# [

###### Abstract

One of the principal mechanisms by which surfaces and interfaces affect microbial life is by perturbing the hydrodynamic flows generated by swimming. By summing a recursive series of image systems we derive a numerically tractable approximation to the three-dimensional flow fields of a Stokeslet (point force) within a viscous film between a parallel no-slip surface and no-shear interface and, from this Green’s function, we compute the flows produced by a force- and torque-free micro-swimmer. We also extend the exact solution of Liron & Mochon (1976) to the film geometry, which demonstrates that the image series gives a satisfactory approximation to the swimmer flow fields if the film is sufficiently thick compared to the swimmer size, and we derive the swimmer flows in the thin-film limit. Concentrating on the thick film case, we find that the dipole moment induces a bias towards swimmer accumulation at the no-slip wall rather than the water-air interface, but that higher-order multipole moments can oppose this. Based on the analytic predictions we propose an experimental method to find the multipole coefficient that induces circular swimming trajectories, allowing one to analytically determine the swimmer’s three-dimensional position under a microscope.

Hydrodynamics of Micro-swimmers in Films]Hydrodynamics of Micro-swimmers in Films
A.J.T.M. Mathijssen, A. Doostmohammadi, J.M. Yeomans and T.N. Shendruk]A.J.T.M. Mathijssen^{†}^{†}thanks: Email address for correspondence: mathijssen@physics.ox.ac.uk,A. Doostmohammadi, J.M. Yeomansand T.N. Shendruk

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## 1 Introduction

Beyond simply containing microbes and their surrounding fluids, surfaces and interfaces alter the behaviours, dynamics and even biological traits exhibited by swimming cells (Bukoreshtliev et al. 2013). Surfaces and interfaces affect micro-swimmer trajectories through hydrodynamics-induced interactions and impact the flows generated by microbes as they move through confined environments. The majority of research has focused on the effect of solid boundaries on swimming dynamics. In particular, the accumulation of bacteria at solid walls has been well demonstrated, both theoretically and in the experiments (Pedley & Kessler 1987; Lauga et al. 2006; Berke et al. 2008; Or & Murray 2009; Crowdy & Or 2010; Li & Tang 2009; Drescher et al. 2009; Li et al. 2011; Spagnolie & Lauga 2012; Molaei et al. 2014; Ishimoto et al. 2016). Likewise, recent studies have demonstrated the interplay between flowing fluids and swimming cells in various geometries for Newtonian (Chacón 2013; Zöttl & Stark 2012, 2013; Costanzo et al. 2012; Masoud et al. 2013; Kantsler et al. 2014; Figueroa-Morales et al. 2015) and non-Newtonian fluids (Karimi et al. 2013; Ardekani & Gore 2012; Mathijssen et al. 2016b).

Less intently studied is the motion of micro-swimmers near fluid-fluid interfaces (Guasto et al. 2010; Di Leonardo et al. 2011; Wang & Ardekani 2013; Lopez & Lauga 2014; Masoud & Stone 2014; Stone & Masoud 2015) and, to the best of our knowledge, theoretical studies of motility in liquid films in contact with solid substrates (Fig. 1) are rarely reported in the literature, despite their natural prevalence (Lambert et al. 2013; Mathijssen et al. 2016a). Innumerable habitats of small organisms are characterised as films that are macroscopically thin but substantially thicker than the characteristic size of swimming microbes and many experimental setups used for studies of various aspects of swimming cell dynamics essentially confine a culture of microorganisms between a substrate and liquid-liquid or liquid-gas interface. Liquid films allow for motility and swarming in order to colonise a wide variety of surfaces including plant and animal tissues (Grimont & Grimont 1978; Harshey & Matsuyama 1994; Bees et al. 2000; Harshey 2003). The motile microbial inoculant P. putida traverses films as it moves through thin aquatic layers in soil (Dechesne et al. 2010), as does E. coli, which is known to swim upstream along crevices (Hill et al. 2007). P. syringae bacteria can swarm on leaves by moving through their own secreted lubricant (Quiñones et al. 2005). In fact, many bacteria secrete extracellular polymeric substances forming self-generated protective surface-bound biofilms (Hall-Stoodley et al. 2004; Givskov et al. 1997; Conrad 2012), within which they then move.

The flow fields generated by the propulsion of micro-swimmers have gathered a large interest from the fluid mechanics community (Blake 1971b; Liron & Mochon 1976; Staben et al. 2003; Crowdy et al. 2011) because they play an indispensable role in their ecological traits such as mechanosensing (Doostmohammadi et al. 2012; Bukoreshtliev et al. 2013), energy expenditure (Guasto et al. 2010), rheology (Ishikawa & Pedley 2007; Guzmán-Lastra & Soto 2012; Gachelin et al. 2013; López et al. 2015), fluid mixing (Kim & Breuer 2007; Leptos et al. 2009; Ishikawa et al. 2010; Kurtuldu et al. 2011; Mino et al. 2011; Karimi & Ardekani 2013; Pushkin & Yeomans 2014; de Graaf & Stenhammar 2016; Jeanneret et al. 2016) and nutrient uptake (Magar et al. 2003; Katija 2012; Jepson et al. 2013). Despite the widespread implications of swimming in films, the underlying hydrodynamics and its impact on the ecology of swimming cells has remained largely unexplored.

In this paper, a detailed hydrodynamic description of swimmer dynamics within viscous films is developed by deriving the three-dimensional flows of a Stokeslet in a liquid film. We consider both a recursive series of image systems (§2.1) and the exact solution using the method developed by Liron & Mochon (1976) (Appendix §A). A multipole expansion of the Stokeslet flow then gives the universal components of the flow field generated by a swimming micro-organism (§2.2 and Appendix §B). Comparing the recursive series and the exact solution demonstrates their respective advantages and disadvantages in various regimes of film thickness (§2.3 and Appendix §C). We conclude that the series solution is more amenable to a hydrodynamic multipole expansion and for numerically computing hydrodynamic interactions with the surfaces when the micro-swimmer is small compared to the film thickness. In §3 these results are used to predict the trajectories of ideal micro-swimmers. We explicitly map the dynamics and boundary accumulation of ideal cells defined by each successive hydrodynamic multipole moment (§3.1–§3.3), where the multipole parameters are directly linked to properties of the micro-organism, including size, shape and propulsion mechanism. Together, these moments allow one to model more physical micro-organisms. Employing our findings (§3.4), we propose an experimental method to determine a swimmer’s rotary multipole coefficient, and its three-dimensional position under a microscope by measuring the radius of curvature of its projected trajectory.

## 2 Flow fields in a film

To derive the flow fields generated by a microbe swimming within a film constrained between a rigid wall and a free surface, we use a multipole expansion of the Stokes flow solution in a film. Performing such a multipole expansion requires a tractable analytic form of the hydrodynamic fields due to a force singularity (Stokeslet) within a viscous incompressible film of height and dynamic viscosity . While the fundamental flow fields between two parallel boundaries have been found using a Fourier transform method (Liron & Mochon 1976), we will present an analytical form that is particularly amenable to including the higher multipole moments required to accurately model micro-swimmers. This method is based on successive image reflections for finding the Stokeslet flow in a film. Previous studies have used a similar framework for studying the flow produced by mobile colloids (Ozarkar & Sangani 2008). To test our recursive series solution and establish the regimes where it is applicable, we derive the exact solution of the Stokeslet flow in a liquid film in Appendix §A by extending the method by Liron & Mochon (1976).

### 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 must respectively be satisfied, in addition to a no-penetration condition (Fig. 1). We aim to solve the Stokes equations

(2.0) | |||||

(2.0) |

Here, the fluid velocity and pressure fields at location and time are due to a point force (Stokeslet) that acts at position . The boundary conditions for a film are the no-slip condition at the solid wall and both impermeability and no-shear at the interface . In an unbounded fluid, the Green’s function is the Oseen tensor (Kim & Karilla 1991)

(2.0) |

where , and is the Kronecker delta. The flow field is then given by , where repeated indices are summed over. The corresponding pressure is with .

If the film height is taken to infinity (), then only the single no-slip boundary remains at and the Stokeslet flow field is altered by the addition of an auxiliary flow field, that can be written in terms of a system of images. This image flow field is given by the Blake tensor (Blake 1971b), which is centered at the position , where the reflection matrix is . The Blake tensor can be written in terms of the Oseen tensor (see Mathijssen et al. 2015) as

(2.0) |

which is a function of and , where the derivatives and are with respect to the force position . The tensor is recorded in the first row of Table 1. The resulting flow field at due to a point force at in the vicinity of a no-slip wall is then .

Similarly, if only a shear-free interface is present at , the auxiliary flow field of the Stokeslet flow is a direct reflection centered at the position , with the corresponding free-slip boundary tensor

(2.0) |

This result is recorded in row two of Table 1.

When both boundaries are present, the image system at corrects the boundary conditions at the solid no-slip wall () but disturbs the boundary conditions at the film interface (), and vice versa for the image at . 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 (or ) can be reflected in the interface (or wall) to form a secondary image system at position (or ), and so on. Hence, the positions of the image systems are

(2.0) | |||||

(2.0) |

where .

Fig. 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 , 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 in the tensor of the previous image system by the appropriate Blake tensor or free-slip boundary tensor given by Eqs. 2.1-2.1, respectively. It is key that all 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 is found via the recursion relations

(2.0) | |||||

(2.0) | |||||

(2.0) | |||||

(2.0) |

where , and the reflection operators B and T act linearly on all the Oseen tensors present in the image system tensor , as defined in Table 1. The foundations of the recursion relations are

(2.0) | |||||

(2.0) |

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

(2.0) |

giving the film Stokeslet flow .

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 Eqs. 2.1-2.1. Furthermore, Staben et al. (2003) 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 achieved for the Stokeslet flow in a liquid film.

(n) | Position | Replace | with |
---|---|---|---|

(0) | — | ||

(1) | |||

(2) | |||

(3) | |||

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

⋮ | ⋮ | ⋮ | ⋮ |

(n) | Image system tensor |
---|---|

(0) | |

(1) | |

(2) | |

(3) | |

(4) | |

(5) | |

(6) | |

(7) | |

⋮ | ⋮ |

### 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 -minor axes and , respectively and aspect ratio . Unless otherwise stated, we use as the characteristic swimmer size throughout the text. The organism is located at position with orientation , swimming in a film of height (Fig. 1). In addition to its own motility (propulsion velocity ), 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 . This swimmer-generated flow field can be written in terms of a multipole expansion of the Stokes flow solution in the film (Eq. 2.1). 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. 2015). Hence, an axisymmetric force-free and torque-free micro-swimmer generates a velocity field

(2.0) |

where is the Stokes dipole, the quadrupole, the source doublet, and the rotlet doublet (Spagnolie & Lauga 2012; Mathijssen et al. 2015). 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.

Each contribution to the multipole expansion can be written in terms of derivatives of the Green’s function in the film (Eq. 2.1). Specifically,

(2.0) | |||||

(2.0) | |||||

(2.0) | |||||

(2.0) |

where the derivatives act on the swimmer position . The multipole coefficients and (, , ) have dimensions of [velocity length] and [velocity length], respectively.

These multipole flow fields (Eqs. 2.2-2.2) are shown in Fig. 2. The dipole arises from the opposing propulsion and drag forces exerted by the swimmer (; Fig. 2a). Pusher-type swimmers such as E. coli have a positive force dipole with (shown in Fig. 2a). Pushers drive fluid out along the swimming direction and draw fluid in from the sides. Puller-type swimmers, on the other hand, have and draw fluid in along their swimming axis (Lauga & Powers 2009). The quadrupole flow field represents the fore-aft asymmetry of the microorganism (; Fig. 2b). For example, the quadrupole describes the weighting of propulsion forces towards the posterior of flagellated bacteria, in which case one expects . The source doublet represents the finite size of the swimmer (; Fig. 2c). For ciliated organisms with a slip velocity at their surface , whereas for non-ciliated swimmers one would expect 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 (; Fig. 2d) (Spagnolie & Lauga 2012; Mathijssen et al. 2015).

In a film of sufficiently large thickness the flow field is relatively unaffected by the boundaries (Fig. 3a). Upon decreasing the height of the film, the flow profiles in the and 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 plane is modified and recirculating flow patterns appear close to the swimming cell (Fig. 3b). Such patterns are reminiscent of those seen for a Stokeslet between two parallel plates (Liron & Mochon 1976). The recirculating regions are enhanced in size as the film thickness is further reduced (Fig. 3c) 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 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 direction due to a swimmer pointing in the direction decay exponentially with the lateral distance if either or or both are equal to three (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 . The Stokeslet has a recirculating pattern of two loops in the 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; Fig. 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 . These thin-film limit expressions could be used, for example, to model swimmer-swimmer interactions in films of thickness comparable to the swimmer size.

### 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 , the film height , the distance to the nearest surface , and the size of the flow region of interest, measured by the distance from the swimmer to where the flow is evaluated .

Fig. 4 shows the flow fields generated by a Stokeslet in a liquid film, using the recursive image series (left panels) and the exact solution (right panels). The Stokeslet is located at a distance from the bottom wall and is oriented in the direction parallel (top panels) and perpendicular (bottom panels) to the film surfaces. Close to the point force, where , the flow in a bulk fluid given by the Oseen result (Eq. 2.1) is recovered. If the film height is increased, with kept constant such that , then the Blake result (Eq. 2.1) is recovered (see details in Appendix §C, Fig. 11). Similarly, if the Stokeslet is located close to the top interface, the local flows can also be described with a single image system (Eq. 2.1). However, for a Stokeslet in the middle of the film (), the image systems above and below the film contribute with equal significance. If one is interested in distant flows, (; Fig. 4, outside the white dashed lines), then all images are approximately equidistant from the point 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 (; Fig. 4, inside the white dashed lines), then there is good agreement between the recursive image series (left panels) and the exact solution (right panels), even with a limited number of images (). 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 Fig. 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 means no images are present, and only the flow generated by the Stokeslet in bulk is considered. As 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 . Both at the bottom wall (Fig. 5; left panels) and the top interface (Fig. 5; right panels), however, the correction to the flow field is small after only a few images. Furthermore, with image systems included the boundary conditions at the bottom wall are satisfied exactly, and similarly with 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 . Since at every image reflection point () 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 Appendix §A.2. For a dipole swimmer the leading terms decay as , so the infinite series of alternatingly opposing dipoles converges more rapidly.

In short, the recursive series can compute flows accurately in the region for any (Fig. 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 to evaluate hydrodynamic interactions with surfaces using the multipole expansion, but it is noteworthy that Spagnolie & Lauga (2012) argue that far-field hydrodynamic interactions give surprisingly accurate results even for small swimmer-wall separations . When these two conditions are satisfied, we find that images are sufficient to describe micro-swimmer flows with an error less than . Therefore in this work, we utilise 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, , 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 Eq. 2.1 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 Swimmer dynamics in a liquid film

Knowing the flow fields that a motile microbe produces within a film allows us to model the hydrodynamic interactions with the bounding surfaces. In this section, we will describe the effects of these hydrodynamic interactions on the swimmer dynamics. Because we focus on the effect of surface accumulation at the bottom wall and the liquid air interface, we consider swimmers much smaller than the film height, . Furthermore, for simplicity we do not include swimmer-swimmer interactions. In this regime, all flows of interest are in the region where the use of the recursive series solution (Eq. 2.1) is appropriate, as discussed in §2.3.

Cells with a swimming velocity in a quiescent film obey the equations of motion

(3.0) | |||||

(3.0) |

where the swimmer position and orientation are and respectively with , hydrodynamic interactions with the surfaces are , and account for steric interactions with the surfaces which, for simplicity, are assumed to be adequately represented by hard-sphere interactions (Zöttl & Stark 2012). Additional terms such as background flows, run-tumble dynamics and thermal noise are neglected here, though they can play important roles in real, biological systems. For convenience of computation we express the swimmer orientation in spherical polar co-ordinates, , and without loss of generality we set at the initial time.

The swimmer’s motion is modified by the flow field it generates (Eq. 2.2) because it is advected and rotated by the reflection of this flow in the boundaries. This reflected (auxiliary) flow field is , where is the swimmer generated flow in the absence of boundaries. The surface-induced translational and rotational velocities are then found by solving the Faxén relations (Kim & Karilla 1991) for the force-free and torque-free swimmer. Writing terms up to second order in particle length gives

(3.0) | |||||

(3.0) |

where the derivatives are with respect to the position , the geometry factor is a function of the aspect ratio of the elongated swimmer, and is the strain rate tensor.

In the following sections, we will describe the effects of individual multipole contributions of the swimmer-generated flow field (Eq. 2.2) on the swimmer dynamics within a film §3.1 - §3.4.

(n) | Pos. | , | , | ||

(1) | 0 | ||||

(2) | 0 | ||||

(3) | 0 | ||||

(4) | 0 | ||||

(5) | 0 | ||||

(6) | 0 | ||||

(7) | 0 | ||||

(8) | 0 | ||||

(9) | 0 | ||||

(10) | 0 | ||||

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

prefactor | , | , |

(n) | Pos. | , | , | ||

(1) | 0 | ||||

(2) | 0 | ||||

(3) | 0 | ||||

(4) | 0 | ||||

(5) | 0 | ||||

(6) | 0 | ||||

(7) | 0 | ||||

(8) | 0 | ||||

(9) | 0 | ||||

(10) | 0 | ||||

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

prefactor | , | , |

(n) | Pos. | , | , | ||

(1) | |||||

(2) | |||||

(3) | |||||

(4) | |||||

(5) |