 a). This can be understood since the source doublet component of the flow field regularises the singular description of the swimmer’s flow field and hence gives the swimmer an effective characteristic hydrodynamic size (Kim & Karilla Reference Kim and Karilla1991; Mathijssen et al. Reference Mathijssen, Pushkin and Yeomans2015), $a^{H}=\sqrt[3]{2\unicode[STIX]{x1D70E}/v^{SW}}$ . When the source doublet contribution is significant, the hydrodynamic size can be substantially larger than the characteristic geometric size of the swimmer $a$ , which can aid escape from boundaries or keep the swimmers from coming into direct contact with surfaces. The corresponding hydrodynamic interactions turn a swimmer away from a no-slip wall, so that the distance of closest approach is $h^{B}=a^{H}/2^{1/3}$ (Mathijssen et al. Reference Mathijssen, Shendruk, Yeomans and Doostmohammadi2016b ). Likewise, close to a free-slip interface, the closest distance of approach is $h^{T}=a^{H}/2$ .

Figure 7.

The effect of source doublet hydrodynamic interactions on the dynamics of a micro-swimmer in a film, as defined in figure 6. (a) A ciliated swimmer with $\unicode[STIX]{x1D70E}=(1/10)^{3}[H^{3}v^{SW}]$ so that the distances of closest approach are $h^{B}=H/10$ and $H-h^{T}=0.937H$ . (b) A flagellated sphere with $\unicode[STIX]{x1D70E}=-(1/10)^{3}[H^{3}v^{SW}]$ . The cyan trajectories are for non-spherical swimmers with an aspect ratio of $\unicode[STIX]{x1D6FE}=3$ with $a/3=b=H/100$ .

For spherical ( $\unicode[STIX]{x1D6FE}=1$ ) ciliated swimmers, the angular velocity vanishes near the top no-shear interface. Such swimmers are only turned away from the bottom boundary and therefore the source doublet leads to accumulation at the top surface. The red and dark blue trajectories in figure 7(b) show this explicitly and the $h$ – $\unicode[STIX]{x1D719}$ phase space shows that all swimmers that initially approach the bottom wall are turned away and accumulate at the film interface where they have the same orientation as that with which they initially approached the bottom wall. Ciliated swimmers that initially approach the interface immediately accumulate there, maintaining their initial angle (figure 7 a).

Elongated ( $\unicode[STIX]{x1D6FE}>1$ ) ciliated swimmers, however, are subject to an additional angular velocity at the top surface, which also turns the swimmer away. Hence, accumulation at both boundaries can be prevented by the action of the source doublet, and the elongated swimmer ( $\unicode[STIX]{x1D6FE}=3$ ) oscillates between the two surfaces (cyan trajectory in figure 7 a).

Non-ciliated organisms with $\unicode[STIX]{x1D70E}<0$ , conversely, are bound more strongly to the surfaces than ciliated swimmers. The source doublet hydrodynamics for any swimmer initially oriented towards the wall act to further orient the swimmer towards a head-on collision with the wall (figure 7 b, red and blue trajectories). This is because a spherical body that translates past a solid wall also rotates in a rolling fashion due to hydrodynamic interactions (Kim & Karilla Reference Kim and Karilla1991). In the case of a swimmer, this effect points the propulsion direction towards the boundary, thus trapping the swimmer. The $h$ – $\unicode[STIX]{x1D719}$ phase space shows that all swimmers initially oriented towards the wall approach the stable fixed point at $h=h^{B}$ , $\unicode[STIX]{x1D719}=\pm \unicode[STIX]{x03C0}/2$ . Spherical swimmers initially oriented towards the top interface move with a roughly constant $\unicode[STIX]{x1D719}$ and keep that orientation, which is similar to ciliated swimmers (figure 7 a). Elongated swimmers, instead, rotate towards the interface (cyan trajectory).

In summary, higher-order hydrodynamic moments are required to fully characterise swimmer dynamics in confined geometries, as large hydrodynamic radii, $a^{H}\gg a$ , can lead to non-trivial trajectories and hence particle distributions.

Figure 8.

The effect of quadrupolar hydrodynamic interactions on the dynamics of a micro-swimmer in a film, as defined in figure 6. (a) A swimmer with a long flagellum and small body, $\unicode[STIX]{x1D708}=(1/10)^{3}[H^{3}v^{SW}]$ and $h^{B}=0.096H$ and $H-h^{T}=0.921H$ . (b) A swimmer with a short flagellum and big body, $\unicode[STIX]{x1D708}=-(1/10)^{3}[H^{3}v^{SW}]$ .

3.3 Quadrupolar hydrodynamic interactions

Next, we characterise the impact of the quadrupolar term in (2.15). This describes the weighting of propulsion forces towards the posterior end of flagellated organisms and therefore accounts for the fore–aft asymmetry. Accounting for the first two images gives

For swimmers with long flagella and small bodies, the quadrupolar coefficient is positive, $\unicode[STIX]{x1D708}>0$ (figure 8 a). For small angles of approach measured from the surface tangent (blue and green trajectories), the swimmer is turned away from the top and bottom surfaces. Such swimmers with $\unicode[STIX]{x1D719}\approx \unicode[STIX]{x03C0}$ perpetually oscillate between the two bounding surfaces. However, for larger angles of approach, the swimmer can get stuck at the fixed point $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2$ pitched downwards (figure 8 b, red trajectory), or pitched upwards at $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ . This mechanism could help prevent the flagella from touching the surface, facilitating an easier escape. For a spherical swimmer (as in figure 8), both fixed points are stable. However, the stability decreases at the bottom wall with increasing elongation $\unicode[STIX]{x1D6FE}$ , which favours top-surface accumulation.

Conversely, swimmers with a large body and small flagella, i.e. $\unicode[STIX]{x1D708}<0$ , are only weakly rotated towards the boundaries (figure 8 b, blue trajectory). Interestingly, such swimmers have different fixed points than the others: The majority of the fixed points in figures 6–8 are at $\unicode[STIX]{x1D719}=0,\pm \unicode[STIX]{x03C0}/2,\pm \unicode[STIX]{x03C0}$ , with the exception of figure 8(b), where swimmers have fixed points at an exceptional angle to the surfaces ( $\unicode[STIX]{x1D719}=\pm \unicode[STIX]{x03C0}/4$ ).

Like the source doublet hydrodynamics, the quadrupolar term turns the swimmer away from both the no-slip and no-shear surfaces for small angles of approach measured from the surface tangent. The distance of nearest approach allowed by the quadrupolar term is $h^{B}=\sqrt[3]{7\unicode[STIX]{x1D708}/8v^{SW}}$ for the no-slip wall and $h^{T}=\sqrt[3]{\unicode[STIX]{x1D708}/2v^{SW}}$ for the no-shear interface. While it is well known that it is energetically favourable for small colloids to reside at interfaces, this is not universally observed for microbes in the presence of air–water interfaces. The source doublet and quadrupolar distance of nearest approach provide a possible explanation for cases when $a<h^{T}$ .

Table 5.

Same as table 3, but for rotlet doublet hydrodynamic interactions.

3.4 Rotlet doublet hydrodynamic interactions

Lastly, we describe the hydrodynamic interactions due to the rotlet doublet flow field of strength $\unicode[STIX]{x1D70F}$ , representing the counter-rotating swimmer head and tail. Table 5 lists the hydrodynamic interactions due to this term in the flow field. A swimmer immediately above a single no-slip boundary moves in circles in the plane parallel to the boundary. E. coli bacteria are observed to move in the clockwise direction, as seen from the liquid side, but in the counter-clockwise direction near a free-slip surface (Berg & Turner Reference Berg and Turner1990; Frymier et al. Reference Frymier, Ford, Berg and Cummings1995; DiLuzio et al. Reference Diluzio, Turner, Mayer, Garstecki, Weibel, Berg and Whitesides2005; Di Leonardo et al. Reference Di Leonardo, Dell’Arciprete, Angelani and Iebba2011; Spagnolie & Lauga Reference Spagnolie and Lauga2012). In a film, these effects are additive. Figure 9(a) shows the radius of curvature of the circular trajectory $R(h,\unicode[STIX]{x1D6FE})$ as a function of the swimmer’s position in the film $h$ and its aspect ratio $\unicode[STIX]{x1D6FE}$ . Using the first two image systems ( $n=2$ ) gives an analytic estimate of the radius of curvature,

where $G=(\unicode[STIX]{x1D6FE}^{2}-1)/(\unicode[STIX]{x1D6FE}^{2}+1)$ . This function is maximised in the middle of the film, at $h=H/2$ , if the swimmer has a spherically symmetric body. The maximum radius of curvature is then $R_{0}=R(H/2,\unicode[STIX]{x1D6FE}=1)\approx v^{SW}H^{4}/6\unicode[STIX]{x1D70F}$ . For an elongated swimmer, the function $R(h,\unicode[STIX]{x1D6FE}>1)$ is no longer symmetric about the film centreline: the radius of curvature is larger near the bottom wall but smaller near the top surface. Consequently, elongated swimmers’ trajectories bend more near the water–air interface (figure 9 b).

Figure 9.

Spiralling trajectories in a film due to the counter-rotation of a swimmer’s head and tail. (a) Radius of curvature $R(h,\unicode[STIX]{x1D6FE})$ of the swimmer moving in circles in the plane parallel to the surfaces for various aspect ratios $\unicode[STIX]{x1D6FE}$ , normalised with respect to $R_{0}=R(H/2,1)$ . (b) An example trajectory of an elongated swimmer ( $\unicode[STIX]{x1D6FE}=4$ ) that has just escaped from the bottom wall, with a small out-of-plane angle $\unicode[STIX]{x1D719}=0.1^{\circ }$ . Colours indicate time passing, ranging from violet to red. The rotlet doublet coefficient is $\unicode[STIX]{x1D70F}=(1/10)[H^{3}v^{SW}]$ , all other multipole coefficients are set to zero, and $n=9$ images have been used.

In microfluidic experiments, the radius of curvature of swimming trajectories can be used to determine whether micro-organisms are located near the top or bottom microscope slide, as was shown qualitatively in experiments by Guidobaldi et al. (Reference Guidobaldi, Jeyaram, Condat, Oviedo, Berdakin, Moshchalkov, Giojalas, Silhanek and Marconi2015). More quantitatively, the knowledge of the function $R(h,\unicode[STIX]{x1D6FE})$ now allows one to compute the vertical position $h$ by inverting (3.14). The rotlet doublet coefficient $\unicode[STIX]{x1D70F}$ can also be determined with this equation using the maximum in radius of curvature in the middle of the film. This may be more reliable and effective than earlier calibration techniques used near a single surface, where an accurate measurement of the swimmer–wall separation is critical and higher-order multipole terms and lubrication effects are hard to separate out.

Between two no-slip walls such as two glass slides, the rotary coefficient $\unicode[STIX]{x1D70F}$ is challenging to determine because there is no maximum radius of curvature $R_{0}$ . However, once the parameters $\unicode[STIX]{x1D6FE}$ , $\unicode[STIX]{x1D70F}$ and $H$ are deduced in the film, the vertical position $h$ between two microscope slides can be computed from experimentally measured radii of curvature, using the equivalent of (3.14) for two parallel no-slip plates:

Moreover, the parameters can be used in different experimental geometries, such as a lubrication layer (solid–liquid–liquid) or a floating oil film (liquid–liquid–air), for which similar expressions for $R(h,\unicode[STIX]{x1D6FE})$ can be computed.