The dynamics of a circular treadmilling low Reynolds number swimmer near a right-angled corner is studied in the asymptotic limit in which the radius of the treadmiller is small. The governing dynamical equations are found explicitly, correct to third order in the swimmer radius, by means of Mellin transform techniques. The existence of arc-like periodic orbits, or ‘hydrodynamic bound states’, in the corner region is identified, and these orbits are found to be attractors in the dynamics for a large range of initial swimmer locations and orientations. The analytical approach may be extended to wedge regions of other fractional angles.
The study of organisms swimming at low Reynolds number near a no-slip wall has received a lot of attention in recent years (Brennen & Winet 1977; Lauga & Powers 2009). Theoretical work involves quantifying the swimming speeds and energetics when swimmers travel near solid boundaries (Reynolds 1965; Katz 1974; Katz et al. 1975; Fauci & McDonald 1995), understanding the attraction of swimmers to no-slip surfaces (Rothschild 1963; Winet et al. 1984; Fauci & McDonald 1995; Woolley 2003), as well as rationalizing the phenomenon of ‘swimming in circles’ (Lauga et al. 2006). Lauga & Powers (2009) provide a useful review. One surprising recent observation concerns the motion of Volvox algae near no-slip walls. It has been found (Drescher et al. 2009) that they interact with those surfaces, and each other, to engage in various types of periodic hydrodynamic bound states. Volvox is a class of organisms operating at low Reynolds numbers and moving, in part, by virtue of the imposition of a tangential surface velocity caused by motion of cilia on a sphere-like body. Drescher et al. (2009) refer to the observed bound states as ‘dancing’, and they include periodic motions of sufficient spatio-temporal complexity that they are dubbed the ‘waltz’ and the ‘minuet’.
Low Reynolds number swimming of bacterial organisms in microchannels, especially those produced with soft lithography, is also an active research area. Galajda et al. (2007) have shown that the motion of a population of bacteria can be controlled by microfabricated ‘walls of funnels’ consisting of walls, in different geometrical configurations, with funnel-shaped openings. Binz et al. (2010) have devised special microstructured environments to test the motility behaviour of bacteria moving into complex, channel-like, closed geometries with different shapes and dimensions. In related studies, Denissenko et al. (2012) have recently looked at the migratory abilities of motile human spermatozoa in assorted microchannel geometries in order to investigate how geometrical confinement contributes to distinguishing those cells that find an egg from those that do not.
As for theoretical contributions, to rationalize the behaviour of low Reynolds number swimmers in confined geometries near no-slip walls, Crowdy & Or (2010) considered a simple circular treadmilling swimmer, of the kind first considered by Blake (1971), driving a flow around it by imposing a tangential velocity 1.1 where the constant sets the time scale of the motion and 0≤β<2π denotes the angular position on the surface. The radius of the swimmer is ϵ and the angle α(t) is that made by some distinguished diameter marked on the swimmer, thought to represent a head–tail axis of the organism. The ‘treadmilling’ velocity (1.1) might be generated by the organized motion of a layer of moving cilia on the swimmer surface. With no walls present, such a swimmer is found to generate a local velocity equivalent to that of a point stresslet of strength 1.2 at the centre of the swimmer, together with a superposed irrotational point quadrupole of strength 2μ(t)ϵ2. Crowdy & Or (2010) proposed the idea of studying the dynamical evolution, not of the finite-area circular treadmiller itself, but of an approximation of it furnished by its free-space effective singularity description of the point stresslet/quadrupole pair. Henceforth, we refer to this approximation as the Crowdy–Or model. This approximation, which affords the advantage of producing an explicit dynamical system, gives excellent qualitative agreement with both numerical (Or & Murray 2009) and laboratory (Or et al. 2011) experiments of swimmers of quite different type, thereby indicating a certain genericity in the types of behaviours that can ensue near a rigid wall, as well as the modelling advantages in using an effective singularity description of a swimmer. The Crowdy–Or model has since been applied to several other confined domains (Crowdy & Samson 2010; Obuse & Thiffeault 2012) having increasing degrees of geometrical complexity.
In a companion paper, Davis & Crowdy (submitted) have carried out a full asymptotic matching procedure to derive the actual swimmer evolution equations, correct to third order in an asymptotic expansion in the swimmer radius, when a swimmer is located in the vicinity of a wall of general shape. The purpose of the present paper is to apply that theory to the physically interesting situation of a treadmiller swimming in a right-angled corner. In this case, a Mellin transform approach determines the fluid flow that, in turn, leads to explicit equations of motion for the swimmer and successfully accounts for any additional effects associated with possible Moffatt (1964) eddies in the corner. From the resulting equations, we find a strong tendency for swimmers to be attracted to the corner region and, once there, to engage in a non-trivial periodic orbit along an arc-like trajectory rocking to and fro between the two walls.
2. Asymptotic analysis
First, we review the recent work of Davis & Crowdy (submitted). Consider small disturbances created by a treadmilling swimmer model in an incompressible, viscous fluid at rest. The two-dimensional geometry has a circle of radius ϵ with centre at (x0,y0), referred to dimensionless Cartesian coordinates (x,y) with origin on the rigid boundary wall C. We assume that C is in the swimmer's far-field, that is, ϵ is small compared with the closest approach of C to (x0,y0). With Stokes flow allowing a quasi-steady solution, we suppose that surface actuators induce a tangential velocity of the form (1.1) with which can be written as , where and α is the instantaneous direction of the swimmer's head. In this way, (ρ,β) are polar coordinates based at the centre of the swimmer. The force and torque-free swimmer has instantaneous velocity Uex+V ey and rotation Ωez, in response to the reflected velocity field generated by the rigid boundary C.
The Stokes equations enable the fluid velocity vector v to be conveniently represented by 2.1 in terms of a streamfunction ψ, which satisfies the biharmonic equation 2.2 The boundary conditions at the swimmer are 2.3 with 2.4 where ∂/∂n denote the derivative normal to C. The leading terms of the inner field satisfy (2.3), yield no flow at infinity and evidently are given by 2.5 whence the leading terms of the outer field have the form 2.6 in which 2.7 The term ψR denotes the streamfunction associated with the leading-order reflected velocity field generated by the rigid boundary C. It can be determined by a variety of analytical and/or numerical techniques (Davis & Crowdy submitted); in this paper, we employ a Mellin transform method to find a closed-form integral expression for it. Note that the stresslet, ψ1, is necessitated by the condition of zero normal velocity in (2.3). The reflected field, ψR, is determined by the no-slip conditions (2.4).
The Taylor expansion of ψR about (x0,y0) facilitates identification of the additional stresslets (Davis & Crowdy submitted), which show that the precise form of ψ3 is 2.8 whence, by reference to (2.7), 2.9 in which denotes the Laplacian with respect to (x0,y0). Substitution of (2.9) into (2.6) facilitates deduction, up to relative order ϵ2, of the swimmer's velocity components, 2.10 the swimmer's rotation, 2.11 and the flux between wall (on right) and swimmer, 2.12
3. Wall forming a right-angled corner
The principal example of C considered here is the pair of semi-infinite straight walls, which bound the first quadrant say, for which (2.4) implies ψ=0=∂ψ/∂y at y=0,x>0 and ψ=0=∂ψ/∂x at x=0,y>0. Figure 1 shows a schematic of the swimmer in such a corner. With 3.1 the ψ=0 (no normal flow) conditions are satisfied by adding image stresslets with orientation −α at ±(x0,−y0) and α at (−x0,−y0). In terms of polar coordinates (r,θ) centred at the origin where the corner is located, this yields 3.2 which evidently vanishes at r=0. However, it is advantageous to subtract from each term its value at r=0 in the construction of the streamfunction, denoted by ψ10, that includes the prescribed stresslet and vanishes on C. Thus 3.3
The remaining step in the construction of ψ1, whose form is given by (2.7), is to eliminate slip on C by adding terms to ψ10. To do so, we define the Mellin transform 3.4 whence (2.4) implies, with the dependence on (r0,θ0) understood, that 3.5 and 3.6 Differentiation of (3.3) yields 3.7 whence 3.8 Evaluation of (3.6) is now achieved by using the Mellin transforms, 3.9 after suitably rearranging the denominators in (3.7) and (3.8). The formulae reduce to 3.10 for −1<ℜ(s)<0. The leftward shift of the strip of regularity can be attributed to the stream function being O(r) larger that its associated velocity field.
The solution form for Ψ(s,θ) is obtained by noting that and deducing from (3.4) that the biharmonic property of [ψ10−ψ1] implies that 3.11 The solution satisfying (3.5) is 3.12 where 3.13 Substitution of (3.10) shows that these numerators are given by 3.14 and 3.15 The former has poles at s=0,−2, with respective residues and , but the latter is regular at s=0,−2.
It now follows from (3.3) and (3.4) that the reflected stream function ψR, defined by (2.7), is given, for −1<γ<0, by 3.16 It is easily seen from (3.12) and (3.13) that Ψ is regular at s=−1 and therefore the strip of regularity can be extended to −2<ℜ(s)<0. These same equations also show that the residue of Ψr−s at s=0 is , which here contributes, on completing the contour to the right, to the far field and cancels the limit value of ψ10.
The asymptotic estimates (2.10) and (2.11) of the swimmer's velocity components and rotation require the evaluation of various derivatives of (3.16) at (x0,y0). This is efficiently accomplished by working with complex coordinates, z=x+iy with z0=r0eiθ0, in terms of which (3.16) is given by 3.17 after substitution of (3.12). It should be noted that because this is the reflected streamfunction of a point stresslet due to the presence of the two walls, and given that there are no lengthscales associated with such a flow, this streamfunction can be shown to be purely a function of the ratio r/r0. The quantities required in the asymptotic equations can then be derived as 3.18 with 3.19 3.20 and 3.21 in which 3.22 and 3.23 and, from (3.13) and (3.15), 3.24 and 3.25
The identification of Moffatt (1964) vortices requires consideration of the small-r expansion of ψ10, given by (3.3), and the poles of the integral in (3.16) in the left-half of the complex s-plane. The former yields 3.26 while the latter yields a residue of Ψ(s,θ;r0,θ0)r−s at s=−2 and these contributions to ψ1 cancel.
The smallest complex-valued poles are zeros of , whose numerical values (Moffatt 1964) are in the left-half plane. The associated pair of complex conjugate residues provide the dominant behaviour of the stream function ψ1 in the neighbourhood of the corner r=0.
By substitution of (3.12), (3.13) and (3.15) into the integral term in (3.16), it may be shown that, for r>r0, the sum of residues of Ψr−s at s=2n(n≥0) and, for r<r0, the sum of residues of −Ψr−s at s=−2n(n>0), together yield, for r≠r0, −ψ10, given by (3.3). This verifies that the boundary conditions are satisfied because the complex-valued poles yield eigenfunctions. However, in the neighbourhood of the point (r0,θ0) of interest here, each sum exhibits terms of the form (xy,y2)/(x2+y2) near (0,0). Evidently, the role of the residues at the complex-valued poles is to remove this non-existence of a limit value and differentiability. Thus, the required reflected velocities and rotation must be evaluated by integration along ℜ(s)=γ, as described already.
It may be argued that the image singularities in (3.16) are redundant, but we choose to retain them for the associated algebraic symmetries and the immediate elimination of normal flow at the walls.
4. Swimmer dynamics in a right-angled corner
The explicit equations for the swimmer evolution derived already are readily solved to find the evolution of the treadmiller in the corner. The Mellin inversion integrals are computed numerically by first making the change of variable from s to η where 4.1 This turns the integral along the Bromwich contour in the s-plane to an integral around the unit circle in the η-plane.
Figure 2 shows the swimmer trajectories from the initial location with r0=1,θ0=π/4 with ϵ=0.1 for different initial orientation angles α(0)=π/4,3π/8,π/2, 11π/16,3π/4. It is clear from the symmetries of the system that it is enough to consider initial orientation angles in the range α(0)∈[π/4,3π/4] because angles in the range [3π/4,5π/4] will give the same trajectories reflected in the line y=x. Indeed, a first check on the solution method is to verify numerically that this is the case. Two distinct types of motion are observed. For a large range of initial angles, the swimmer is attracted to the corner and, as it draws close, appears to engage in a periodic motion rocking back and forth along an arc-like trajectory symmetric about the x=y line. Only the case with initial condition α(0)=π/4 leads to rectilinear motion along the x=y line ending up at a stationary point on the periodic orbit. For a range of other initial orientation angles, the swimmer is attracted to the walls but it escapes the corner region by travelling in a rectilinear trajectory near parallel to the wall and away from the corner. While in this rectilinear motion, the swimmer's distance from the wall is found to be of the order of, but just greater than, its radius ϵ. Strictly speaking, the validity of the asymptotic approximations underlying the analysis of Davis & Crowdy (submitted) breaks down during this stage of the evolution. In some cases, close to the critical initial orientation between eventual entrapment in the corner and eventual escape, some orbits are found to draw close to the wall and move away from the corner before eventually reversing their direction and being attracted to the periodic orbit in the corner. It is clear from these simulations that the swimmer has a strong proclivity for being attracted to the corner region.
Figure 3 shows the swimmer location and orientation r0(t),θ0(t) and α(t) as functions of time for a simulation where the initial condition is taken from one of the trajectories of figure 2 in the basin of attraction of the periodic orbit near the corner and after sufficient time has elapsed that the swimmer is on (or close to) the periodic orbit. Further simulation of the dynamics shows that the swimmer remains on this periodic orbit with its orientation also oscillating periodically.
Figure 4 reconfirms qualitatively similar dynamical behaviour for a larger swimmer with ϵ=0.2. Again, there appears to be an attracting periodic orbit, located near the corner, with a large basin of attraction. Figure 5 illustrates the attracting periodic orbits for three different values of ϵ=0.1,0.2 and 0.3; each has a similar arc-like shape symmetric about the line y=x.
Figure 6 shows the trajectories for swimmers that are initially located off the line y=x and provides further evidence that the corner periodic orbit is a strong attractor in the dynamics.
5. Semi-infinite barrier
It is instructive to point out that a similar, but simpler, use of the Mellin transform is suitable when C is a half line, the negative x-axis say, for which (2.4) implies ψ=0=∂ψ/∂y at y=0,x<0. With 5.1 the reflected streamfunction ψR is readily constructed in the form 5.2 whence, for , 5.3 The small r expansion of ψ1 proceeds in powers of r1/2, starting with r3/2 if the velocity at r=0 is zero and the stresses there are integrable. The strip of regularity is narrowed to −1<γ<−1/2 so that the residue at r=−1 can cancel the O(r/r0) term in ψS. With the constant removed, the application of conditions (2.4) at θ=±π to (5.3) yields 5.4 and 5.5 whose respective inversions give, after further use of (3.9), 5.6
The solution of (3.11) that satisfies (5.6) is 5.7 Evidently, the residue of Ψ(s,θ)r−s at s=−1 is which confirms that (5.3) yields zero velocity at r=0. The asymptotic estimates (2.10) and (2.11) of the swimmer's velocity components and rotation require the evaluation of various derivatives of (5.2) at . This is efficiently accomplished by substitution of (5.7), either directly or by working with complex coordinates, in terms of which (5.7) is given by 5.8 With no other length scales defined in the problem, these are or times functions of θ, as indicated by (5.7), and this suggests exact evaluation of the inversion integrals. By completion of the contour along the line ℜ(s)=γ−1/2, it is shown that For , as here, n=−2, which gives 5.9 and hence, by successive differentiation, 5.10 for . Then, after much algebraic manipulation, the required derivatives of (5.2) at (x0,y0) are given by the explicit formulae 5.11 5.12 5.13 5.14 and 5.15
Obuse & Thiffeault (2012) have recently applied the Crowdy–Or model to study the dynamics of a treadmiller in this semi-infinite wall geometry. They employed the complex variable approach of Crowdy & Or (2010) and determined the coefficients f0(z0,α),f1(z0,α) and g0(z0,α). It has been verified numerically that the above-mentioned explicit formulae give the same results as those obtained by that alternative approach. It is not clear, however, that those complex variable methods can be readily generalized to the case of a right-angled corner; indeed, it is even suggested in the monograph by England (1971) that it is not possible using such methods. The Mellin transform method, on the other hand, applies naturally to the latter scenario, as we have shown.
It is interesting to compare how the additional asymptotic terms typically affect the dynamics compared with those computed by Obuse & Thiffeault (2012). In figure 7, we compute the evolution, according to the asymptotic equations correct to , of the swimmer dynamics for several of the same initial conditions considered in fig. 1 of Obuse & Thiffeault (2012). The dotted lines show the evolution computed using the Crowdy–Or model while the solid lines show the evolution as computed from equations (5.11)–(5.15). As might be expected, the swimmer evolution is qualitatively similar in all cases but can differ quantitatively after the swimmer has come close to the wall. In the first and third cases, the dynamics over most of the trajectory is qualitatively similar, but the second case shows a dramatic difference: the full asymptotic equations show that the swimmer is trapped at the wall while the Crowdy–Or model predicts that the swimmer will travel in a nonlinear periodic orbit along the wall. Interestingly, the authors have observed that a similar ‘trapping’ of the swimmer occurs, at the same order of approximation, when it is placed in a circular tank (Davis & Crowdy submitted).
6. Wall forming a corner of angle π/N
The Mellin transform method presented here can be extended to allow C to be the pair of semi-infinite straight lines that subtend an angle π/N, where N>2 is an integer, for which (2.4) implies ψ=0=∂ψ/∂θ at θ=0,π/N,r>0. Now the ψ=0 (no normal flow) conditions are satisfied by adding image stresslets of strength with orientation −α at (r0,−θ0+2πn/N),0<n<N−1 and α at (r0,θ0+2πn/N),1<n<N−1. This yields which evidently vanishes at r=0. The extension of (3.3) is then readily deduced.
The asymptotic equations for the evolution of a treadmilling swimmer in a right-angled corner, and near a semi-infinite wall, have been derived correct to third order in the swimmer radius. The asymptotic approach is valuable because, as we have shown, in many cases it is easier to compute the evolution according to these asymptotic equations than to determine the full solution from the boundary value problem for the Stokes flow generated by the swimmer in different geometries. For a swimmer in a corner, Mellin transform techniques have been usefully applied to find explicit evolution equations for small swimmers that remain sufficiently far from the wall.
As with any asymptotic approximation, it loses applicability in certain regimes—in this case, when the swimmer is within a distance from the wall comparable to its radius. Then, a full analysis of the unapproximated swimmer evolution problem becomes necessary. The asymptotic equations are, nevertheless, still useful because they are expected to give valuable insights into the dynamical behaviour of the full system. They are known to do so faithfully—for example, for a treadmiller near an infinite straight wall (Davis & Crowdy submitted) by virtue of the availability of an exact solution for comparison in that case (Crowdy 2011).
The results here suggest that there is a strong tendency for small treadmilling swimmers to be attracted into the corner region and, once in this vicinity, to engage in a non-trivial periodic motion along an arc-like orbit. Similar ‘hydrodynamic bound states’ have been observed for a swimmer evolving near a gap in a wall (Crowdy & Samson 2010) and are qualitatively reminiscent of the ‘dancing’ gaits exhibited by Volvox algae near no-slip walls (Drescher et al. 2009). In a fully three-dimensional situation, it might be envisaged that a swimmer trapped in a corner region of this type might be able to travel along the corner region (i.e. in or out of the page in our analysis) while rocking back and forth between the two walls. Indeed, recent experiments involving human spermatozoa in microchannels show precisely this tendency for ‘swimming along corners’ (Denissenko et al. 2012).
D.G.C. acknowledges partial financial support from EPSRC Platform grant EP/I019111/1 and an EPSRC Mathematics Small grant. This research was initiated while D.G.C. was a Visiting Professor at UC San Diego between July and December 2010.
- Received April 18, 2012.
- Accepted July 20, 2012.
- This journal is © 2012 The Royal Society