Focusing of Active Particles in a Converging Flow
Mykhailo Potomkin, Andreas Kaiser, Leonid Berlyand, Igor Aranson
FFocusing of Active Particles in a Converging Flow
Mykhailo Potomkin , Andreas Kaiser , Leonid Berlyand , and IgorAranson Department of Mathematics, Pennsylvania State University, UniversityPark, 16803, USA Materials Science Division, Argonne National Laboratory, 9700 South CassAvenue, Argonne, Illinois 60439, USA Department of Biomedical Engineering, Pennsylvania State University,University Park, 16803, USA * corresponding author: [email protected] 7, 2018 Abstract
We consider active particles swimming in a convergent fluid flow in a trapezoidnozzle with no-slip walls. We use mathematical modeling to analyze trajectories ofthese particles inside the nozzle. By extensive Monte Carlo simulations, we show thattrajectories are strongly affected by the background fluid flow and geometry of thenozzle leading to wall accumulation and upstream motion (rheotaxis). In particular, wedescribe the non-trivial focusing of active rods depending on physical and geometricalparameters. It is also established that the convergent component of the backgroundflow leads to stability of both downstream and upstream swimming at the centerline.The stability of downstream swimming enhances focusing, and the stability of upstreamswimming enables rheotaxis in the bulk.
Active matter consists of a large number of self-driven agents converting chemical energy,usually stored in the surrounding environment, into mechanical motion [1, 2, 3]. In the lastdecade various realizations of active matter have been studied including living self-propelledparticles as well as synthetically manufactured ones. Living agents are for example bacteria[4, 5], microtubules in biological cells [6, 7], spermatozoa [8, 9, 10] and animals [11, 12, 13].Such systems are out-of-equilibrium and show a variety of collective effects, from clustering[14, 15, 16, 17] to swarming, swirling and turbulent type motions [3, 4, 5, 13, 18, 19, 20, 21],reduction of effective viscosity [22, 23, 24, 25, 26, 27, 28], extraction of useful energy [29,1 a r X i v : . [ c ond - m a t . s o f t ] J u l
0, 31], and enhanced mixing [32, 33, 34]. Besides the behavior of microswimmers in thebulk the influence of confinement has been studied intensively in experiments [35, 36] andnumerical simulations [37, 38, 39, 40]. There are two distinguishing features of swimmersconfined by walls and exposed to an external flow: accumulation at the walls and upstreammotion (rheotaxis). Microorganisms such as bacteria [41, 42, 43, 44, 45] and sperm cells[46] are typically attracted by no-slip surfaces. Such accumulation was also observed forlarger organisms such as worms [47] and for synthetic particles [48]. The propensity of activeparticles to turn themselves against the flow (rheotaxis) is also typically observed. While forlarger organisms, such as fish, rheotaxis is caused by a deliberate response to a stream tohold their position [49], for micron sized swimmers rheotaxis has a pure mechanical origin[50, 51, 52, 53, 54].These phenomena observed in living active matter can also be achieved using syntheticswimmers, such as self-thermophoretic [55] and self-diffusiophoretic [56, 57, 58, 59] micronsized particles as well as particles set into active motion due to the influence of an externalfield [60, 61, 62].Using simple models we describe the extrusion of a dilute active suspension through atrapezoid nozzle. We analyze the qualitative behavior of trajectories of an individual activeparticle in the nozzle and study the statistical properties of the particles in the nozzle. Theaccumulation at walls and rheotaxis are important for understanding how an active sus-pension is extruded through a nozzle. Wall accumulation may eliminate all possible benefitscaused by the activity of the particles in the bulk. Due to rheotaxis active particles may neverreach the outlet and leave the nozzle through the inlet, so that properties of the suspensioncoming out through the outlet will not differ from those of the background fluid.The specific geometry of the nozzle is also important for our study. The nozzle is a finitedomain with two open ends (the inlet and the outlet) and the walls of the nozzle are notparallel but convergent, that is, the distance between walls decreases from the inlet to theoutlet. The statistical properties of active suspension (e.g., concentration of active particles)extruded in the infinite channel with parallel straight or periodic walls are well-established,see e.g., [63] and [64], respectively. The finite nozzle size leads to a “proximity effect”, i.e.,the equilibrium distribution of active particles changes significantly in proximity of both theinlet and the outlet. The fact that the walls are convergent, results in a “focusing effect”,i.e., the background flow compared to the pressure driven flow in the straight channel (thePoiseuille flow) has an additional convergent component that turns a particle toward thecenterline. Specifically, in this work it is shown that due to this convergent component of thebackground flow both up- and downstream swimming at the centerline are stable. Stabilityof the upstream swimming at the centerline is somewhat surprising since from observationsin the Poisueille flow it is expected that an active particle turns against the flow only whileswimming towards the walls, where the shear rate is higher. This means that we find rheotaxisin the bulk of an active suspension.
To study the dynamics of active particles in a converging flow, two modeling approaches areexploited. In both, an active particle is represented by a rigid rod of length (cid:96) swimming in2igure 1: Sketch of a trapezoid nozzle filled with an dilute suspension of rodlike activeparticles in the presence of a converging background flow.the xy -plane. In the first - simpler - approach, the rod is a one-dimensional segment whichcannot penetrate a wall, whereas in the second - more sophisticated - approach we use theYukawa segment model [65] to take into account both finite length and width of the rod, aswell as a more accurate description of particle-wall steric interaction.The active particle’s center location and its unit orientation vector are denoted by r =( x, y ) and p = (cos ϕ, sin ϕ ), respectively. The active particles are self-propelled with avelocity directed along their orientation v p . The active particles are confined by a nozzle,see Fig. 1, which is an isosceles trapezoid Ω, placed in the xy -plane so that inlet x = x in andoutlet x = x out are bases and the y -axis is the line of symmetry:Ω = (cid:8) x in < x < x out , α x − y > (cid:9) . (1)The nozzle length, the distance between the inlet and the outlet, is denoted by L , i.e., L = | x out − x in | . The width of the outlet and the inlet are denoted by w out and w in , respectively,and their ratio is denoted by k = w out /w in .Furthermore, the active particles are exposed to an external background flow. We ap-proximate the resulting converging background flow due to the trapezoid geometry of thenozzle by u BG ( r ) = ( u x ( x, y ) , u y ( x, y )) = ( − u ( α x − y ) /x , − u y ( α x − y ) /x ) , (2)where u is a constant coefficient related to the flow rate and α is the slope of walls of thenozzle. Equation (2) is an extension of the Poiseuille flow to channels with convergent walls .Active particles swim in the low Reynolds-number regime. The corresponding over-damped equations of motion for the locations r and orientations p are given by:d r d t = u BG ( r ) + v p , (3)d p d t = (I − pp T ) ∇ r u BG ( r ) p + (cid:112) D r ζ e ϕ . (4) In order to recover the Poiseuille flow (for channels of width 2 H ) from Eq. (2), take x = H/α , u = H /α and pass to the limit α →
0. Note that the walls of the nozzle are placed so that they intersect at the origin,so in the limit of parallel walls, α →
0, both the inlet and the outlet locations, x in and x out , go to −∞ . D r ; ζ is an uncorrelated noise with theintensity (cid:104) ζ ( t ) , ζ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ), e ϕ = ( − sin ϕ, cos ϕ ). Equation (4) can also be rewritten forthe orientation angle ϕ : d ϕ d t = ω + ν sin 2 ϕ + γ cos 2 ϕ + (cid:112) D r ζ. (5)Here ω = 12 (cid:18) ∂u y ∂x − ∂u x ∂y (cid:19) , ν = 12 (cid:18) ∂u y ∂y − ∂u x ∂x (cid:19) = ∂u y ∂y = − ∂u x ∂x , and γ = 12 (cid:18) ∂u y ∂x + ∂u x ∂y (cid:19) are local vorticity, vertical expansion (or, equivalently, horizontal compression; similar toPoisson’s effect in elasticity) and shear.The strength of the background flow is quantified by the inverse Stokes number, whichis the ratio between the background flow at the center of the inlet and the self-propulsionvelocity v . Specifically, σ = u x ( x in , v = u α v | x in | , (6)where ( x in ,
0) denotes the location at the center of the inlet.In the first modeling approach we include the particle wall interaction in the followingway: an active particle is not allowed to penetrate the walls of the nozzle. To enforce this,we require that both the front and the back of the particle, r ( t ) ± ( (cid:96)/ p , are located insidethe nozzle. In numerical simulations of the system (3)-(5) this requirement translates intothe following rule: if during numerical integration of (3)-(5) a particle penetrates one ofthe two walls, then this particle is instantaneously shifted back along the inward normalat the minimal distance, so its front and back are again located inside the nozzle while itsorientation is kept fixed.Unless mentioned otherwise, in this modeling approach we consider a nozzle whose inletwidth w in = 0 . w out = 0 . L = 0 . L = 0 . L = 1 . (cid:96) = 20 µ m, they swim with a self-propulsion velocity v = 10 µ m s − and their rotationaldiffusion coefficient is given by D r = 0 . − .All active particles are initially placed at the inlet, x (0) = x in , with random y -component y (0) and orientation angle ϕ (0). The probability distribution function for initial conditions y (0) and ϕ (0) is given by Ψ ∝ (cid:96) , width λ and the corresponding aspect ratio a = (cid:96)/λ is discretized into n r spherical segments with n r = (cid:98) a/ (cid:101) ( (cid:98) x (cid:101) denotes the nearest integer function). Theresulting segment distance is also used to discretize the walls of the nozzle into n w segmentsin the same way. Between the segments of different objects a repulsive Yukawa potential isimposed. The resulting total pair potential is given by U = U (cid:80) n r i =1 (cid:80) n w j =1 exp[ − r ij /λ ] /r ij ,where λ is the screening length defining the particle diameter, U is the prefactor of theYukawa potential and r ij = | r i − r j | is the distance between segment i of a rod and j of thewall of the nozzle, see Fig. 2. 4igure 2: Sketch of a discretized active rod (red) of length (cid:96) and width λ which is propelledwith a velocity v along its orientation p and is exposed to a converging background flow u BG in the presence of a trapezoid nozzle confinement of length L and with an inlet of size w in and outlet of size w out (blue). To study a system with a packing fraction ρ = 0 . f T and f R which can be decomposed into parallel f (cid:107) , perpendicular f ⊥ and rotational f R contributions which depend solely on the aspect ratio a [68]. For thisapproach we measure distances in units of λ , velocities in units of v = F /f (cid:107) (here F isan effective self-propulsion force), and time in units of τ = λf (cid:107) /F . While the width of theoutlet w out is varied, the width of the inlet w in as well as the length of the nozzle L is fixedto 100 λ in our second approach. Initial conditions are the same as in the first approach. Toavoid that a rod and a wall initially intersect each other, the rod is allowed to reorient itselfduring an equilibration time t e = 10 τ while its center of mass is fixed.Furthermore, we use the second approach to study the impact of a finite density ofswimmers. For this approach we initialize N active rods in a channel confinement whichis connected to the inlet of the nozzle, see Fig. 2. Inside the channel we assume a regular(non-converging) Poiseuille flow [69]. We restrict our study to a dilute active suspension witha two dimensional packing fraction ρ = 0 .
1. To maintain this fraction, particles which leavethe simulation domain are randomly placed at the inlet of the channel confinement.
Here we characterize the properties of the particles leaving the nozzle at either the outletor the inlet. Specifically, our objective is to determine whether particles accumulate at thecenter or at walls when they pass through the outlet or the inlet.We start with the first modeling approach. Figure 3 shows the spatial distribution ofactive particles leaving the nozzle at the outlet for various inverse Stokes number σ and threedifferent lengths L of the nozzle, while the width of the inlet and the outlet are fixed. For small5igure 3: Histograms of the outlet distribution for y | out for given inverse Stokes number σ andlength L of the nozzle. The histograms are obtained from numerical integration of (3)-(5).Figure 4: Outlet distribution histograms for ( y, ϕ ) | out computed for given inverse Stokesnumber σ and nozzle length L = 0 . σ , the background flow is negligible compared to the self-propulsionvelocity. Active particles swim close to the walls and peaks at walls are still clearly visiblefor σ = 0 . L , see Fig. 3(a). For σ = 1, the self-propulsion velocity andthe background flow are comparable; in this case the histogram shows a single peak at thecenter of the outlet, see Fig. 3(b). Further increasing the inverse Stokes number from σ = 1to σ = 9 leads to a broadening of the central peak and then to the formation of two peakswith a well in the center of the outlet, see Fig. 3(c)-(e). Finally, for an even larger inverseStokes number σ , the self-propulsion velocity is negligible and the histogram becomes closeto the one in the passive (no self-propulsion, v = 0) case, see Fig. 3(f). Here the histogramfor a nozzle length L = 0 . y -component and the orientation angle ϕ of the active particlesreaching the outlet are depicted in Fig. 4(a)-(c). While active particles leave the nozzlewith orientations away from the centerline for small inverse Stokes number, σ = 0 .
5, theyare mostly oriented towards the centerline for larger values of the inverse Stokes number.In Fig. 4(c), one can observe that the histogram is concentrated largely for downstreamorientations ϕ ≈ ϕ ≈ ± π . These local peaks for ϕ ≈ ± π away from walls are evidence of rheotaxis in the bulk. These peaks are visible forlarge inverse Stokes numbers only and the corresponding active particles are flushed out ofthe nozzle with upstream orientations.Figure 5: (a) Probability of active particles to reach the outlet for various inverse Stokesnumber σ (horizontal axis) and given lengths of the nozzle L . Insets: Trajectories for thecase of L = 0 . y | in computed for given reduced flow velocities σ and nozzle lengths L .Due to rotational diffusion and rheotaxis it is possible that an active particle can leave thenozzle through the inlet. We compute the probability of active particles to reach the outlet.This probability, as a function of the inverse Stokes number σ for the three considered nozzlelengths L , is shown in Fig. 5(a), together with selected trajectories, see insets in Fig. 5(a).The figure shows that the probability that an active particle eventually reaches the outletmonotonically grows with the inverse Stokes number σ . Note that a passive particle alwaysleaves the nozzle through the outlet. By comparing the probabilities for different nozzlelengths L it becomes obvious that an active particle is less likely to leave the nozzle throughthe outlet for longer nozzles. Due to the larger distance L between the inlet and the outletan active particle spends more time within the nozzle, which makes it more likely to swim7pstream by either rotational diffusion or rheotaxis. In Fig. 5(b)-(d) histograms for activeparticles leaving the nozzle through the inlet are shown. In the case of small inverse Stokesnumber, σ = 0 .
5, the majority of active particles leaves the nozzle at the inlet. Specifically,most of them swim upstream due to rheotaxis close to the walls, but some active particlesleave the nozzle at the inlet close to the center. These active particles are oriented upstreamdue to random reorientation. By increasing the inverse Stokes number σ ≥
1, active particlesare no longer able to leave the nozzle at the inlet close to the center.Figure 6: Examples of two trajectories for L = 1 mm and σ = 1 .
0. The red trajectorystarts and ends at the inlet (the endpoint is near the lower wall). The blue trajectory has azigzag shape with loops close to the walls; the particle that corresponds to the blue trajectorymanages to reach the outlet.Let us now consider specific examples of active particles’ trajectories, see Fig. 6. The firsttrajectory (red) starts and ends at the inlet. Initially the active particle swims downstreamand collides with the upper wall due to the torque induced by the background flow. Close tothe wall it exhibits rheotactic behavior, but before it reaches the inlet it is expelled towardsthe center of the nozzle due to rotational diffusion, similar to bacteria that may escape fromsurfaces due to tumbling [70]. Eventually, the active particle leaves the nozzle at the inlet.As for the other depicted trajectory (blue), the active particle manages to reach the outlet.Along its course through the nozzle it swims upstream several times but in the end the activeparticle is washed out through the outlet by the background flow. For larger flow rates thetrajectories of active particles are less curly, since the flow gets more dominant, see insets ofFig. 5(a).Next we present results of the second modeling approach which is based on the Yukawa-segment model. So far we have concentrated on fixed widths of the inlet and outlet. Herewe consider nozzles with fixed length L and inlet width w in and vary nozzle ratio k . Westudy the behavior of active rods with varied aspect ratio a . As shown in Fig. 7, neitherthe aspect ratio a , see Fig. 7(a), nor the nozzle ratio k , see Fig. 7(b), have a significantimpact on the probability P out which measures how many active rods leave the nozzle at theoutlet. However, the aspect ratio a is important for the location where the active rods leavethe nozzle at the inlet and the outlet, see Fig. 8. For short rods ( a = 2) and small inverseStokes numbers ( σ ≤
1) the distribution of active particles shows just a single peak locatedat the center. This peak broadens if the inverse Stokes number increases, which is in perfectagreement with the results obtained by the first approach, cf. Fig. 3. It is more likely forshort rods than for long ones to be expelled towards the center due to rotational diffusion.Hence the distribution of particles at the outlet for long rods ( a = 10) shows additional peaksclose to the wall. These peaks become smaller if the inverse Stokes number increases. Thedistribution of particles leaving the nozzle at the inlet is similar to our first approach. Whilethe distribution is almost flat for small inverse Stokes numbers, increasing this number makes8igure 7: (a) Probability for an active particle to reach the outlet of the nozzle P out as afunction of inverse Stokes number σ for three given aspect ratios a of self-propelled rods and(b) for a fixed aspect ratio a and three given ratios of the nozzle k . Insets show close-ups.Figure 8: Comparison of the spatial distribution of active particles at (top row) the outletand (bottom row) the inlet of the nozzle for given inverse Stokes numbers σ and aspect ratios a , an outlet width w out = 50 λ and an inlet width w in = 100 λ .it impossible to leave the nozzle close to the center at the inlet. Similar to the outlet the wallaccumulation at the inlet is more pronounced for longer rods.9igure 9: Outlet distribution histograms for ( y, ϕ ) | out computed for given inverse Stokesnumbers σ and a nozzle with an outlet width of w out = 50 λ for active rods with an aspectratio (top row) a = 2 and (bottom row) a = 10.By comparing the orientation of the particles at the outlet, the influence of the actuallength of the rod becomes visible, see Fig. 9. As seen before for short rods, a = 2, forsmall inverse Stokes numbers σ there is no wall accumulation. Hence most particles leavethe nozzle close to the center and are orientated in the direction of the outlet. This profilesmears out if the inverse Stokes number is increased to σ = 1. For larger inverse Stokesnumbers the figures are qualitatively similar to the one obtained by the first approach, cf.Fig. 4(c). Particles in the bottom half of the nozzle tend to point upwards and particlesin the top half tend to point downwards. The same tendency is seen for long rods a = 10and small inverse Stokes number. However for long active rods, this is because they slidealong the walls. The bright spots close to the walls for long rods and large inverse Stokesnumbers indicate that particles close to the walls are flushed through the outlet by the largebackground flow even if they are oriented upstream. In addition, there are blurred peaksaway from the walls for large inverse Stokes numbers σ . The corresponding particles crossedthe outlet with mostly upstream orientations. This is similar to Fig. 4(c), where particlesexhibiting in-bulk rheotactic characteristics were observed at the outlet of the nozzle.By comparing the results for individual active rods, see again Fig. 9, with those forinteracting active rods at a finite packing fraction ρ = 0 .
1, see Fig. 10, we find that wallaccumulation becomes more pronounced. Mutual collisions of the rods lead to a broaderdistribution of particles. For long rods, a = 10, the peaks at ϕ ≈ ϕ ≈ ± π remain close10o the walls and the blurred peaks at the center vanish.Figure 10: Outlet distribution histograms for ( y, ϕ ) | out computed for given inverse Stokesnumbers σ and a nozzle with an outlet width w out = 50 λ for active rods with an aspect ratio(top row) a = 2 and (bottom row) a = 10 for a packing fraction ρ = 0 . Here we study the properties of the active particles in more detail and provide insight intothe nozzle geometry, the background flow and the size of the swimmers that should be usedin order to optimize the focusing at the outlet of the nozzle.For this purpose we study three distinct quantities. The averaged dwell time (cid:104) T (cid:105) , thetime it takes for an active particle to reach the outlet, the mean alignment of the particlesmeasured by (cid:104) cos ϕ out (cid:105) and the mean deviation from the center y = 0 at the outlet (cid:104)| y out |(cid:105) .As depicted in Fig. 5, for increasing inverse Stokes number the probability for active particlesto reach the outlet increases. However they are spread all over the outlet. This is quantifiedby the (cid:104)| y out |(cid:105) . Small values of (cid:104)| y out |(cid:105) correspond to a better focusing. If particles leavethe nozzle with no preferred orientation, their mean orientation vanishes, (cid:104) cos ϕ out (cid:105) = 0; incase of being orientated upstream we obtain (cid:104) cos ϕ out (cid:105) = − (cid:104) cos ϕ out (cid:105) = 1 if theparticles are pointing in the direction of the outlet. Obviously in an experimental realizationa fast focusing process and hence small dwell times T would be preferable.The numerical results obtained by the first modeling approach are depicted in Fig. 11.While the dwell time hardly depends on the size ratio k of the nozzle, obviously the strength of11he background flow has a huge impact on the dwell time and large inverse Stokes numbers σ lead to a faster passing through the nozzle of the active particles, see Fig. 11(a). Thealignment of the active particles, (cid:104) cos ϕ out (cid:105) , becomes better if the nozzle ratio k is large andthe flow is slow, see Fig. 11(b). The averaged deviation from the centerline (cid:104)| y out |(cid:105) increaseswith increasing nozzle ratio k since the width of the outlet becomes larger. As could alreadybe seen in Fig. 3, the averaged deviation from the centerline is non-monotonic as a functionof the inverse Stokes number and shows the smallest distance from the centerline for allnozzle ratios if the strength of the flow is comparable to the self-propulsion velocity of theswimmers, σ = 1.Figure 11: (a) Dwell time (cid:104) T (cid:105) ; (b) mean alignment at the outlet, (cid:104) cos ϕ (cid:105) ; (c) mean deviationfrom center y = 0 at the outlet (cid:104)| y out |(cid:105) .Let us now study how these three quantities depend on the aspect ratio of the swimmer.To this end, we use the second modeling approach. We consider all three parameters asa function of the inverse Stokes number σ . Longer rods have a shorter dwell time so thatthey reach the outlet faster, see Fig. 12(a). Increasing the flow velocity obviously leads to adecreasing dwell time. The same holds for the mean alignment – it decreases for increasinginverse Stokes number, see Fig. 12(b). Moreover, for small inverse Stokes numbers, σ ≤ a = 10 are washed out with almost random orientation, however short rods a = 2 are slightlyaligned with the flow. Short rods are focused better for small inverse Stokes numbers, σ ≤ k with fixed simmers’ aspect ratio a , we obtain that smaller ratios k lead tosmaller dwell times [Fig. 12(d)] and better alignment [Fig. 12(e)]. For narrow outlets (small k ) the active particles leave the outlet closer to the center, see Fig. 12(f). We discuss the stability of particles around the centerline y = 0 in the presence of a back-ground flow and confining walls if they are converging with a non-zero slope α . This stabilityis in contrast to a channel with parallel walls, where an active particle swims away from thecenterline provided that its orientation angle ϕ is different from nπ , n = 0 , ± , ± , . . . .12igure 12: (a,e) Dwell time (cid:104) T (cid:105) , (b,f) the mean alignment, (cid:104) cos ϕ out (cid:105) and (c,f) mean deviationfrom center y = 0 at the outlet (cid:104)| y out |(cid:105) for (top row) a fixed outlet width of w out = 50 λ andgiven aspect ratios a of the swimmers and (bottom row) fixed aspect ratio a = 2 and variednozzle ratio k , whereby the width of the outlet changes.Indeed, in the case of a straight channel, α = 0, the background flow is defined as u x = u ( H − y ), u y = 0 (Poiseuille flow; u is the strength of the flow, 2 H is the distancebetween the walls). Then the system (3)-(5) reduces to˙ ϕ = u y (1 − cos 2 ϕ ) (7)˙ y = v sin ϕ. (8)Here we omit the equation for x ( t ) due to invariance of the infinite channel with respect to x and neglect orientation fluctuations, that is D r = 0. The phase portrait for this system isdepicted in Fig. 13(a). Dashed vertical lines ϕ = nπ , n = 0 , ± , ± , . . . consist of stationarysolutions: if an active particle is initially oriented parallel to the walls, it keeps swimmingparallel to them. If initially ϕ is different from nπ , then the active particle swims away fromthe centerline, y ( t ) → ±∞ as t → ∞ .When the walls are converging, α >
0, the y -component of the background flow is non-zero and directed towards the centerline. For the sake of simplicity we take u y = − αy , α > u x as in the Poiseuille flow, u x = u ( H − y ). In this case, the system (3)-(5) reduces to˙ ϕ = − ( α/
2) sin 2 ϕ + u y (1 − cos 2 ϕ ) (9)˙ y = − αy + v sin ϕ. (10)The corresponding phase portrait for this system is depicted in Fig. 13(b). Orientations ϕ = nπ represent stationary solutions only if y = 0. In contrast to the Poiseuille flow ina straight channel, see Eqs. (7) and (8), these stationary solutions ( ϕ = πn, y = 0) areasymptotically stable with a decay rate α (recall that α is the slope of walls). In additionto these stable stationary points there are pairs of unstable (saddle) points with non-zero y (provided that v > | y | does notchange, since a particle is oriented away from centerline, so the propulsion force moves theparticle away from the centerline and this force is balanced by the convergent component ofthe background flow, u y , moving the particle toward the centerline. The orientation angle ϕ does not change since the torque from the Poiseuille component of the background flow, u x ,is balanced by the torque from the convergent component, u y .Figure 13: Phase portraits ( ϕ, y ) for v = 0 . H = 1 . u = 0 .
6. (a) System (7)-(8), describingPoiseuille flow in a straight channel; dashed lines consist of stationary points. (b) System (9)-(10) describinga simplified convergent flow with α = 0 . πn,
0) (in red) and pairs of saddleswith non-zero y (in blue). Trajectories near the centerline converge to a stationary solution in the centerline.(c) System (3)-(5) with the convergent flow u BG = ( u x , u y ) used in Section 3.1 with x = − H/α = − . We also draw the phase portrait for the converging flow u BG = ( u x , u y ) introduced inSection 2, Fig. 13(c). One can compare the phase portraits Fig. 13(b) and Fig. 13(c) aroundthe stationary point ( ϕ = 0 , y = 0) to see that the qualitative picture is the same: thisstationary point is stable and it neighbors with two saddle points.The asymptotic stability of ( ϕ = 0 , y = 0) means that if a particle is close to the cen-terline and its orientation angle is close to 0 (particle is oriented towards the outlet), it willkeep swimming at the centerline pointing toward the outlet, whereas in Poiseuille flow theparticle would swim away. The asymptotic stability of ( ϕ = ± π, y = 0) is evidence of thatin the converging flow there is rheotaxis not only at walls but also in the bulk, specifically atthe centerline. Another consequence of this stability is the reduction of effective rotationaldiffusion of an active particle in the region around the centerline, that is the mean square14ngular displacement (cid:104) ∆ ϕ (cid:105) is bounded in time due to the presence of restoring force comingfrom the converging component of the background flow (cf. diffusion quenching for Janusparticles in [48]). Finally, we note that the nozzle has a finite length L and thus, the conclu-sions of the stability analysis are valid if the stability relaxation time, 1 /α s, does not exceedthe average dwell time (cid:104) T (cid:105) . We introduce a lower bound ˜ T for the dwell time (cid:104) T (cid:105) as thedwell time of an active particle swimming along the centerline oriented forward, ϕ = 0:˜ T = Lk/ ( σv (1 − k )) ln | σ (1 − k ) / ( k ( σ + 1)) | . Our numerical simulations show that ˜ T underestimates the average dwell time by a factorlarger than two. Using this lower bound, we obtain the following sufficient condition forstability: kw in σv ln (cid:12)(cid:12)(cid:12)(cid:12) σ (1 − k ) k ( σ + 1) (cid:12)(cid:12)(cid:12)(cid:12) ≥ In this work we study a dilute suspension of active rods in a viscous fluid extruded througha trapezoid nozzle. Using numerical simulations we examined the probability that a particleleaves the nozzle through the outlet - which is the result of the two counteracting phenomena.On the one hand, swimming downstream together with being focused by the converging flowincreases the probability that an active rod leaves the nozzle at the outlet. On the otherhand, rheotaxis results in a tendency of active rods to swim upstream.Theoretical approaches introduced in this paper can be used to design experimental setupsfor the extrusion of active suspensions through a nozzle. The optimal focusing is the resultof a compromise. While for large flow rates it is very likely for active rods to leave thenozzle through the outlet very fast, their orientation is rather random and they pass throughthe outlet close to the walls. The particles are much better aligned with the flow for smallflow rates and focused closer to the centerline of the nozzle, however the dwell time of theparticles becomes quite large. Based on our findings the focusing is optimal if the velocity ofthe background flow and the self-propulsion velocity of the active rods are comparable. Toreduce wall accumulation, the rods should have a small aspect ratio.We find that rheotaxis in bulk is possible for simple rigid rodlike active particles. Wealso established analytically the local stability of active particle trajectories in the vicinityof the centerline. This stability leads to the decrease of the effective rotational diffusion ofthe active particles in this region as well as the emergence of rheotaxis away from walls. Ourfindings can be experimentally verified using biological or artificial swimmers in a convergingflow.
Acknowledgements
The work was supported by NSF DMREF grant DMS-1628411. A.K. gratefully acknowl-edges financial support through a Postdoctoral Research Fellowship (KA 4255/1-2) from theDeutsche Forschungsgemeinschaft (DFG). 15 uthor contributions statement
Simulations have been performed by M.P. and A.K., the research has been conceived by L.B.and I.S.A. and all authors wrote the manuscript.
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