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5 July 2019
Abstract

The swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. The validity of the far-field description is probed for a selection of model swimmers of varying geometry and propulsive activity by comparison with full numerical simulations. The reduced model is then used to deliver simple but accurate predictions of hydrodynamically generated wall attraction and pitching dynamics, and may help to explain a number of experimental results.

Hydrodynamics of self-propulsion near a boundary]Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations S. E. Spagnolie and E. Lauga]SAVERIO E. SPAGNOLIE ANDERIC LAUGA

1 Introduction

The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. This activity takes place at extremely low Reynolds numbers, a regime in which fluid motion is dominated by viscous dissipation. An important factor for swimming at such scales is the long range nature of hydrodynamic interactions, either between immersed bodies, or between an immersed body and a surface (Lauga & Powers 2009). When the swimming dynamics of an organism vary near such boundaries a question arises naturally: is the change in behavior biological, fluid mechanical, or perhaps mediated by other physical laws? E. coli cells, for instance, have been observed to swim in large circles when in the presence of a solid boundary, which has been accounted for in a purely fluid mechanical consideration by Lauga et al. (2006). Other organisms have been shown to reverse direction at boundaries by inverting the orientation of flagellar rotation, resulting in a departure from the boundary which is clearly not a passive hydrodynamic effect (Cisneros et al. 2007). In an attempt to help differentiate such observations, we seek a general framework for determining the extent to which fluid mechanics can passively alter the swimming trajectories of microorganisms near surfaces.

Surface effects on motility lead to varied and important consequences in a number of engineering and biological systems. Van Loosdrecht et al. (1990) note that surfaces are the major site of microbial activity in natural environments, and refer to Harvey & Young (1980) who showed that almost all detectable bacteria in a marsh estuary were associated with particles. Correspondingly, the attraction of certain microorganisms to surfaces has a major impact on the development of biofilms, which can begin with the adhesion of individual cells to a surface (Van Loosdrecht et al. 1990; O’Toole 2000). Biofilms are responsible for numerous microbial infections, and can play an important role in such phenomena as biological fouling (Lynch, Lappin-Scott & Costerton 2003; Harshey 2003; Kolter & Greenberg 2006). A recent review on the mathematical modeling of microbial biofilms has been presented by Klapper & Dockery (2010). Meanwhile, in a lab setting it is common that microorganisms are in near contact with microscope slides or are directed through microfluidic channels in which boundaries can play significant roles. The migration of bacteria through small-diameter capillary tubes was studied by Berg & Turner (1990), and that of infectious bacteria along medically implanted surfaces was considered by Harkes, Dankert & Feijen (1992). More recently, Evans & Lauga (2010) have shown that the presence of a wall can lead to a change in the waveform expressed by actuated flagella, which in turn results in an increase or decrease in its propulsive force depending on the type of actuation.

The study of microbial attraction to surfaces reaches back to the observations of Rothschild (1963), who measured the distribution of bull spermatozoa swimming between two glass plates and found the cell distribution to be nonuniform with the cell density strongly increasing near the walls. By modeling swimmers as dipolar pushers, it has been argued by Berke et al. (2008) that this hydrodynamic consideration alone can account for the attraction. Immersed boundary simulations of swimming bodies with undulating flagella have also shown a hydrodynamic attraction towards a wall (Fauci & McDonald 1995). More recently, Smith et al. (2009) have explored numerically the wall effects on geometrically accurate swimming human spermatozoa. They have demonstrated that hydrodynamic interactions can trap the body in a stable orbit near a boundary, in some cases with counter-intuitive orientation and at finite separation distance from the wall. The numerical results of Shum, Gaffney & Smith (2010) and Giacché, Ishikawa & Yamaguchi (2010) show the existence of a stable swimming distance from the boundary in swimming E. coli that depends upon the shape of the cell body and the flagellum. Goto et al. (2005) have also detected an equilibrium pitching angle for a given wall separation distance. Recent experiments showing the upstream swimming of bacteria in a shear flow by Hill et al. (2007) suggest that the geometry and orientation of hydrodynamically bound swimming organisms can be important. Meanwhile, Drescher et al. (2011) have measured experimentally the flow field generated by the swimming of an individual E. coli bacterium near a solid surface and have shown that steric collisions and near-field lubrication forces dominate any long-range fluid dynamical effects on these length scales.

Other recent studies on the dynamics of swimming bodies near walls of a more theoretical nature includes work by Zargar, Najafi, & Miri (2009), who studied the dynamical motion of a three-sphere swimmer near a wall, and Crowdy & Or (2010), who studied a simple two-dimensional model of a swimmer using methods of complex analysis (see also Crowdy 2011). A different avenue of inquiry has also seen much recent activity, the effect of boundaries on swimming suspensions of microorganisms. For instance, Hernandez-Ortiz et al. (2009) have studied model swimmers composed of dipolar pushing beads, and have shown that the additional length scales introduced by confinement can suppress the onset of large scale structures in the suspension.

Frequently it is the case that the surface of interest does not impose a no-slip condition on the fluid velocity, for instance at a free boundary between water and air. Tuval et al. (2005) have considered the development of large scale fluid structures driven by a competition between oxygen-taxis near the surface of a sessile drop and gravitational effects. Di Leonardo et al. (2011) considered the hydrodynamic interactions of a swimming bacterium with a stress-free surface, which can be analyzed by placing a mirror image of the swimming organism opposite the free-surface. The circular trajectories studied by Lauga et al. (2006) were found to be reversed in this setting.

Outside of the fundamental benefits of scientific inquiry, a more complete understanding of the hydrodynamic interactions between self-propelled bodies will continue to drive the development of engineering applications as well. Synthetic swimming particles have been designed to perform tasks at an exceptionally small length scales, including chemically driven bimetallic nano-rods (Paxton et al. 2004; Fournier-Bidoz et al. 2005; Rückner & Kapral 2007), magnetic nanopropellers (Ghosh & Fischer 2009; Pak et al. 2011), and undulatory chains of magnetic colloidal particles (Dreyfus et al. 2005) (see also Wang 2009). Sorting and rectification devices which lean upon asymmetries in microbial interactions with walls have been explored by Galajda et al. (2007), while Di Leonardo et al. (2010) have considered the driven motion of gear-like ratchets in bacterial suspensions. Another application of more recent interest is in the production of biofuels, where suspensions of algae are shuttled through long channels (see Bees & Croze 2010). Exploring the hydrodynamic interactions between self-propelled bodies and surfaces not only allows us to understand the biological realm with greater sophistication, but may also allow for the development of manmade devices of increasing complexity and creativity.

In the present study, we utilize a multipole representation of self-propelled organisms in order to improve our understanding of swimming behaviors near a surface from a generalized perspective. The modeling of swimming organisms by Stokeslet dipole singularities has become commonplace, but here we take one systematic step further in the far-field expansion of the flow generated by self-propelling bodies. The inclusion of higher order singularities will be shown to have important consequences on swimming trajectories. Full scale simulations of the Stokes equations are used as a benchmark to explore the regions of validity and limitations of the reduced model for two types of model swimmers, namely ellipsoidal Janus particles with prescribed tangential surface actuation and bacteria-like spheroid-rod swimmers. The far-field approximation leads to very good quantitative agreement with the full simulation results in some cases down to a tenth of a body length away from the wall. Exploiting the quantitative predictions from our singularity approach, the reduced model is further shown to provide good predictive power for the initial attraction/repulsion to the wall and the rotation induced by the presence of the wall, and even surface scattering in the particular case of spheroidal squirmers.

This paper is organized as follows. In §II, the Stokeslet and higher order singularity solutions of the Stokes equations are introduced, and a general axisymmetric swimmer is described in terms of a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The wall effects on the trajectory of a swimming body are described through the contribution of each singularity in §III. In §IV we address the question of how accurately this multipole representation of swimming trajectories captures the real wall effects observed in full numerical solutions of the Stokes equations. The reduced (singularity) model is used to provide a simple description of the wall-induced rotation for model Janus particles, as well as to describe the complete swimming dynamics of a squirming spheroid in §V. In §VI we consider model polar swimmers that are bacteria-like in their geometry, and develop an approximate Faxén Law for their study. In §VII we then employ the reduced model to study a transition in the wall-induced rotations experienced by bacteria-like swimmers for a critical flagellum length. We finally discuss the accuracy and limitations of the reduced model in describing the geometry and dynamics of trapped self-propelled bodies near surfaces.

2 Singularity representation of motion in a viscous fluid

2.1 The Stokes equations and singularity solutions

The length and velocity scales which describe the locomotion of microorganisms are extremely small. The fluid flow generated by their activity is dictated almost entirely by viscous dissipation, as summarized by Purcell (1977). The Reynolds number describing the flow, is likewise very small, where is the fluid density, is the dynamic viscosity, and and are length and velocity scales characteristic of the organism. The swimming of E. coli, for example, is characterized by a Reynolds number (see Childress 1981). The fluid behavior is therefore described well by the Stokes equations,

(2.0)
(2.0)

where is the Newtonian fluid stress tensor, is the pressure, is the fluid velocity, is the identity operator, and is the symmetric rate-of-strain tensor. The fluid velocity is assumed to decay in the far-field, and the boundary condition assumed on the swimming body depends on the specific organism, as will be described below. In situations where we include the presence of a plane wall of infinite extent at , we shall also assume a no-slip condition there . Now classical treatises on zero Reynolds number fluid dynamics have been written by Happel & Brenner (1965) and Kim & Karrila (1991).

The linearity of the Stokes equations allows for the introduction and exploitation of Green’s functions. The description of the fluid behavior far from an actively motile organism, for instance, can be described accurately using only the first few terms in a multipole expansion of fundamental singularities, which will be our approach here. The utilization of fundamental singularities allowed for a series of exact solutions to fundamental problems in Stokesian fluid dynamics to be derived by Chwang & Wu (1975).

A free-space Green’s function for the Stokes equations is derived by placing a point force in an otherwise quiescent infinite fluid, (where is the Dirac delta function centered at ). With the point force directed along the unit-vector (and defining ), the solution to the forced system produces the so-called Stokeslet singularity,

(2.0)

where

(2.0)

is the -directed Stokeslet, and . Derivatives of the Stokeslet singularity produce other higher-order singularity solutions of the Stokes equations. The first three such singularities are the Stokeslet dipole, quadrupole, and octupole, described by

(2.0)
(2.0)
(2.0)

respectively, where the gradient () acts on the singularity placement . The vectors and indicate the directions along which each derivative is taken. As the solutions above are regular outside of the singular point there are many possible identities that may be observed by rearranging the order in which these derivatives are taken. Tensorial expressions of the singularities above are provided by Pozrikidis (1992) (see also Chwang & Wu 1975), and we have included the full vector expressions of these singularities in Appendix A. In addition to the solutions above, there are singular potential flow solutions to the Stokes equations which are associated with Laplace’s equation ( in 2.1). The source, source dipole, source quadrupole, and source octupole singularity solutions are, respectively,

(2.0)
(2.0)
(2.0)
(2.0)

The source singularities are related to the force singularities through the relation

(2.0)

and its derivatives. A notable combination of the above singularities has been named alternately the couplet or rotlet,

(2.0)

where the vectors form an orthonormal basis with . A rotlet dipole may then be written simply as

(2.0)

Finally, a combination of the above singularities that we will require later is termed the Stresslet, which may be written as

(2.0)
(2.0)

representing the fluid force on a plane with normal corresponding to the -directed Stokeslet velocity field.

2.2 Far-field description of a swimming body

In the present study, we will consider only microorganisms which are to a good approximation axisymmetric along a direction indicated by the unit-vector . Such an organism, with its centroid at a point , generates fluid motion in the far-field of the form

(2.0)

Here we have introduced the shorthand notation . The coefficient has units of , while and have units of . The values of the coefficients , and must be determined for each microorganism, and depend on the specific body geometry and propulsive mechanism.

An illustration of the singularity decomposition above is provided in Fig. 1. At leading order, a swimming E. coli organism can be modeled as a force dipole (decaying as ). This leading order representation has been used by many authors to consider the effects of nearby walls (Berke et al. 2008), and many-swimmer interaction dynamics (see for instance Hernandez-Ortiz et al. 2005; Saintillan & Shelley 2008; Hohenegger & Shelley 2010). A swimmer such as the one illustrated in Fig. 1, in which a flagellar propeller pushes a load through the fluid, is generally referred to as a pusher, in contrast to such organisms as Chlamydomonas which pulls a cell body through the fluid with a pair of flagella. At the next order (decaying as ), the flow in the far-field varies due to the length asymmetry between the backward-pushing propeller and the forward-pushing cell body (producing a Stokeslet quadrupole), due to the finite size of the cell body (producing a source dipole), and due to the rotation of the flagellum and counter-rotation of the cell body (producing a rotlet dipole). Vector field cross-sections of the Stokeslet dipole, source dipole, and Stokeslet quadrupole singularities are shown in Fig. 2. The strengths of these singularities have been measured experimentally for the organisms Volvox carteri and Chlamydomonas reinhardtii by Drescher et al. (2010), and for E. coli by Drescher et al. (2011). The effects of the Stokeslet quadrupole component of spermatozoan swimming has been suggested by Smith & Blake (2009), and force-quadrupole hydrodynamic interactions of E. coli have been studied by Liao et al. (2011).

Figure 1: The fluid velocity far from a swimming E. coli is modeled at leading order as that of a Stokeslet dipole. At the next order, the flow in the far-field varies due to the length asymmetry between the backward-pushing propeller and the forward-pushing cell body (producing a Stokeslet quadrupole), the finite size of the cell body (producing a source dipole), and the rotation of the flagellum and counter-rotation of the cell body (producing a rotlet dipole).

While the flow field is set up instantaneously in Stokes flow upon the variation of an organism’s geometry, the means of propulsion of a particular organism might be unsteady. In general the singularity strengths can be time-dependent, varying for example with the different phases of a swimming stroke pattern. As an example, the highly time dependent flow field generated by the oscillating motions of C. reinhardtii has been examined by Guasto et al. (2010). Nevertheless, for a first broad look at the far-field representation above we will restrict our attention to constant values of the singularity strengths for the remainder of our study. Also, we have assumed in the description given by (2.0) that there are no net body forces or torques on the organism, which would require the inclusion of Stokeslet and rotlet singularity terms as well (as explored for the organism Volvox by Drescher et al. 2009). While some organisms are not neutrally buoyant and do experience a body force or torque due to gravity, many others (including most bacteria) live on such a scale that such effects are negligible. In addition, we assume that there is no mass flux through such mechanisms as fluid extrusion, as studied by Spagnolie & Lauga (2010), which can present a source singularity in addition to those included in the expression above.

Figure 2: Velocity field cross-sections of (a) a Stokeslet (force) dipole, which decays as ; (b) a source dipole, which decays as ; and (c) a force quadrupole, which decays as , all in free-space. Arrow intensity correlates with the magnitude of the velocity. The effects of a nearby boundary may be intuited by imagining the wall to follow the streamlines.

2.3 The surface effect: Faxén’s Law

In a fluid of infinite extent, the fluid velocity in the far-field generated by an active body behaves as described in (2.0). When a boundary such as a plane wall is present, however, the velocity everywhere is altered due to the additional boundary condition. Borrowing an approach which has seen a long history in electrodynamics, the boundary condition on the surface can be satisfied by the placement of additional singularities at the image point inside the wall (where is the unit vector normal to the surface).

The image singularities required to cancel the effects of Stokeslet singularities placed parallel or perpendicular to a no-slip wall have been presented by Blake & Chwang (1974), each requiring a Stokeslet, Stokes doublet, and source dipole, as described in Appendix B. The image system for a “tilted” Stokeslet (a Stokeslet directed at an angle relative to the wall) is simply a linear combination of the wall-parallel and wall-perpendicular image systems. The image systems for higher order singularities, however, are not simply linear combinations of wall-parallel and wall-perpendicular image systems. The images for each of the axisymmetric singularities in (2.0) will be denoted by an asterisk (see Appendix B). For instance, the effect of a Stokeslet dipole along with its image, evaluated on the wall surface , returns . Likewise, we denote by the fluid velocity generated by the entire collection of image singularities needed to cancel, on the no-slip wall, the swimmer-generated velocity description in (2.0). Magnaudet et al. (2003) have recently taken a similar approach to studying the deformation and migration of a drop moving near a surface, and have provided a valuable review of Faxén’s technique.

The flow generated by the image singularities indicates the alteration to the fluid motion everywhere due to the presence of the wall. The effects of this induced fluid motion on the swimming trajectory are provided by Faxén’s Law, which can be written exactly for a prolate ellipsoidal body geometry (Kim & Karrila 1991). For an ellipsoid of major and minor axis lengths and , with the major axis aligned with the unit vector , the translational velocity and rotational velocity induced on the swimmer due to its experience of the flow may be written as

(2.0)
(2.0)

where is the position of the body centroid, , and , with the body aspect-ratio. Denoting the swimming speed attained by the organism in free-space by , we have therefore that the body swims with velocity and changes swimming direction via .

At the order of our consideration in this paper (via 2.0) the strength of the singularities representing the body motion are not changed by the presence of the wall. If the singularities required to represent the motion differed in rate of decay by more than one degree of separation (we currently include only terms decaying at order and ), then Faxén’s Law above would indicate a problematic interaction of the wall effect with the measurement of the singularity strengths. The approach above, then, must be handled with more care in the event that a Stokeslet singularity is required, or if higher order terms than those considered here are to be included in (2.0). So long as the distance of the body to the wall is sufficiently large relative to the body size, the higher order terms in (2.0)-(2.0) may be neglected.

The expressions above do not extend easily to geometries that are not ellipsoidal. In order to study a body geometry more like that illustrated in Fig. 1 we will need to develop an approximate “Faxén Law.” First, however, let us consider the consequences of singularity images on a prolate ellipsoidal body to develop some intuition.

3 The surface effect, singularity by singularity

A swimming body first begins to experience the hydrodynamic effects of a wall through the singularities which decay least rapidly in space. We now list each singularity in the multipole representation and describe the corresponding effect on the swimming speed and orientation of the body. First, however, the system is made dimensionless by scaling velocities upon the free-space swimming speed, , lengths upon the body semi-major axis length , and forces upon . Henceforth all variables are understood to be dimensionless. The unit vector is normal to the wall and points into the fluid, and the dimensionless distance between the wall and the body centroid is denoted by (see Fig. 3). The pitch angle with respect to the wall is denoted by ; the body is swimming directly away from the wall when , directly towards the wall when , and parallel to the wall when .

Figure 3: Schematic representation of the generic problem studied in this paper: an inclined swimmer near a solid surface. The distance of the body centroid from the wall is denoted by , measured along the direction normal to the wall . The pitch angle of the body’s director with respect to the wall is denoted by ; the body is swimming directly away from the wall when , directly towards the wall when , and parallel to the wall when .

3.1 Force dipole

Since we will not consider external body forces or torques, the least rapidly decaying singularity (in space) generated by the activity of an organism is a Stokeslet dipole directed along the swimming direction, , which induces a dimensionless attraction to the wall (or repulsion from the wall) by (2.0) of the form

(3.0)

Further details are provided in Appendix B, along with the effects induced not by a wall but instead by a stress-free surface such as a fluid-air interface. From (3.0), the surface-induced velocity of a pusher () is towards the wall when . For small orientation angles (when the body swims almost parallel to the wall, and ), combining the wall effect above with the vertical component of the free-space swimming speed, , we see that the body will move towards the wall when . Hence, for a pusher that is swimming nearly parallel to the wall, the first effect of the hydrodynamic interaction with the wall is an attraction. It has been argued by Berke et al. (2008) that this hydrodynamic consideration can account for observations of the entrapment of E. coli near surfaces, as well as the observations of Rothschild (1963), who measured the distribution of bull spermatozoa swimming between two glass plates and found the cell density to increase near the walls. Experimental measurements of force dipole strengths generated by swimming E. coli were evaluated by Drescher et al. (2011), who found rotational diffusion to dominate hydrodynamic effects in that particular regime.

Now, what of the body orientation dynamics? The pitch angle is assumed to be constant in time absent the presence of a wall, so the variation in is due only to wall-induced rotational effects. The leading order effect is again that of the slowly decaying Stokeslet dipole term, which generates the rotation rate

(3.0)

where the approximation given is appropriate when the body is nearly parallel to the wall (see Appendix B for the full expression). Hence, for the induced rotation acts to align the body with the wall for (pushers), and perpendicular to the wall for (pullers), with no qualitative dependence upon the aspect-ratio of the body. The rotation induced by the force dipole may be intuited based on a consideration of the velocity fields shown in Fig. 2a; imagining a wall to follow the streamlines, the body is seen to be drawn into the wall, and based on velocity gradients to rotate towards . The nature of the wall effect is even more predictable for the source dipole and force quadrupole from a similar consideration of Figs. 2b-c, as we will show. As we now proceed to consider the next order of singularities, we shall find that the leading order wall effects described above can be rather deceptive if they are used to predict the full trajectory of a given swimmer.

3.2 Source dipole

As a swimming body comes into closer contact with a wall, or when the body is swimming parallel to the wall, higher order singularities will begin to affect the trajectory of the swimming organism. The source dipole singularity, which enters due to the presence of a cell body of finite size (such as the spherical head in Fig. 1) contributes an induced attraction/repulsion relative to the wall of the form

(3.0)

For a swimmer such as that shown in Fig. 1 it is common to have , since motion of an inert sphere through a viscous fluid can be represented by a Stokeslet singularity and a source dipole with placed at its center (Kim & Karrila 1991). In this case, the source dipole term contributes a wall repulsion when the body is pitched away from the wall (“nose up”), and contributes an attraction to the wall when the body is pitched down (“nose down”).

When , the Stokeslet dipole term no longer determines the rate of rotation; instead, it is set by the higher order singularity structure. The rotational velocity induced (for ) by the source dipole is

(3.0)

which acts to rotate the nose downward towards the wall if . In general, these effects will compete with those generated by the Stokeslet dipole and quadrupole to set the trajectory and equilibrium states of the self-propelled body.

If instead a body swims by activity on its surface (as is the case for ciliated organisms, often modeled as so-called squirmers), we can find , as will be shown in the following section. The noted effects above are thus reversed for such an organism, which we will explore in greater detail in §V.

3.3 Force quadrupole

At the same order of decay as the source dipole, the Stokeslet quadrupole enters and induces a wall-perpendicular velocity of the form

(3.0)

and contributes an induced rotation rate of

(3.0)

for . The attraction/repulsion and induced rotation rate depend on the sign of , which itself depends upon the fore-aft body asymmetry, indicated in Fig. 1. From studying swimmers with exact singularity expressions (to be described below), we expect for such swimmers as shown in Fig. 1 with large cell bodies and short flagella, and for those with small cell bodies and long flagella. Like the source dipole, this singularity also acts to rotate the swimmer when .

3.4 Rotlet dipole

The rotlet dipole term can account for at least one surprising behavior of locomotion near surfaces, the circular swimming trajectories of E. coli as studied by Lauga et al. (2006). For a body swimming parallel to the wall, , the rotation about the axis is given by

(3.0)

the effect disappearing for infinitely slender swimmers (for fixed ). Fixing the distance to the wall , the body is thus predicted to swim in circles with a (dimensionless) radius

(3.0)

The cell bodies of E. coli bacteria rotate clockwise as seen from the distal end (behind the organisms) during their forward swimming runs, and the net torque is balanced by the counter-clockwise rotation of the propelling flagellar bundle. This situation corresponds to , and hence (3.0) predicts a large clockwise circular trajectory (as seen from above) parallel to the plane of the wall, which is consistent with the experimental observations of Lauga et al. (2006).

Note that the same organism moving near a stress-free boundary (like an air-water interface) experiences a passive rotation in the opposite direction (see Appendix B), as studied by Di Leonardo et al. (2011). While the rotlet dipole contributes to three dimensional swimming dynamics, this component of the propulsion has no bearing on the wall-attraction/repulsion or pitching dynamics of a swimmer: and . This is assured by the kinematic reversibility of Stokes flow. We will focus here on wall-attraction/repulsion and pitching dynamics, and thus for the remainder of our consideration we will set in (2.0).

4 Where is the multipole singularity representation valid?

The central question that we wish to answer in this paper is: how accurately are the wall effects predicted by the multipole singularity representation (2.0) and accompanying Faxén Law (2.0)-(2.0)? In order to provide an answer, we must have at our disposal a means of computing the full fluid-body interaction. For this purpose we will utilize the method of images with regularized Stokeslets, as derived by Ainley et al. (2008), in which the boundary integral formulation of the Stokes equations is accompanied by image singularity kernels which cancel the fluid velocity on the wall. In this approach the wall need not be discretized. We will make a necessary adjustment to this framework to allow for the inclusion of a slip velocity on the body surface. Equipped with a means of computing solutions to the full Stokes equations in the half-space bounded by a no-slip wall, we can study the accuracy of the multipole representation and wall effects for a selection of model swimmers.

4.1 Model swimmers

In order to probe the accuracy of the far-field representation we will enlist the help of model “Janus swimmers,” as illustrated in Fig. 4. The model swimmers are prolate ellipsoids, chosen so that the Faxén expressions (2.0)-(2.0) may be applied exactly. For a given pitch angle , the surface is parameterized as , with and , and where form an orthonormal basis with . Recall that is the body aspect-ratio. The unit tangent vector everywhere on the surface is denoted by , and the unit normal is denoted by .

Figure 4: A selection of model swimmers are illustrated. Spheres () and ellipsoids () are shown with activity lengths , and 1. Each body shown shuttles fluid along the active part of the surface to the left (the propulsive activity is indicated by arrows), and thus swims to the right. The gray regions are inert, where a no-slip condition is assumed.

The propulsive mechanism is a prescribed axisymmetric distribution of a slip velocity which acts tangentially to the body surface for , where is a dimensionless “activity length,” while the remainder of the body surface is inert (where a no-slip condition is applied). The entire body surface is active for (a squirmer). More specifically, the prescribed slip velocity distribution is chosen to be

(4.0)

The constant is selected so that the free-space (no wall) swimming velocity is unity, and is determined numerically. A no-penetration condition is applied on the entire body surface.

The squirmer model of ciliated organisms was introduced by Lighthill (1952) and extended by Blake (1971). Squirmer models (where either the slip velocity or the surface stress is specified) have been used to study multiple-organism interactions by Ishikawa, Simmonds, & Pedley (2006) and Kanevsky, Shelley, & Tornberg (2010), hydrodynamically bound states by Drescher et al. (2009), efficiency optimization in ciliary beating by Michelin & Lauga (2010), fluid stirring effects by Lin, Thiffeault, & Childress (2011), and motion in a polymeric fluid by Zhu et al. (2011). Swimmers with partially activated surfaces () have recently been designed and studied with great enthusiasm; see for instance the work of Paxton et al. (2004), Golestanian, Liverpool, & Ajdari (2007) and Jiang, Yoshinaga, & Sano (2010), where the activity is generated by self-phoretic and thermophoretic surface effects. Migration of similar ‘slip-stick’ spheres in an ambient flow has been studied by Swan & Khair (2008).

4.2 Full numerical simulation approach

An application of Green’s theorem to the Stokes equations (2.1) reveals a representation of the fluid velocity everywhere based solely on integrations of the stress and velocity on the immersed boundaries (the swimming body, in this case) (Pozrikidis 1992). Accounting for the presence of the wall by including image singularities, the fluid velocity everywhere may be written as

(4.0)

where

(4.0)
(4.0)
(4.0)
(4.0)

with the dimensionless fluid force per unit area, the Stokeslet singularity (2.1), the Stresslet singularity (2.0), and the differential surface area element for the integration variable . The image singularities and ensure that the no-slip condition is satisfied on the wall at , and are provided in Appendix B. The single-layer integral is weakly singular, which presents both theoretical and numerical difficulties. One approach to computing this integral is through the use of a regularized kernel, , where a small regularization parameter is introduced. This is the approach of Ainley et al. (2008), who derive the necessary adjustments which must be made to account for this regularization, and who also derive the image-singularities () which must accompany such an approach when a wall is present. This is the approach taken in the present work, though we must include the double layer integral and its image in order to accommodate the slip velocity .

We briefly recount the method of images with regularized Stokeslets as presented by Ainley et al. (2008) (itself a modification to the method derived by Cortez 2002). The surface is discretized by points, located at for . For a given point in the fluid or on the body or wall, we define , and . Absorbing the surface integration into the force so that we may simply write , and choosing a blob function of (used to spread the singular effect of a point force to a small finite area), it may be shown that

(4.0)
(4.0)

where is a regularization parameter (discussed below), , , and

(4.0)
(4.0)

Having subtracted off the velocity at the target point , the integrand of the double layer integral is finite with a jump discontinuity at (Power & Miranda 1987; Pozrikidis 1992). Insertion of rigid body motion velocities into the integrals returns zero, so only the tangential slip velocity need be considered. Since the integrands in and are known and finite, they are computed using adaptive quadrature to computer precision accuracy.

Having absorbed the surface integration into the definition of , the net (dimensionless) force on a boundary is computed simply as

(4.0)

which must return zero in the case of self-propelled swimming (where body forces such as gravity have been neglected, as previously noted).

For a given body position and orientation, a linear system must be solved to determine the swimming velocity and rotation rate, and , along with the scaled force , via the boundary integral relation in (4.0). The linear system is closed by requiring that the boundary conditions hold as follows. Denoting the inert part of the body by and the active part of the body by ,

(4.0)
(4.0)

The no-slip condition on the wall is satisfied automatically by the inclusion of the image kernels. Continuing to follow Ainley et al. (2008), the ellipsoidal surface is discretized by dividing the azimuthal angle into points, for . At each station , the polar angle is discretized into points, for , where is the smallest integer larger than . Taking as a representative discretization size , with the total number of gridpoints and the (dimensionless) ellipsoidal surface area, the regularization parameter chosen for all the problems considered herein is , and we set to capture the swimming behavior with sufficient accuracy for our purpose.

4.3 Computing the singularity strengths: , and

In order to compare the full system with the singularity representation in (2.0) we must determine the singularity strengths for the swimming bodies under consideration. Having recovered numerically, these singularity strengths may be computed as follows. Assuming that the body is directed along and centered about , for the sake of presentation, we have by an involved inspection (expanding (4.0) for in free-space, and matching the term for the dipole and the term and terms for the higher order singularities):

(4.0)
(4.0)
(4.0)

For purely rigid body motion, , the integrals involving vanish, so we need only insert the slip velocity into these integral expressions. Note that the expressions above are geometry dependent; other expressions would need to be derived for non-ellipsoidal body shapes.

The singularity strengths , and computed for a range of aspect-ratios and activity lengths are shown in Fig. 5. For all such model swimmers as shown in Fig. 4, we find that (the swimmers are pushers). However, both and change sign, as indicated by dashed lines. For a body that is primarily inactive (), the source dipole term is negative, , which can be predicted by considering the exact singularity solution for a sedimenting solid ellipsoid as derived by Chwang & Wu (1975). However, when the body is primarily active , we find , which can be predicted from the exact solution for a squirming ellipsoid as derived by Keller & Wu (1975). The source dipole term decreases in magnitude with , and also decreases in magnitude as the body becomes less geometrically active (). Meanwhile, is negative for bodies with small active surface areas and positive for bodies with large active surfaces .

Figure 5: The singularity strengths , and are shown for the model swimmers illustrated in Fig. 4, but for a wide range of aspect-ratios and activity lengths . The Stokeslet dipole strength, , is positive for all such swimmers (they are all pushers). The source dipole strength, , and the Stokeslet quadrupole strength, , change sign where indicated by dashed lines.

As a simple example, consider a slender rod of dimensionless length which satisfies a no-slip condition for arc-lengths and has specified active forcing for arc-lengths . A leading order approximation (in the small aspect-ratio of the body) of the force on the no-slip part of the body is simply . From (4.0)-(4.0) we find that , , and . The swimmer is a pusher for , has no source dipole term due to its slenderness, and generates a Stokeslet quadrupole term which changes sign precisely when , distinguishing bodies which are more active than inert or vice versa. The direction of rotation induced on such a swimmer by the presence of the wall thus depends here critically upon the asymmetry of the propulsive mechanism [through (3.0)], and not the body geometry, which is symmetric about its centroid. Namely, when the swimmer is parallel to the wall () then for and for . A long inert segment pushed by a short active segment ) leads to an upward pitching motion, while a short inert segment pushed by a long active segment leads to a downward pitching motion, with the situation reversed if the swimmer is a puller ().

The singularity strengths for a squirming (potential flow, ) spheroid are even simpler, with , and . Keller & Wu (1975) have shown that such a swimmer may be described in free-space exactly as an integration over source doublet singularities distributed along the body centerline. Specifically, writing

(4.0)
(4.0)

where is the ellipsoidal focal length, then on the body boundary the velocity satisfies the rigid body motion (with speed unity) and the tangential slip velocity prescribed in (4.0) with . For a spherical squirmer we have , and in (4.0), and the term above is precisely zero. A squirming potential flow spheroid therefore always rotates away from the wall surface [through (3.0)], which we will explore in greater detail in §V.

4.4 How accurate is the far-field representation?

Given the computed singularity strengths for the model Janus swimmers we can compare the far-field representation of the swimming dynamics with the full simulation results. Figure 6 shows the “horizontal” (wall-parallel, in the direction ) and “vertical” (wall-perpendicular, in the direction ) velocities and the rotational velocities of the eight model swimmers shown in Fig. 4, for a range of distances from the wall, and with pitching angle fixed at . Here is the minimal distance of the centroid to the wall without penetration, . The far-field approximation from (2.0) generally matches the full simulation results with great accuracy in this range. Inspection of log-log plots (not shown here) affirm that the singularity strengths have been computed correctly, or alternatively verify the accuracy of the full numerical simulations. All the bodies with the exception of the squirmers (for which ) swim horizontally with greater velocity due to the presence of the wall at this pitch angle. Also note that the spherical squirmer is the only swimmer of the eight that rotates away from the wall at this pitching angle; the rest all passively rotate towards the wall.

Importantly, we find that the rotation rates are very small relative to the swimming speed, even when the swimmer is as close to the wall as one body length away. For swimming organisms directed towards the wall a collision may be inevitable. Meanwhile, when a body is swimming parallel to a wall, this small rotation and trajectory adjustment may be sufficient to entirely determine whether or not the swimmer collides with the wall and enters a hydrodynamically or otherwise bound state, or possibly swims away.

Figure 6: The wall-parallel (“horizontal,” ) and wall-perpendicular (“vertical,” ) velocities and rotational velocities () of the eight model swimmers shown in Fig. 4 are shown for a range of distances to the wall , with computed values indicated by symbols and far-field predictions indicated by lines. The pitching angle is fixed to . Top row: spherical swimmers. Bottom row: spheroidal swimmers. The far-field approximation is seen to reproduce the full simulation results, and generally remains valid for the range considered. The scaling is the smallest value of for which there is no wall contact (defined in the text).
Figure 7: The horizontal, vertical, and rotational velocities of the eight model swimmers shown in Fig. 4, just as in Fig. 6, but for smaller distances from the wall. Computed values are indicated by symbols and far-field predictions by lines. The far-field approximation is surprisingly accurate, in some cases all the way down to one-tenth the body radius distance from the wall. Other velocities begin to vary not only quantitatively but qualitatively as the body approaches wall contact.

How close to the wall can the far-field expressions be extended? Figure 7 shows the same measurements but for a range of even smaller body distances from the wall. We find that the far-field approximation is surprisingly accurate, in some cases all the way down to a distance of one-tenth a body length from the wall. In particular, the horizontal swimming velocities of both the spherical and ellipsoidal swimmers is generally matched quite well, as are the vertical and rotational velocities of the spherical bodies. The behaviors of the ellipsoidal bodies are still captured qualitatively in this regime, but we begin to see a departure of the far-field approximation from the full simulation results. This is to be expected, as the far-field approximation does not capture effects such as lubrication, and higher order terms in the multipole expansion eventually become comparable to those kept in (2.0). For the swimmers considered here, however, the far-field approximations appear to be surprisingly accurate, and we expect that much can be predicted using the simple framework encapsulated by (2.0). This said, the predictions can become qualitatively inaccurate when the body is almost touching the wall, yielding non-physical results in some cases, so care clearly must be taken when attempting to apply the far-field theory to near-wall or near-swimmer interactions.

5 Employment of the reduced model: Janus swimmers

5.1 Equilibrium pitching angles

Our interest now turns to the rotations induced on a self-propelled body by the presence of a wall. Absent any boundaries, the swimmer is assumed to move along a straight path, with . When a wall is present, the body will pitch up away from the wall or pitch down towards the wall due to hydrodynamic interactions. For a given fixed distance from the wall, , this rotation will persist until an equilibrium angle is reached. Stable equilibrium angles so found using the full numerical simulations are shown in Fig. 8a, where we have fixed the centroid distance to one body length away from the wall, . Figure 8b shows the same plot but generated using the predictions of the far-field theory. Both the full simulations and reduced model predict that there exists a class of swimming bodies for which the hydrodynamic interaction with the wall results in a negative equilibrium pitching angle. Namely, those that are simultaneously slender and more geometrically active than inert, though only to a point. For potential flow squirmers, , we find that for any aspect-ratio the pitching equilibrium angle is oriented directly away from the wall. Inclusive of the potential flow squirmers, we detect a barrier above which the only equilibrium angle is , and we note that the transition from a small orientation angle to no equilibrium other than is extremely sensitive to the geometry of activation along this curve (a small change in can produce a dramatically different behavior).

Figure 8: Comparison of the full simulations to the far-field predictions for fixed distance . (a) Computed contours of the stable equilibrium pitching angle at fixed distance . (b) The same, as predicted with the far-field theory. (c) Analytically predicted equilibrium angle using the far-field approximation, linearized about (taking the minimum value between the prediction in (5.0) and ). Bodies that are both slender and sufficiently active (but not completely active, ) exhibit pitching equilibria with their noses down towards the wall. Bodies that are not sufficiently slender or not sufficiently active exhibit pitching equilibria with their noses turned up away from the wall.

In order to derive a simple analytical estimate of the equilibrium angle, we linearize the rotational contribution of each singularity about , yielding

(5.0)

and hence an equilibrium swimming angle (for a fixed distance ) is predicted to exist at

(5.0)

The estimate from the linearized far-field theory is show in Fig. 8c. The results match the full far-field theory and simulations well where the equilibrium angle is small, as expected, showing an overestimate in the angle for bodies which are less geometrically active (). That the completely active bodies () have no stable equilibrium pitching angle outside of , in which the body is swimming directly away from the wall, is explored in greater detail in the following section.

The dashed curve marking the transition from pitched-up to pitched-down equilibria also separates bodies that will rotate away from the wall and those that will rotate towards the wall when (which may be inferred by continuity). Pausing to consider this rotation rate for wall-parallel swimming (), we refer to Fig. 5 and note that the rotation is towards the wall in a region where and , and also where . For , Fig. 5 shows that grows more rapidly than as the aspect-ratio is increased. For , then, both and will become negative for sufficiently large aspect-ratio, and will become positive [see (5.0)]. In other words, the effect of the source dipole will overwhelm the effects of the Stokeslet quadrupole for sufficiently large aspect-ratio when . When the situation is reversed: decreases more rapidly than does as the aspect-ratio is increased. In this regime the Stokeslet quadrupole effects dominate those of the source dipole, both and are negative in (5.0), and thus .

To summarize, bodies that are both slender and sufficiently active exhibit pitching equilibria with their noses down towards the wall, though completely active bodies () always rotate away from the wall. Bodies that are not sufficiently slender or are not sufficiently active exhibit pitching equilibria with their noses turned up away from the surface. In addition, there is a boundary in parameter space beyond which the only stable equilibrium angle is , with the body swimming directly away from the wall.

5.2 Full swimming trajectories of squirmers

Unsurprisingly, the far-field approximation cannot generally be counted on for quantitative (and in many cases qualitative) predictions of the entire swimming behavior when the body is in very near contact with the boundary. The exact form of the propulsive mechanism (in this case the form of the tangential slip velocity) will specify the nature of near-wall contact, be that a hydrodynamically trapped state or a trajectory that leads to a departure from the surface. However, we have found one class of swimmers for which the far-field approximation can be used to predict the entire interaction with the boundary, namely for squirmers (), which we now describe in detail. That the far-field theory provides an accurate depiction of the full dynamics of a treadmilling swimmer was found in a two dimensional setting by Crowdy & Or (2010) and Crowdy (2011). The interaction of a squirmer with a wall has also been studied recently by Llopis & Pagonabarraga (2010).

We comment briefly on the numerical method. Time does not enter into the Stokes equations explicitly, and since the means of propulsion studied here is steady there are no variations in the dynamics with time outside of the trajectories described by the distance of the centroid from the wall, , and the pitching angle, . An adaptive time-stepping algorithm for stiff systems (ode15s in Matlab) is used to integrate the swimming trajectory to small enough error tolerance so that the sole error in the dynamics is due to discretization errors and the associated quadrature errors in evaluating (through the regularization parameter ). Hydrodynamic interactions with the wall are sufficient to prevent body-wall collisions in some but not in all cases. Following Brady & Bossis (1985) and Ishikawa & Pedley (2007), we include a screened electrostatic-type body repulsion force which acts only at very small distances from the boundary of the form , where is the minimum distance between the body surface and the wall, and we take , . These values are selected so that the body does not come closer than approximately , where the numerical method for computing the fluid velocity just begins to lose accuracy (though not dramatically; see Ainley et al. 2008). For elongated squirmers () an associated torque is included as well. More physically realistic near-contact interaction effects have been discussed by Poortinga et al. (2002).

We first describe the results of the full simulations by focusing on a spherical squirmer (). By scanning the parameter space of distances and pitching angles , we have observed that the body rotates away from the wall regardless of its distance from the boundary and orientation (save for the special case of swimming directly towards the wall at , though this orientation is found to be unstable). Setting the body initially at a distance one body radius from the wall (), for initial pitching angles the body rotates as it moves through the fluid and does not come into close contact with the surface. It then settles into a final pitching angle as it swims away from the wall, never to return. This scattering angle, as determined from full simulations, is shown in Fig. 9a as a solid line. For initial angles the body swims towards the wall, which increases the rotational effect on the body and hence increases the final pitching angle once it departs, accounting for the non-monotonicity of the scattering curve. Three trajectories have been picked out with the intention of illustrating this non-monotonicity and are included as Fig. 9b.

Figure 9: (a) The scattering angle exhibited by a squirming sphere ( and ), for an initial centroid distance . Solid line: results from the full simulation. Circles: analytical far-field prediction, neglecting the surface attraction/repulsion, from (5.0) and (5.0). (b) The trajectories of spherical squirmers with three different initial pitching angles, illustrating the non-monotonicity indicated in (a). A trajectory that takes the body nearer to the wall leads to a greater net rotation before the swimmer escapes.

Meanwhile, for initial pitching angles the squirmer “impacts” the wall (realized here by blocks of time in which there is negligible wall-perpendicular velocity), but continues to rotate while in near-wall contact. Eventually the squirmer escapes the surface and swims away until settling to a final pitching angle . For non-impacting trajectories, the swimming trajectory must be symmetric about the point of parallel swimming, , by the time-reversibility of the Stokes equations (reversing the direction of time is indistinguishable from reversing the direction of surface activation and swimming speed). In such a case we must have that the pitching angle reaches a value when the swimmer has returned to the distance on its journey away from the surface. Having determined in this case that a critical initial pitching angle for wall-impact is given by , it might then have been predicted correctly that all wall-impacting trajectories in this case lead to final pitching angles of . The near-wall contact behavior simply acts to remove any information about the initial pitching angle until the trajectory matches that of the critical case (outside of a horizontal shift along the wall).

For the spherical squirmer that swims with speed unity in a quiescent fluid we have (see 4.0), and the dynamics predicted by the reduced model are set by

(5.0)

Can we predict the scattering behavior described above analytically? In order to derive a simple estimate of the swimming behavior, let us linearize the motion about and assume . Then the translational swimming speed is not varied by the wall beyond its effect on the swimming angle , and we have

(5.0)

with and , where the geometrical dependence has been absorbed into the singularity strength, . Integrating this system, we find (for trajectories that do not impact the wall) that

(5.0)

or equivalently,

(5.0)

Taking , we know from the simulations that all squirmer trajectories have and ; upon insertion, we find a final pitch angle for non-impacting squirmers of

(5.0)

The distance to the wall here corresponds to wall impact. First we ask: which initial distances and orientations lead to wall impact? Writing the pitching angle at the time of impact as , the centroid will be located at a distance at that time. Inserting and into (5.0) gives

(5.0)

Given the simple expression for the translational velocity , and that at all times, a wall-impacting squirmer must have . A curve in space separating initial conditions for which a squirmer does or does not impact the wall may then be deduced by setting , leaving

(5.0)

or