swimming dynamics near a wall in a weakly elastic...
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J Nonlinear Sci (2015) 25:1153–1167DOI 10.1007/s00332-015-9253-x
Swimming Dynamics Near a Wall in a WeaklyElastic Fluid
S. Yazdi1 · A. M. Ardekani2 · A. Borhan1
Received: 16 September 2014 / Accepted: 15 April 2015 / Published online: 16 May 2015© Springer Science+Business Media New York 2015
Abstract We present a fully resolved solution of a low-Reynolds-number two-dimensional microswimmer in a weakly elastic fluid near a no-slip surface. The resultsillustrate that elastic properties of the background fluid dramatically alter the swim-ming hydrodynamics and, depending on the initial position and orientation of themicroswimmer, its residence time near the surface can increase by an order of magni-tude. Elasticity of the extracellular polymeric substance secreted by microorganismscan therefore enhance their adhesion rate. The dynamical system is examined through aphase portrait in the swimming orientation and distance from the wall for four types ofself-propulsion mechanisms, namely: neutral swimmers, pullers, pushers, and stirrers.The time-reversibility of the phase portraits breaks down in the presence of polymericmaterials. The elasticity of the fluid leads to the emergence of a limit cycle for pullersand pushers and the change in type of fixed points from center to unstable foci for amicroswimmer adjacent to a no-slip boundary.
Keywords Microorganisms · Locomotion · Low-Reynolds-number swimming ·Viscoelastic fluid · No-slip boundary
Communicated by Paul Newton.
B A. M. [email protected]
1 Department of Chemical Engineering, The Pennsylvania State University, University Park,PA 16802, USA
2 School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
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Mathematics Subject Classification 76Z99 · 76D07 · 74F10
1 Introduction
Hydrodynamic interactions between microorganisms and confining boundaries in thelow-Reynolds-number regime have been the subject of active research due to theirimportant role in a number of biological and environmental issues (Lauga and Pow-ers 2009; Brennen and Winet 1977). For over half a century, theoretical studies havemostly focused on the locomotion of microorganisms, and their swimming speed andefficiency near boundaries in a Newtonian fluid (Katz 1974; Katz et al. 1975). Thesestudies have been inspired by experimental observations of bacteria colonizing bioticand abiotic surfaces (Harshey 2003), attraction of microorganisms to surfaces (Roth-schild 1963; Fauci andMcDonald 1995;Winet et al. 1984), and the circularmotion of ahelical swimmer near a solid surface (Berg and Turner 1990;Maeda et al. 1976) and anair–liquid interface (Lemelle et al. 2010). Near-wall accumulation of microorganismsis caused by the combined effects of hydrodynamic interaction of swimming microor-ganisms with a surface (Berke et al. 2008; Lauga et al. 2006; Leonardo et al. 2011),collisions with the surface, and rotational Brownian motion (Li and Tang 2009). Usinglubrication theory, Wang and Ardekani (2013) quantified the reduction in swimmingvelocity of an archetypal swimming model for ciliated bodies, called “squirmer,” nearan air–liquid interface. Lopez and Lauga (2014) modeled a helical swimmer near aninterface covered by surfactant using a far-field hydrodynamic approach and showedthat the boundary conditions at the interface significantly affect the direction of rota-tion of the swimmer. Spagnolie and Lauga (2012) investigated the effects of geometryand propulsive mechanism of a microswimmer on the surface attraction and pitchingdynamics. They showed that in comparison with full numerical solution of the Stokesequation, the far-field approximation for a microswimmer with axisymmetric surfacevelocity generally provides a good model, depending on the swimmer’s geometry,but does not capture lubrication effects. Hydrodynamic interaction of a suspension ofsquirmers between twowalls has been numerically studied by Li andArdekani (2014).They investigated the inertial effect on the squirmer–surface interaction, and demon-strated that different swimming modes occur near a wall, depending on the swimminggait of the squirmer. Along with these studies on the kinematics of locomotion, anumber of studies have also focused on the control and dynamical aspect of swim-ming near boundaries. Numerical investigations by Or and Murray (2009) reportedon nonlinear periodic orbits along the wall in addition to the steady state trajecto-ries. These results were later confirmed via an experimental study on small roboticswimmers (Zhang et al. 2010). The same periodic behaviour was reported for a two-dimensional treadmilling swimmer near a no-slip boundary (Crowdy and Or 2010;Crowdy 2011). Ishimoto and Gaffney (2013) characterized the dynamical system ofa swimmer with an axisymmetric tangential surface motion and reported on stable(unstable) swimming next to a no-slip boundary (free surface in the small capillarynumber limit). Their study for a three-dimensional swimmer suggests that free-surfacedeformation is a requirement for stable swimming, consistent with a previous reporton two-dimensional swimmers by Crowdy et al. (2011).
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Despite extensive previous work on locomotion in Newtonian fluids, more stud-ies are needed to capture the influence of the microorganisms’ natural habitats sincemicroorganisms in nature ubiquitously interact with complex fluids. Some examplesincludemicrobes colonizing in viscoelastic mucus in human and animal bodies (Naka-mura et al. 2006), bacteria aggregating in oceanic gels, spermatozoa moving throughthe uterotubal junction (Suarez and Pacey 2006), and biofilms growing on naturaland manmade surfaces (Costerton et al. 1995). Early studies modeling swimmers asinfinite waving sheets concluded that depending on the Reynolds number, elasticityof the fluid can either increase or decrease the swimming speed (Chaudhury 1979;Sturges 1981). In recent years, dependence of swimming speed and efficiency incomplex fluids on the rheological properties of the fluid, propulsion mechanism, andmicroswimmer geometry have been investigated (Normand and Lauga 2008; Elfringet al. 2010; Pak et al. 2010, 2012). For example, for a Taylor’s waving sheet (Taylor1951) as an idealized model of flagella beating, the swimming speed in a viscoelasticfluid is lower than that for an infinite waving sheet of small amplitude in a Newtonianfluid (Lauga 2007). In contrast, for a finite undulatory sheet of large amplitude in aviscoelastic fluid, the swimming speed is higher than the aforementioned Newtonianvalue (Teran et al. 2010). A reduction in swimming speed compared to the Newtoniancounterpart was also reported for a spherical squirmer in a viscoelastic (Giesekus) fluid(Zhu et al. 2012), for an infinitely long cylinder with prescribed beating pattern (Fuet al. 2007), and for nematode Caenorhabditis elegans due to stretching of polymericmolecules near hyperbolic stagnation points (Shen and Arratia 2011). In concentratedpolymeric solutions, however, an enhancement of approximately 65% in the nema-tode’s swimming speed (compared to that in semi-dilute solutions) was reported, dueto the local shear-induced anisotropy (Gagnon et al. 2013). A computational study byThomases and Guy (2014) suggested that the source of the propulsion stroke in anundulatory swimmer (either the head or tail) can justify the underlying mechanism forthe increase or decrease in the swimming speed of an undulating sheet in a viscoelasticfluid. Other studies also suggest strong correlation between swimming speed and fluidelasticity for the cylindrical version of Taylor’s sheet (Dasgupta et al. 2013), similar toa model helical flagellum in a Boger fluid (Liu et al. 2011). A recent numerical studyby Spagnolie et al. Spagnolie and Lauga (2012) suggests that elasticity decreases theswimming speed for helical bodies with small pitch angle, but follows the oppositetrend for large pitch angles.
The diversity in the results of these studies suggests that the influence of elasticityof the surrounding fluid on the motility of microorganisms can vary depending onthe swimmer geometry and rheological properties of the fluid. In addition to complexfluids, confining boundaries are present in a microorganism’s natural habitat. There-fore, investigating the effect of polymers on the motion of microswimmers near asurface will lead to a more profound understanding of many biological processes. Liet al. (2014) numerically studied the effect of a no-slip boundary on the propulsion ofspherical microswimmers in a viscoelastic fluid characterized by the Giesekus consti-tutive equation and showed that fluid elasticity leads to the entrapment of pushers nearthe wall. Even though a neutral squirmer escapes from the wall in a viscoelastic fluid,the near-wall residence time increases significantly and reaches a maximum at Weis-senberg number of unity, where the Weissenberg number is defined as the product of
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the polymer relaxation time and effective shear rate. In our previous work (Yazdi et al.2014), we investigated the time-averaged locomotion of a two-dimensional swimmerwith time-reversible tangential surface motion near a no-slip boundary in an Oldroyd-B fluid. Our results showed the existence of a thin “attraction” layer in the vicinity ofthe wall. Both pullers and pushers swim toward the wall if they are initially placedwithin the “attraction” layer. Otherwise, their time-averaged motion repels them fromthe wall. Based on Purcell’s scallop theorem (Purcell 1976), however, zero net motionis expected for a swimmer with a time-reversible propulsion mechanism in the Stokesregime in a Newtonian fluid. In this study, a time-independent boundary condition atthe swimmer’s surface is used in order to compare the swimming dynamics and sta-bility of the swimmer in Newtonian and viscoelastic fluids. The role of fluid elasticityon the near-wall residence time is quantified for a wide range of initial positions andorientations.
2 Problem Formulation
We model the microorganism as a two-dimensional “squirmer” that propels itself viaa tangential surface velocity (Blake 1971; Lighthill 1952), us , described as
us = [B1 sin(θ − φ) + B2 sin(2(θ − φ))] eθ , (1)
where θ is the polar angle and φ is the swimming direction (both measured from thex-direction along the wall, as shown in Fig. 1), eθ is the tangential unit vector, and“ . ” denotes dimensional variables. The surface velocity described in (1), representingthe truncated form of Blake’s original boundary condition (Blake 1971), is commonlyused in the literature to model the metachronal beating of ciliated microorganisms andJanus particles (Lauga and Powers 2009; Crowdy 2013). Here, B1 and B2 = β B1,
x
yFluid
Wall
R
d
θ
φ
UΩ
Fig. 1 (Color online) Schematic of the system geometry. The red circle indicates the orientation of thesquirmer’s thrust
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respectively, represent the amplitudes of the first and second modes of Blake’s surfacevelocity. For β > 0 (< 0), thrust is generated in front of (behind) the microorganism,as in the case of bacteria (algae), and the squirmer is called a puller (pusher). In thecase of β = 0, the squirmer is termed a neutral swimmer and generates a fore-aftsymmetric velocity field, as in the case of Volvox. For the special case of |β| → ∞,known as a “stirrer,” the squirmer is unable to propel in an unbounded fluid, and onlymixes the fluid around it. As a result of the tangential surface motion (1), the squirmermoves with translational and rotational locomotion velocities U and �, respectively. Inthis paper, we quantify the role of elasticity of the surrounding fluid on the kinematicsof propulsion of the squirmer near a no-slip boundary and interpret the potentialaftermath. The tangential velocity on the surface of the swimmer described in (1) isassumed to be steady, i.e., B1 is constant.
Due to the highviscosity (μ) ofmicroorganisms’ natural habitats and their small size(radius R) and locomotion velocity (U or R�), inertial forces are negligible. Hence,the locomotion is characterized by low Reynolds number, Re � 1, which greatlysimplifies the governing equations. In the case of locomotion in a Newtonian fluid, thelinearity of the governing equations facilitates the analysis of the flow generated bythe microorganism, and a far-field approximation for the flow can be obtained fromsuperposition of Stokes singularity solutions. Recent experimental studies show thatthe superposition of point-force singularities overestimates the near-field velocity ofswimming organisms (Drescher et al. 2011). In addition, these solutions are limited toNewtonian fluids and cannot be extended to polymeric fluids due to the nonlinearityof the polymeric stress terms. Here, we use a perturbation analysis to examine thelocomotion kinematics of a two-dimensional squirmer near a no-slip boundary in aweakly elastic fluid.
The flow around the swimmer is described by the dimensionless differential equa-tions
∇.u = 0 (2)
∇ · σ = 0, (3)
where the stress tensor σ can be written as
σ = −p I + τ , (4)
with σ , τ , u, and p, denoting the total stress, deviatoric stress, velocity, and pressuredistributions, respectively. TheWeissenberg number, which is the ratio of the polymerrelaxation time (λ1) to a flow time scale, is defined as Wi = λ1 B1/R. In these equa-tions, the characteristic scales R, B1, and μB1/R are used to render length, velocity,and stress dimensionless, respectively. In what follows, all variables and parametersare in dimensionless form, unless noted otherwise.
For the constitutive model, we use the single-relaxation-time Oldroyd-B relation
τ + Wi∇τ = E + λWi
∇E, (5)
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which is based on a spring dumbbell model for the polymer molecules (Bird et al.1987). In (5), E = ∇u + ∇uT denotes the rate of strain tensor. The upper-convected
derivative of tensor a is defined as∇a = ∂a/∂t + u · ∇a − (∇uT · a + a · ∇u), where
t is time and λ = λ2/λ1 is the ratio of the fluid’s retardation and relaxation times λ2and λ1, respectively. The Oldroyd-B model, with its shear-rate-independent viscosity,serves well for the purposes within the scope of this study due to the small range ofWi. In the limit of small Weissenberg number, all field variables are perturbed usingexpansions of the form
{u, p,E} = {u0, p0,E0} + Wi {u1, p1,E1} + · · · . (6)
For a smallWi, we canwrite τ = E+WiΣ[u]. The nonlinear stress term,Σ[u], arisesfrom polymeric stresses modeled by the constitutive relation in (5). The leading-ordercontribution of the nonlinear stress term is then given by
Σ[u0] = (λ − 1) ×[∂E0
∂t+ u0 · ∇E0 − (∇uT0 · E0 + E0 · ∇u0)
]. (7)
The kinematics of locomotion in the leading-order solution is the same as that fora Newtonian fluid and can be determined by applying the Lorentz reciprocal theoremfor a force-free and torque-free microswimmer, namely
F′ · U0 + L′ · �0 = −∫∫
Sn · σ ′ · us dS, (8)
where S represents the surface of the swimmer characterized with an outward unitnormal vector n, all primed quantities correspond to the auxiliary problem of low-Reynolds-number motion of a cylinder next to a wall in a Newtonian fluid (Jeffreyand Onishi 1981), with F′ and L′ denoting the force and torque on the cylinder,respectively. The leading-order translational and rotational velocities resulting fromthe evaluation of (8) have been reported in our previous work (Yazdi et al. 2014). TheO(Wi) contribution from (8) takes the form (Leal 1980)
∫∫Sn · σ ′ · u1 dS −
∫∫Sn · σ 1 · u′ dS =
∫∫∫V
Σ[u0] : ∇u′ dV, (9)
where V is the volume of fluid outside of S. Equation (9) can be further simplifiedby applying the boundary condition u1 = U1 + �1 × xs + us on the surface of thesquirmer, and satisfying the force-free and torque-free conditions to yield
F′ · U1 + L′ · �1 =∫∫∫
VΣ[u0] : ∇u′ dV . (10)
The Newtonian flow field obtained in our earlier work (Yazdi et al. 2014) is used toevaluate the volume integral on the right-hand side of (10); namely, the solution of thebiharmonic equation, satisfying the no-slip condition on thewall andBlake’s boundary
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condition (Blake 1971) on the cylinder surface, is used to evaluate the leading-ordervelocity contributions to the integrand in (10). Finally, using the force and torquereported by Jeffrey and Onishi (1981) for the auxiliary problem, the kinematics oflocomotion at O(Wi) is found to be
(U1,x ,U1,y,�1
) = (1 − λ)
4π
(Λ1,Λ2,Λ3
), (11)
where Λi(β, y, φ)(i = 1, 2, 3) are the components of the dimensionless integral onthe right-hand side of (10), subscripts x and y indicate the components of the velocityvector parallel and normal to the wall, respectively, and y ≡ d/R is the dimensionlesscenter-to-wall distance. Note that Λi is identical to that obtained in Yazdi et al. (2014)since the contribution of ∂E0
∂t to the volume integral in (10) can be shown to vanish,by applying the Reynolds transport theorem, the Lorentz reciprocal theorem, and theforce-free and torque-free conditions (Yazdi 2015).
3 Results and Discussion
Having evaluated the kinematics of locomotion atO(Wi), we canwrite the autonomoussystemof equations describing the location (x(t), y(t)) and swimmingorientationφ(t)of a two-dimensional squirmer near a wall in a weakly elastic fluid as
∂x(t)
∂t= U0,x (y, φ;β) + Wi U1,x (y, φ;β),
∂ y(t)
∂t= U0,y(y, φ;β) + Wi U1,y(y, φ;β),
∂φ(t)
∂t= �0(y, φ;β) + Wi �1(y, φ;β). (12)
As expected, the dynamical system of equations is invariant under translation ofthe coordinate system along the wall, i.e., independent of x(t). The resulting phaseportraits for different types of squirming motion in a Newtonian (Wi = 0) and a vis-coelastic (Wi = 0.1) fluid are shown in Fig. 2. Note that due to the linearity of theStokes equation, the leading-order solution is time-reversible, which can be leveragedin further simplifying the analysis for pushers and pullers. Namely, under the transfor-mation β → −β, φ → −φ, and t → −t , the dynamical system for a Newtonian fluid(U0,y and �0) remains unchanged. However, the system of equations resulting fromtheO(Wi) solution (U1,y and�1) changes in sign (but not magnitude) under the sametransformation. Therefore, the dynamical system described in (12) results in differentbehaviors for pullers and pushers. The analysis can be further simplified by reducingthe range of swimming orientations to −π/2 ≤ φ ≤ π/2, owing to the symmetry ofthe problem.
The dynamical system in (12) has two fixed points for neutral swimmers (β = 0),six fixed points for pullers and pushers, and five fixed points for stirrers (|β| → ∞).For a neutral swimmer, the fixed points indicated by the red filled circles in Fig. 2consist of two saddle points at (±π/2, 1) for both Newtonian and viscoelastic fluids.
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−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(a) β = 0, Wi = 00 5 10 15 20 25 30
t
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(e) β = 0, Wi = 0.10 5 10 15 20 25 30
t
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(b) β = 2, Wi = 00 5 10 15 20 25 30 35
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(f) β = 2, Wi = 0.10 100 200 300 400 500
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(c) β = −2, Wi = 00 5 10 15 20 25 30 35
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(g) β = −2, Wi = 0.10 50 100 150 200 250 300 350
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(d) β = ∞, Wi = 00 5 10 15 20 25
−π/2 −π/3 −π/6 0 π/6 π/3 π/21
2
3
4
φ
y
(h) β = ∞, Wi = 0.10 20 40 60 80 100 120 140
Fig. 2 (Color online) Phase portrait of a squirmer in a Newtonian fluid (first column) and an Oldroyd-Bfluid with Wi = 0.1 (second column). Saddle, center, and focus points are shown with red filled circle,triangle and square symbols, respectively. The color map represents values of dimensionless time, startingfrom the squirmer’s initial position which is indicated by an asterisk symbol. The initial points of all spiraltrajectories are the unstable foci. Red dotted lines forming triangles are the separatrices
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For both pullers and pushers in a Newtonian fluid, four out of the six fixed points aresaddle points, indicated by red circles, and the remaining two are centers, indicated byred filled triangles. For a puller (pusher) with positive (negative) swimming orientationin a Newtonian fluid, the center trajectories are confined to a triangular region confinedto three saddle points as shown in Fig. 2b, c. The trajectories surrounding the othercenter are not necessarily closed. If the initial point is far from the center, the swimmercannot rotate its orientation back to the wall in a finite time. This is due to the fact thatthe swimmer’s rotational velocity � asymptotically approaches zero as y increases.
Interestingly, the addition of elastic effects changes the two centers to unstablefoci, indicated by red filled squares, via a Hopf bifurcation which, in the case ofthe positive (negative) focus point of a puller (pusher), leads to the emergence of astable limit cycle. The appearance of stable limit cycles has also been reported fornon-spherical squirmers near a surface in a Newtonian fluid (Ishimoto and Gaffney2013; Or et al. 2011). The stable limit cycle is bounded by the wall below and close totwo separatrices, represented by red dotted lines, on the sides, as shown in Fig. 2f, g.The computed spiral trajectories starting from the unstable focus point inside the limitcycle, and an initial point outside the limit cycle, but inside the separatrices, are shownin Figs. 2f, g. We associate the emergence of a limit cycle with the nonlinearity causedby polymeric stresses. Emergence of a limit cycle for swimming microorganisms inpolymeric fluids has been previously reported in a different context for a suspensionof motile cells in a vortical flow (Ardekani and Gore 2012).
The time evolutions of the swimming orientation (φ) and the dimensionless distancefrom the wall (y) are depicted in Fig. 3, using selected initial points in Fig. 2. Theresults are shown for three types of squirming gait (β = 0,±2) in a Newtonian(Wi = 0) and a viscoelastic fluid (Wi = 0.1). For a neutral squirmer, there is nosignificant change in the time evolutions of orientation and distance from the wallin a Newtonian or a weakly elastic fluid, in contrast to the behavior of pullers andpushers. For a puller and a pusher with initial conditions (φ0 = 0.92, y0 = 1.06)and (φ0 = 0.6, y0 = 1.35), respectively, the swimmer in a Newtonian fluid follows acenter trajectory with a constant amplitude for periodic evolution of orientation anddistance. However, in an elastic fluid, due to the change of center points to foci, theswimmer trajectories become spiral; the puller moves on a limit cycle for the choseninitial point, and the pusher escapes from the wall.
Another consequence of nonlinear stresses is the change in the location of the fixedpoints, with the exception of those at (±π/2, 1) (Fig. 2). The most significant positionchange occurs for the saddle point that marks the upper limit of the triangular regionformed by the separatrices. This point dictates how far a squirmer can swim away fromthe wall before ultimately returning toward the wall. The other unstable focus pointpositioned at negative (positive) φ for a puller (pusher) does not have a limit cycle andthe phase-portrait trajectories initiated from points close to it are spirals. The locationof fixed points, except those at (±π/2, 1), is also a function of β. As |β| increases,in both Newtonian and viscoelastic fluids, the vertical position of the unstable saddlepoint that dictates the upper limit of the separatrices increases and ultimately reachesinfinity for a stirrer. While the locations of the saddle points at (±π/2, 1) remainunchanged with increasing values of |β|, the fourth saddle point moves to (0, 1) for a
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-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
t
φ
β = 0, Wi = 0.1, φ0 = −1.42, y0 = 4.00β = 0, Wi = 0.0, φ0 = −1.42, y0 = 4.00β = 2, Wi = 0.1, φ0 = 0.92, y0 = 1.06β = 2, Wi = 0.0, φ0 = 0.92, y0 = 1.06β = -2, Wi = 0.1, φ0 = 0.60, y0 = 1.35β = -2, Wi = 0.0, φ0 = 0.60, y0 = 1.35
0 50 100 150 200 250 300 350 400
0 50 100 150 200 250 300 350 4001.0
1.5
2.0
2.5
3.0
3.5
4
t
y
Fig. 3 (Color online) Time evolution of swimming orientation (φ) and dimensionless distance from thewall(y) for different squirming gaits, i.e., a neutral swimmer (β = 0), a puller (β = 2), and a pusher (β = −2),with various initial conditions (φ0, y0) in a Newtonian (Wi = 0) and a viscoelastic fluid (Wi = 0.1)
stirrer, and the positions of the two centers—unstable foci in the case of viscoelasticfluids—become symmetric with the same values of y and |φ| (Fig. 2).
In addition to affecting the swimming dynamics and stability, the presence of aboundary has been suggested to influence the cell–surface scattering and the experi-
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mentally observed near-wall residence time, both of which play essential roles in theinitial stages of biofilm formation. Although long-range hydrodynamic interactionshave negligible effects on bacteria–surface scattering, it has been shown that theycould partially contribute to the experimentally observed long residence time as thebacteria get very close to the wall (Drescher et al. 2011). Given the fact that extracel-lular polymeric substances and DNA molecules in the fluid prior to biofilm formationchange the rheology of the fluid, the question arises: Does the elasticity of the fluidincrease the hydrodynamic contribution to cell–surface scattering and residence time?and if so, how much? The analytical solution for the locomotion allows us to answerthese questions while taking into account the lubrication effects.
Here, we define the residence time τ as the time during which the squirmer swimsat a distance smaller than a body length from the wall (<1–3µm). Drescher et al.(2011) observed that on average an individual E. coli cell spends almost 1min nearthe wall once it gets very close (approximately 1–3µm) to the surface. Figure 4 showsthe phase diagrams for the ratio of the residence time in a viscoelastic fluid to that ina Newtonian fluid, rτ ≡ τV/τN, for a puller and a pusher. The results in this figurecorrespond to the residence times of a microswimmer in the region y < 1.1 near thewall. Also, note that for the cases with center/spiral trajectories, τ is calculated for onlyone cycle during which the swimmer stays in close proximity (y < 1.1) of the wall.Overall, a pusher is more likely to swim at a close distance to the wall, based on theregion depicted by white, where rτ → 0/0 (see Fig. 4). For both pullers and pushers,however, spiral trajectories for the focus point with no limit cycle bring the squirmercloser than a body length to the wall, behavior that is not possible for a squirmer oncenter trajectories in a Newtonian fluid. This is shown by the dark red grids in Fig. 4,for which rτ → ∞. For the rest of the domain under study, the residence time for asquirmer is either not affected by fluid elasticity (0.98 < rτ < 1.1) or increases up toan order of magnitude.
Fluid elasticity also has a significant effect on the scattering orientation of the swim-mer as it swims away from the wall. Figure 5 shows the final angle of a squirmer’sthrust versus its initial swimming direction φ0 for initial location y0 = 2.1 in New-tonian and viscoelastic fluids. Here, we define the final orientation angle φf as that atwhich the magnitude of the rotational velocity |�| becomes less than 10−6. The barsin Fig. 5 indicate that a squirmer with initial condition (φ0, 2.1) undergoes either acenter or a spiral trajectory (bounded by a limit cycle), and the size of the bar is anindication of the range of angles φf to which the squirmer’s swimming direction isconfined. While the elasticity of the background fluid has a minor effect on the finalangle of a neutral squirmer, it changes the final angles of a puller and a pusher. Thisdifference is due to the change of center trajectories to spirals in the presence of fluidelasticity, which leads to the squirmer escaping from a closed loop near the wall in aviscoelastic fluid.
4 Conclusions
Using a perturbation analysis in the limit of small Weissenberg number (Wi), weinvestigated the effect of aweakly elastic fluid environment on the swimmingdynamicsof a two-dimensional (2D) squirmer. Whereas in a previous study (Yazdi et al. 2014),
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−π/2 −π/3 −π/6 0 π/6 π/3 π/21.0
1.5
2.0
2.5
3.0
φ
y
(a) β = 2, Wi = 0.1 rτ
0.00
0.60-0.70
0.98-1.10
1.10-1.20
1.20-1.50
1.50-2.50
2.50-4.00
>10
∞
−π/2 −π/3 −π/6 0 π/6 π/3 π/21.0
1.5
2.0
2.5
3.0
φ
y
(b) β = -2, Wi = 0.1 rτ
0.00
0.60-0.97
0.97-1.10
1.10-1.20
1.20-1.50
1.50-2.00
2.00-2.50
2.50-3.00
>10
∞
Fig. 4 (Color online) Ratio of the residence time, rτ , for a squirmer in a viscoelastic fluid to the corre-sponding value in a Newtonian fluid for a a puller and a b pusher. White color grids correspond to caseswith zero residence time in both Newtonian and viscoelastic fluids
we examined the motion of a 2D swimmer with a time-reversible gait, in this work, wehave considered the gait of the swimmer to be constant in time. Our results show thata neutral swimmer eventually escapes the wall in both Newtonian and weakly elastic
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−π/2 −π/3 −π/6 0 π/6 π/3 π/2
−π/2
−π/3
−π/6
0
π/6
π/3
π/2
φ0
φf
(a) β = 0
NewtonianViscoelastic
−π/2 −π/3 −π/6 0 π/6 π/3 π/2
φ0
(b) β = 2
−π/2 −π/3 −π/6 0 π/6 π/3 π/2
φ0
(c) β = −2
Fig. 5 (Color online) Final orientation angle, φf , of a a neutral squirmer, b a puller, and c a pusher forlocomotion in a Newtonian fluid (circle) and an Oldroyd-B fluid with Wi = 0.1 (triangle). The initialposition is (φ0, y0 = 2.1). The bars indicate that a squirmer follows either a center or a spiral trajectory(bounded by a limit cycle), and the size of the bar is an indication of the range of final angles to which thesquirmer’s orientation is confined
fluids. Both pullers and pushers can have periodic trajectories in a Newtonian fluid.These trajectories change to spirals in the presence of elasticity in the backgroundfluid, which leads to a hopf bifurcation in the dynamical system. We have also shownthat elasticity of the surrounding fluid may contribute to an increase in the residencetime of a 2D microswimmer in the vicinity of a wall, depending on the swimmer’sinitial orientation and distance from the wall.
Most of the important effects of extracellular polymers, and other molecules suchas DNA and proteins that alter the rheology of the fluid, are manifested throughchanges in dynamical system and stability analysis. In particular, emergence of limitcycles leads to an enhancement in the near-wall residence time and consequentlyadhesion rate. In practice, short-range non-hydrodynamic forces, e.g, van der Waalsand electrostatic forces, can increase the near-wall residence time even further. Thechange in trajectories from center to spiral due to elastic effects may enhance cell–cellinteraction as cell trajectories approach each other in a viscoelastic fluid, providingmore time for extracellular communication such as quorum sensing.
Acknowledgments This work was partially supported by the NSF under Grant No. DMR-0820404through the Penn State Center for Nanoscale Science. AMA acknowledges support from Grant No. CBET-1445955-CAREER.
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