aa r X i v : . [ phy s i c s . f l u - dyn ] M a y Pursuit and Synchronization in Hydrodynamic Dipoles
Eva Kanso and Alan Cheng Hou Tsang
Aerospace and Mechanical Engineering, University of Southern California854 Downey Way, Los Angeles, CA 90089-1191
July 23, 2018
Abstract
We study theoretically the behavior of a class of hydrodynamic dipoles. This study is motivatedby recent experiments on synthetic and biological swimmers in microfluidic
Hele-Shaw type geometries.Under such confinement, a swimmer’s hydrodynamic signature is that of a potential source dipole, andthe long-range interactions among swimmers are obtained from the superposition of dipole singularities.Here, we recall the equations governing the positions and orientations of interacting asymmetric swimmersin doubly-periodic domains, and focus on the dynamics of swimmer pairs. We obtain two families of‘relative equilibria’-type solutions that correspond to pursuit and synchronization of the two swimmers,respectively. Interestingly, the pursuit mode is stable for large tail swimmers whereas the synchronizationmode is stable for large head swimmers. These results have profound implications on the collectivebehavior reported in several recent studies on populations of confined microswimmers.
Active systems, i.e., systems driven internally by self-propelled individual units, often exhibit rich collectivebehavior at the system’s scale; a scale that is typically several orders of magnitude larger than the scale ofthe individual unit. Such collective behavior naturally arises in disparate biological systems, from schoolsof fish [1] to suspensions of motile bacteria [2] and assemblages of sub-cellular extracts [3]. It also emergesin inanimate systems such as driven and self-propelled droplets and reactive colloids [4, 5], and provide anattractive paradigm for reconfigurable smart materials [6] and biomedical devices [7].The question of how these highly-coordinated collective motions arise from piecewise interactionsamong individual units has been the subject of intense research in the past few years. A well-studiedexample is the behavior of self-propelled particles in a viscous fluid, [8]. Most of this work has focused onthe instabilities and spatiotemporal fluctuations in three-dimensional (3D) systems. However, motivatedby recent technological advances in producing and manipulating large ensembles of particles in microfluidicdevices [5, 9, 10], attention began to shift to the collective dynamics of particles confined in quasi two-dimensional (2D) geometries. Geometric confinement changes drastically the nature of the hydrodynamicinteractions among particles, [9]. The long-ranged hydrodynamic interactions in 3D are driven by the forcedipoles exerted by self-propelled particles on the fluid medium, [8]. In quasi-2D geometries, the solid wallsscreen the force dipole contribution, making it subdominant in comparison with the potential dipole arisingfrom incompressibility, [10]. As a result, the long-range interactions among swimmers can be obtained fromthe superposition of dipole singularities.In this paper, we revisit the hydrodynamic dipole model proposed by Brotto et al. [11] for asymmet-ric dumbbell swimmers in confined Hele-Shaw type geometries. The head-tail asymmetry causes a givenswimmer to reorient, not only in response to the flow gradient as anticipated by Jeffery’s equation, but alsoin response to the flow velocity itself. This result is rooted in the fact that the lubrication forces betweenthe swimmer and the solid walls hinder its advection by the fluid, inducing unequal translational motilitycoefficients at the swimmer’s head and tail. In [11], Brotto et al. derived a kinetic theory-type model fora population of interacting swimmers and predicted a novel long-wave linear instability that leads to theemergence of large-scale directed motion and polarization in isotropic populations of confined large headswimmers. Lefauve and Saintillan [12] and Tsang and Kanso [13] used numerical simulations to explore1he implications of these instabilities on the collective behavior in finite-sized populations of interactingswimmers.The present paper examines the detailed dynamics of a pair of asymmetric (head-tail) swimmers indoubly-periodic domains, where the orientation dynamics is dominated by the flow field itself, thus neglectingreorientation in response to the flow gradient as done in [11, 12]. We particularly focus on a special classof solutions where the two swimmers move with constant speed and at constant orientation. We find twofamilies of these “equilibrium-like” solutions: (1) both swimmers swim side by side in a synchronized way;and (2) one swimmer tailgates the other. We analyze their stabilities and find that they depend on thedetails of the head-tail asymmetry. We conclude this work by discussing the significance of these results tothe behavior of populations of swimmers.Note that a dynamical theory of dipole interactions has also been pursued in two additional contexts.One motivation stems from the desire to obtain low-order representations of two-dimensional, inviscid andincompressible fluids in terms of interacting particles such as point vortices and point dipoles, see, e.g., [14–18]. A shortcoming of these models is that the dipole’s self-propelled speed is ill-defined; thus, a dipole,unless properly desingularized, induces infinite velocity through its center. Another motivation for dipolemodels that is closer to the focus of this paper grew out of efforts to examine the role of hydrodynamiccoupling in fish schooling. It is a well-known result in fluid dynamics that the leading order flow of a self-propelled body is that of a potential source dipole. Kanso and co-workers proposed a finite dipole dynamicalsystem that captures the far-field hydrodynamic interactions of self-propelled bodies [19, 20]. Each dipoleconsists of a pair of equal and opposite strength point vortices separated by a finite constant distance. Byconstruction, the self-propelled speed is well-defined. These finite dipoles are advected by the local flow andreorient in response to the local flow gradient. In particular, the finite dipole reorients according to velocitygradient in the direction transverse to the dipole orientation, as opposed to reorienting in response to theflow gradient along the dipole’s direction predicted by Jeffery’s equation for slender bodies in viscous fluids.Intrigued by the similarities between the finite dipole model [19, 20] and the dipole model of [11, 21], Kansoand Tsang [22] presented a unified framework for deriving two point dipole models: a dipole consistent withthe finite dipole model, appropriate for bluff bodies (fish) in potential flows, and another consistent withJeffery’s equation for slender bodies and equivalent to the microswimmer model employed in [11, 21]. Theyfurther showed that, in unbounded domains, dipole pairs can synchronize their motions for a range of initialconditions; however, the details of the synchronized motion differ between the two models.The organization of this paper is as follows: In Section 2, we formulate the equations of motions fora systems of dipoles in unbounded and in doubly-periodic domains. A detailed treatment of the dynamicsof dipole pairs is conducted in Section 3. These results are discussed in Section 4 in light of the large-scalesimulations performed on the same system in [12, 13].
Microswimmer model.
Consider a microswimmer composed of two connected disks of radii R tail and R head located at z tail and z head respectively, where z = x + i y is the complex coordinate (i = √− ℓ . The equations of motion for the swimmer’s tail( z tail ) and head ( z head ) can be written in complex notation as (see [11])˙¯ z tail = U o e − i α o + µ tail ¯ w ( z tail ) + λ tail e − i α o , ˙¯ z head = U o e − i α o + µ head ¯ w ( z head ) − λ head e − i α o . (1)Here, U o is the swimmer’s self-propelled velocity, α o its orientation angle, and w ( z ) is the velocity field ofthe ambient fluid. The bar notation denotes the complex conjugate, ¯ z = x − iy . The coefficients µ tail , µ head are the translational mobility coefficients whereas λ tail , λ head are unknown Lagrange multipliers that enforcethe constraint | z head − z tail | = ℓ . In particular, the translational mobility coefficients µ tail and µ head arisefrom the balance of hydrodynamic drag and wall friction acting on the tail and head, and are decreasingfunctions of R tail and R head respectively, with values less than 1, see, e.g. [11, 21].We define the hydrodynamic center of the swimmer to be z o = ( λ tail z head + λ head z tail ) / ( λ tail + λ head ).Our goal is to rewrite the equations of motion (1) in terms of the swimmer’s hydrodynamic center z o and2rientation α o . Let ℓ ≫ R tail , R head , and use Taylor series expansion to expand the flow velocity at the tailand head ¯ w ( z tail ) = ¯ w ( z o ) + e i α o λ tail λ tail + λ head d ¯ wdz (cid:12)(cid:12)(cid:12)(cid:12) z o + . . . ¯ w ( z head ) = ¯ w ( z o ) + e i α o λ head λ tail + λ head d ¯ wdz (cid:12)(cid:12)(cid:12)(cid:12) z o + . . . . (2)Substitute (2) into (1) to get that the equation governing the translational motion of the swimmer’s center˙¯ z o = U o e − i α o + µ ¯ w ( z o ) , (3)where µ = ( λ head µ tail + λ tail µ head ) / ( λ head + λ tail ). To obtain the equation governing the rotational motionof the swimmer, note that, by definition, ℓ e i α o = z head − z tail , which gives, upon differentiating both sideswith respect to time and further simplifications,˙ α o = Re (cid:20) ( ˙¯ z head − ˙¯ z tail )ie i α o ℓ (cid:21) . (4)Here, Re denotes the real part of the expression in bracket. Now substitute (1) and (2) into (4) to get˙ α o = Re (cid:20) ν dwdz ie α o + ν w ie i α o (cid:21) . (5)where dwdz and w are evaluated at z o and the constant parameters ν and ν are given by ν = ( λ head µ head + λ tail µ tail ) λ head + λ tail , ν = ( µ head − µ tail ) /ℓ. (6)The sign of ν dictates how the swimmer orients in local flow: it aligns to the local flow when ν > µ head − µ tail > µ head/tail is a decreasing function of R head/tail , [9]), and opposite to the local flow when ν < µ head − µ tail < Hydrodynamic interactions of multiple microswimmers.
Consider the interaction of multiple mi-croswimmers in an unbounded fluid domain. By virtue of (3) and (5), the dynamics of N swimmers, allhaving the same self-propelled velocity U , can be expressed in concise complex notation˙ z n = U e − i α n + µw ( z n ) , ˙ α n = Re (cid:20) ν dwdz ie α n + ν w ie i α n (cid:21) . (7)Here, z n and α n denote the position and orientation of each swimmer ( n = 1 , . . . , N ). To close the model,one needs to obtain an expression for the fluid velocity field w ( z ). Recalling that each swimmer inducesa far-field velocity which is that of a potential source dipole [9], the far-field flow of a microswimmer j located at z j = x j + i y j and oriented at an arbitrary angle α j can be described by the complex velocity w ( z ) = u x − iu y = σ e i α j / ( z − z j ) , where σ is the dipole strength. Note that σ = R U , where R is theeffective radius of the swimmer. A microswimmer n responds to the flow induced by all microswimmers inthe fluid domain, namely, w ( z n ) = N X j = n j =1 σ e i α j ( z n − z j ) . (8)3 π /4 π /2 3 π /4 π π /4 π /23 π /4 π π /43 π /2 α θ stable unstableunstable stable pursuit synchronization large taillarge head ν >0ν <0 (a) (b) Stability Analysis α α θ c Figure 1: (a) Two modes of solutions are obtained from (11) : a pursuit mode (blue) where the swimmers trail one anotherand a synchronization mode (red) where they swim side by side. The shown solid curves correspond to ω = − i ω = 5 and thedashed lines to ω = − i ω = 10 . The separation distance c is set to c = 4 . (b) Summary of the stability analysis for these twomodes. Microswimmers in doubly-periodic domains.
When the swimmers are placed in a doubly-periodicdomain, one needs to take into account, not only the velocity field induced by the swimmers themselvesbut also the effect of their image system. A given swimmer n has a doubly-infinite set of images. Thus,evaluating w ( z ) requires the evaluation of conditionally-convergent, doubly-infinite sums of terms that decayas 1 / | z | . These sums are evaluated using an approximate numerical approach in [12]. In [13], we offered aclosed-form analytic expression for these infinite sums in terms of the Weierstrass elliptic function, namely, w ( z ) = N X n =1 σρ ( z − z n ; ω , ω )e i α n . (9)The Weierstrass elliptic function ρ ( z ) is given by ρ ( z ; ω , ω ) = z + P k,l (cid:16) z − Ω kl ) − kl (cid:17) , with Ω kl =2 kω + 2 lω , k, l ∈ Z − { } , and ω and ω being the half-periods of the doubly-periodic domain. Thisfunction has infinite numbers of double pole singularities located at z = 0 and z = Ω kl , corresponding tothe 1 / | z | singularities induced by the potential dipoles. Equations (7) and (9) form a closed system for N swimmers in a doubly-periodic domain.We conclude by writing the system of equations (7) and (9) in dimensionless form using the swimmersradius R as a length scale and R/U as a time scale. That is, we introduce the dimensionless spatial variable˜ z = z/R and time variable ˜ t = tU/R . We then drop the tilde notation assuming all variables are non-dimensional. Equations (7) and (9) have the same form but the parameters U and σ are now normalized toone, that is, U = 1 and σ = 1. The parameter values µ , ν and ν are also non-dimensional. We consider two microswimmers in a doubly-periodic domain, and focus on their dynamic response when ν = 0, that is, when their alignment with the flow gradient is negligible. In this case, the orientationdynamics is dominated by alignment with the flow due to head-tail hydrodynamic asymmetry. Periodic solutions and relative equilibria.
We look for special solutions where the two swimmers moveat the same velocity and orientation for all time. That is, we look for solutions where ˙ z = ˙ z = constantand ˙ α = ˙ α = 0. To obtain the initial conditions that lead to this behavior, it is convenient to rewrite the4 -10 -5 0 5 10 y -10-50510 x -10 -5 0 5 10 y -10-50510 (a) (b) Figure 2:
Aperiodic behavior of two dipoles in doubly-periodic domain. The parameter values are α = π/ , z (0) = − iθ ) , z (0) = 2 exp( iθ ) , while θ is obtained by numerically solving the first condition in (11) . The positions of the dipoles are markedby ‘ × ’ at t = 0 and by ‘ o ’ at the end of the integration. As time progresses, the two trajectories densely fill the whole domain. equations of motion (7,9) in terms of the reduced coordinate z − z which we set to z − z = β = c e i θ (seeinset of Figure 1(a)). To this end, one gets˙¯ β = e − i α − e − i α + µρ ( β )(e i α − e i α ) , ˙ α = ˙ α = ν Re[ie i( α + α ) ρ ( β )] . (10)The translation equation for ˙ β is identically zero when α = α = α . Whereas to guarantee ˙ α = ˙ α = 0,one must satisfy the condition Re[ ρ ( c e − i θ )] = 0 therefore α = π , π α, θ ) = (cid:26) ( α, α )( α, α + π/ ρ ( c e − i θ )]Re[ ρ ( c e − i θ )] = − tan 2 α, Re[ ρ ( c e − i θ )] = 0 , (11)A total of ten strict relative equilibria of the two swimmers are depicted schematically in Figure 1(a). Namely,the five solutions given by α = 0 , π , π , π , π and θ = α correspond to the two dipoles moving parallel toeach other in a “pursuit” mode (blue arrows), whereas the five solutions given by α = 0 , π , π , π , π and θ = α + π/ α, θ ) for which Im[ ρ ( c e − i θ )] / Re[ ρ ( c e − i θ )] = − tan 2 α and Re[ ρ ( c e − i θ )] = 0, are not analytically available and need to be computed numerically. Figure 1(a)shows the values of ( α, θ ) that satisfy these conditions – clearly, two branches of solutions are obtained.These solutions depend implicitly on the domain size ( ω , ω ) and on c , the separation distance between thetwo swimmers. In other words, for a choice of domain size and separation distance c , ( α, θ ) are computedaccordingly such that the two dipoles move at the same constant velocity and orientation for all time. Thetwo branches shown in Figure 1(a) correspond to two modes of behavior: a pursuit mode where one swimmertrails the other, and a synchronization mode where the two swimmers move side by side. These solutions,while they correspond to the dipoles moving at constant velocity and orientation, can exhibit two distincttypes of dynamical behavior due to the doubly-periodic nature of the domain, namely, they could lead toaperiodic and periodic motion of the dipoles. Aperiodic motion refers to the case where the paths of thedipoles densely fill the whole domain, as shown in Figure 2. This seems to be the generic behavior forarbitrary initial conditions. Periodic behavior refers to trajectories that satisfy the condition¯ z ( T ) = ¯ z (0) + 2 pω + 2 qω , ¯ z ( T ) = ¯ z (0) + 2 pω + 2 qω (12)5
10 −5 0 5 10−10−50510 x y −10 −5 0 5 10−10−50510 x y (a) (b) Figure 3: (a) Pursuit and (b) synchronization in two dipoles undergoing periodic motion. The parameter values are α =tan − (3) , z (0) = − iθ ) , z (0) = 2 exp( iθ ) , while θ is obtained by numerically solving the first condition in (11) . Thepositions of the dipoles are marked by ‘ × ’ at t = 0 and by ‘ o ’ at the end of the integration time t = 80 . −10 −5 0 5 10−10−50510 x y −10 −5 0 5 10−10−50510 x y (a) (b) Figure 4:
Pursuit and synchronization modes as attracting modes. Two dipoles hone in on quasi periodic trajectories where:(a) one dipole is in pursuit of the other for ν = 0 . . (b) the two dipoles synchronize and move along parallel trajectories for ν = − . . The initial conditions are z (0) = α (0) = α (0) = 0 while z (0) = 1 . in (a) and z (0) = 2 + 1i in (b). where p and q are integers and T is the period of the motion. This amount to the additional condition α (0) = α (0) = tan − ( qp ) . (13)The ratio of q/p indicates the ratio of the number of times the dipole crosses the y and x axes in one period T . Figure 3 depicts the periodic behavior of two dipoles in pursuit and synchronization modes for q/p = 3. Stability analysis.
We analyze the linear stability of the pursuit and synchronization modes by consideringsmall perturbations δβ = δβ x + i δβ y , δα and δα about β = c e i θ ( β x = c cos θ , and β y = c sin θ ) and α = α = α , with ( α, θ ) satisfying (11). We linearize equations (10) accordingly. The linearized equationscan be written in matrix form as follows: ddt δβ x δβ y δα δα = M δβ x δβ y δα δα , (14)6here the Jacobian matrix M is given by M = − sin α − µ Re[ie i α ρ ( β )] sin α + µ Re[ie i α ρ ( β )]0 0 − cos α − µ Im[ie i α ρ ( β )] cos α + µ Im[ie i α ρ ( β )] ν Re[ie i2 α ρ ′ ( β )] − ν Re[e i2 α ρ ′ ( β )] − ν Re[e i2 α ρ ( β )] − ν Re[e i2 α ρ ( β )] ν Re[ie i2 α ρ ′ ( β )] − ν Re[e i2 α ρ ′ ( β )] − ν Re[e i2 α ρ ( β )] − ν Re[e i2 α ρ ( β )] . (15)We compute the eigenvalues numerically and find that, for large tail swimmers ν >
0, the pursuit modeis stable, whereas for large head swimmers ν <
0, the synchronization mode is stable. Our findings aresummarized in Figure 1(b).We test our results numerically by integrating the nonlinear equations (7, 9) for arbitrary choicesof initial conditions. Interestingly, the pursuit and synchronization modes seem to be globally attractingmodes in the case of large tail and large head swimmers, respectively. Figure 4(a) shows a depiction oftwo large-tail swimmers honing in on quasi-periodic pursuit trajectories, while (b) depicts two large-headswimmers synchronizing their motion in finite time to swim side by side.
The limit of unbounded domain.
We conclude this section by noting that in the limit of infinite domain,the solutions (11) of the doubly-periodic system (7, 9) converge to the relative equilibria of the unboundedsystem (7, 8). In the unbounded system, the relative equilibria can be obtained either by symmetry argumentsor by analytical manipulation of the equations of motion. Namely, one has two families of relative equilibria θ = α and θ = α + π/ ω , ω → ∞ , the Jacobian matrix M converges to M ∞ = − sin α + µc sin( α − θ ) sin α − µc sin( α − θ )0 0 − cos α − µc cos( α − θ ) cos α + µc cos( α − θ )2 ν c sin(2 α − θ ) 2 ν c cos(2 α − θ ) − ν c cos(2 α − θ ) − ν c cos(2 α − θ )2 ν c sin(2 α − θ ) 2 ν c cos(2 α − θ ) − ν c cos(2 α − θ ) − ν c cos(2 α − θ ) . (16)The corresponding eigenvalues are [0, 0, 0, ∓ ν /c ]. The eigenvalue − ν /c corresponds to the pursuitmode where α = θ , whereas +2 ν /c corresponds to the synchronization mode. This means that, for largetail swimmers with ν >
0, the pursuit mode is linearly stable and the synchronization mode is unstable,whereas for large head swimmers when ν >
0, the opposite is true, thus confirming the results obtainedabove for finite-sized doubly-periodic domains.
We revisited the hydrodynamic dipole model governing the interaction of asymmetric microswimmers inHele-Shaw confinement, [11]. Following [13, 20], we obtained a closed-form expression for the velocity fieldinduced by the swimmers and their image system in doubly-periodic domains. We treated in details thedynamics of two interacting swimmers, and found two special solutions that correspond to pursuit andsynchronization of the two dipoles. The pursuit mode is stable and attracting for large tail swimmers whilethe synchronization mode is stable and attracting for large head swimmers. By attracting, we mean that,starting from arbitrary initial conditions, large tail swimmers tend to tailgate each other while large headswimmers tend to synchronize their motion in finite time to swim side by side. These results are particularlyinteresting in light of the collective behavior reported in [12, 13] on populations of such swimmers. In theseworks, large tail swimmers were observed to “develop active lanes” [12] and “tail-gate each other” [13], asshown in Figure 5(top row), which suggests that the pursuit mode remains stable as the system size increases.Note that, to generate Figure 5, we integrate equations (7) and (9) for a population of 400 dipoles, startingfrom a uniform isotropic distribution and using the parameter values µ = 0 . ν = 0 and ν = 1 (large tail)or ν = − . y −30 −20 −10 0 10 20 30−30−20−100102030 xy −30 −20 −10 0 10 20 30 x −30 −20 −10 0 10 20 30 x −30 −20 −10 0 10 20 30 x t=0t=0 t=285 t=335t=10 t=30 t=80t=195 Figure 5:
Emergent collective behavior in large tail swimmers (top row) and large head swimmers (bottom row) starting froma uniform isotropic distribution. Large tail swimmers tend to tailgate each other, thus forming active lanes, while large headswimmers tend to form stationary clusters. Parameter values are µ = 0 . , ν = 1 (top row) and ν = − . (bottom row) in atotal population of swimmers. Populations of large head swimmers were shown to form heavily polarized sharp density waves in [12],consistent with predictions based on linear stability analysis of a kinetic-type continuum model [11]. Onecould conjecture that the synchronization mode observed here in pairs of large head swimmers may beresponsible for the global polarization observed in [12]. However, this thinking is too simplistic. Theemergence of global polarization patterns in finite size populations is not intuitive given the nature of dipolarinteractions among the swimmers. Further, these polarized density waves were not observed in the detailedparametric study reported in [13, Figure 7]. Instead, [13] reported, in agreement with unpublished resultsby Levaufe and Saintillan, that large head swimmers tend to form stationary clusters (see Figure 5(bottomrow)), which are not predicted by the linear stability analysis of [11]. All this is to say that the global patternsof the finite size systems in [12] and [13] are in agreement, except for the global polarization pattern. Thisinconsistency may be due to differences in the system size – thousands of particles in [12] versus hundreds inFigure 5 and in [13] – or to differences in the details of the numerical implementation. In [12], the point dipolemodel is desingularized and hydrodynamic interactions are approximated for fast computations, whereas [13]use a local repulsion potential for collision avoidance and accurately account for hydrodynamic interactionsand the doubly-infinite image system. While the difference in numerical implementation may play a role, thesystem size may be the main reason why polarized waves are not observed in [13]. Brotto et al. [11] predictedthis behavior for a continuous kinetic-like model, therefore it is not surprising that it is not reproduced bya fully nonlinear model with only a few hundred swimmers. Irrespective of the reason, the results reportedin this study suggest that the global polarization mode in large head swimmers is not “robust” to systemperturbances, whereas the pursuit mode in large tail swimmers is.
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