Front-back asymmetry controls the impact of viscoelasticity on helical swimming
V. Angeles, F.A. Godinez, J.A. Puente-Velazquez, R. Mendez, E. Lauga, R. Zenit
aa r X i v : . [ phy s i c s . f l u - dyn ] D ec Front-back asymmetry controls the impact of viscoelasticity on helicalswimming
V. Angeles, F.A. God´ınez,
2, 3
J.A. Puente-Velazquez, R. Mendez, E. Lauga, ∗ and R. Zenit
5, 1, †1
Instituto de Investigaciones en Materiales,Universidad Nacional Aut´onoma de M´exico,Apdo. Postal 70-360, M´exico Distrito Federal 04510, M´exico Instituto de Ingenieria, Universidad Nacional Aut´onoma de M´exico,Apdo. Postal 70-360, M´exico Distrito Federal 04510, M´exico Polo Universitario de Tecnolog´ıa Avanzada,Universidad Nacional Aut´onoma de M´exico,Apodaca 66629, Nuevo Leon, M´exico Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge CB3 0WA, United Kingdom Center for Fluid Mechanics, School of Engineering,Brown University, Providence RI 02912, USA (Dated: December 4, 2020)
Abstract
We conduct experiments with force-free magnetically-driven rigid helical swimmers in Newto-nian and viscoelastic (Boger) fluids. By varying the sizes of the swimmer body and its helical tail,we show that the impact of viscoelasticity strongly depends on the swimmer geometry: it can leadto a significant increase of the swimming speed (up to a factor of five), a similar decrease (alsoup to a factor of five) or it can have approximately no impact. Analysis of our data along withtheoretical modeling shows that the influence of viscoelasticity on helical propulsion is controlledby a snowman-like e ff ect, previously reported for dumbbell swimmers, wherein the front-backasymmetry of the swimmer leads to a non-Newtonian elastic force that can either favor or hinderlocomotion. ∗ [email protected] † [email protected] . INTRODUCTION Microorganisms swim in an environment in which inertial e ff ects are negligible [1]and therefore they employ locomotion strategies very di ff erent from those of fish andhumans. The scallop theorem [2], that states that a simply-articulated time-reversibleswimmer cannot achieve locomotion in a Stokes flow, provides a clear illustration of theimplications of living in a viscous-dominated environment.There are several methods exploited by microorganisms to cope with environmentsdominated by viscosity. In particular, the majority of motile bacteria, simple single-cellorganisms, exploit helical flagellar filaments in order to achieve locomotion [3]. Thesesemi-rigid filaments can either be used in isolation (monotrichous bacteria) or, for cellswith several helical filaments (peritrichous bacteria), they can bundle together to form asingle helical structure. In all cases, propulsion of the cell is enabled by the rotation ofsemi-rigid helical filaments in the viscous fluid. Since a helix is a chiral shape, a rotationaround the helical axis bypasses the constraints of the scallop theorem and it is able togenerate viscous thrust along its axis.The mechanics of helical swimming is well understood in the case of Newtonianflows [4]. However many of the fluids in which microorganisms move are not Newto-nian, ranging from mucus and complex suspensions to biological tissues. As with mostflows in which such fluids are involved, the dynamics of swimming microorganisms issignificantly a ff ected by viscoelasticity, the presence of shear-dependent stresses, or both.Numerous studies have recently been devoted to the subject [5–13], with some resultswhich appear to be in contradiction with each other, and thus a number of fundamentalissues remain.One possible starting point to capture the e ff ect of viscoelasticity is the theoreticalstudy in Ref. [5] which extended the classical Taylor swimming sheet result to the case ofviscoelastic Oldroyd-like fluids. The swimming speed of the sheet, U NN , normalized byits Newtonian value, U N , was calculated at leading order in the waving amplitude to be U NN U N = + κ De + De , (1)where κ = µ S / µ ≤ = τω is theDeborah number, where τ is the fluid relaxation time and ω the angular frequency of the2ave, a dimensionless parameter measuring the relative importance of viscoelasticity ina given flow. Since κ ≤
1, the result in Eq. (1) predicts that the swimming speed in aviscoelastic fluid will be smaller that its Newtonian equivalent for any value of De. Whilethis result reignited interest in the field, its validity is restricted to the case of small waveamplitude ( ak <<
1, where a and k are the amplitude and wave number of the oscillation)and to the case in which the wave is not a ff ected by the nature of the surrounding fluid(fixed-kinematics). In that limit, both experiments [7] and numerical simulations [11]have shown that this prediction is correct.In contrast with the result above, numerical computations found that when the am-plitude of oscillation was not small, the swimming speed in the viscoelastic fluid couldbe larger than that in the Newtonian fluid [14]. Several experimental reports have sub-sequently confirmed that a faster speed in viscoelastic media was in fact possible [6, 9].The possibility of obtaining both decrease and increase in swimming was reported inRef. [13] where experimental measurements for the ratio of swimming speeds for threedi ff erent swimming strategies at fixed De number showed that the swimming ratio couldbe smaller, larger or approximately one depending on the swimming kinematics. In otherwords, the swimming speed in a viscoelastic fluid does not depend solely on the value ofthe De number. A recent analysis of the e ff ect of the swimming gait on locomotion in non-Newtonian media obtained theoretical predictions in good agreement with experimentsso far [15].Given the complexity that arises from having swimming in which the waving shape ofthe appendages might depend on the flow itself via mechanical feedback, it is simpler tofocus first on the case for which the kinematics are fixed. The biological example wherethe shape is known to be essentially rigid and unchanged by the the fluids is the rotatinghelical filaments of swimming bacteria. The work in Refs. [16, 17] extended the Taylorswimming sheet result from Ref. [5] to the case of helix in the limiting case of a smallpitch angle θ (i.e. the angle between the helix axis and the local tangent along the helixcentrerline). They obtained the same decreasing trend of the normalized swimming speedwith Deborah number as in Eq. (1). Subsequent experiments with force-free helices drivenin rotation showed, in contrast, that the helical swimming speed could, be larger than thatin the Newtonian case [6]. Specifically, the swimming speed was shown to depend on boththe value of the Deborah number and the shape of the helix and helices with larger pitch3ngle produced more pronounced increase in swimming. However, only two values of thepitch angles were tested experimentally [6]. Subsequent numerical simulations confirmedthat the normalized swimming speed could be smaller or larger than one, depending onboth Deborah number and the geometry of the helix [11]. Related work showed that thedrag force on slender cylinders in viscoelastic fluids – the required building blocks tounderstand force generation for rotating helices – depend strongly on their orientationrelative to the main flow and their drag experiences strong tip e ff ects [18, 19]. Recentexperimental measurements using live bacteria showed that shear thinning e ff ects leadto higher swimming speeds that those with Newtonian fluids [20], in agreement withpast work [21, 22]; however, the fluid viscoelasticity did not a ff ect the swimming speeddirectly but instead the unsteady bundling / unbundling dynamics of the bacteria flagella.It is therefore clear that, in addition to the expected dependence on the value ofthe Deborah number, the geometrical properties of a helix impact its free swimmingspeed in a non-Newtonian fluid. In this paper, we conduct experiments with force-freemagnetically-driven rigid helical swimmers in Newtonian and viscoelastic (Boger) fluids.We measure the swimming speeds for helices with many di ff erent geometries and relativehead sizes. In accordance with previous studies, we found that depending on the helicalgeometry their swimming speeds can either increase significantly (up to a factor of five),decrease (also up to a factor of five) or remain approximately unchanged. The increase vsdecrease of the normalized swimming speed for all of our experimental results appearsto be correlated to the front-back asymmetry in size: when the helix has a larger diameterthat the head, a swimming speed larger than the Newtonian value is observed, and vice-versa. The impact of viscoelasticity on helical swimming appears thus to be controlled bythe snowman e ff ect, proposed theoretically [23] and corroborated experimentally [24] inpast work, wherein an elastic force driven by normal stress di ff erences is generated in theviscoelastic fluid by the rotation of the swimmer. Adapting the modeling from Ref. [23],we show, in agreement with our experiments, that this elastic force can then either hinderor favor propulsion depending on the ration between the size of the swimmer’s body andthat of its helical tail. 4 IG. 1. Sketch of the design parameters of the magnetically-driven rigid helical swimmers. Forthe helix, R is the radius, λ is the wavelength (pitch), L T is the projected length, d is the diameterof the helical filament. For the head, L H is the length and D H is the diameter. II. EXPERIMENTAL SETUP
The experimental design is similar to that previously used in Ref. [22]. A force-freeswimmer consisting of a tubular plastic head with a rigid helix tail is placed inside atest fluid. By inserting a small permanent magnet inside the head, the swimmers canbe rotated under the action of an external rotating magnetic field [25]. The shape of theswimmers is depicted schematically in Fig. 1. A right-handed rigid helix was placed atthe other end of the cylindrical head. In all cases, both the size of the head (length L H anddiameter D H ) and the helix (contour length L , projected length L T , radius R , wavelength λ and filament diameter d ) were varied in order to explore the e ff ect of geometry as widelyas possible. The values of the geometrical parameters for all swimmers used in this studyare shown in Table I. The first five swimmers (F1 and R1 to R4) had tails made of steelwire (Young’s modulus E ≈
207 GPa). The second set of swimmers (A1 to A5) were 3Dprinted and the tail fabricated with a polymeric resin. Note that the pitch angle of thehelix, θ , defined as tan θ = π R / λ , varies in our experiments from 29 ◦ to 77 ◦ .The rotation of the head, when combined with a chiral tail shape, produces the thrustforce that propels the swimmer through the fluid. The rotation frequency of the externalmagnetic field, measured with a digital tachometer, ranged from 0.41 Hz to 5.8 Hz, witha di ff erent range for each swimmer. All experiments are conducted below the step-outfrequency when the swimming no longer rotates with the external frequency.5 wimmer L H D H L d λ R R / λ θ D ∗ = R / D H L T F1 ( △ , ▲ ) 14.3 4.0 58 0.3 7.6 3.5 0.23 0.97 (55 ◦ ) 0.88 35.7R1 ( (cid:7) , (cid:8) ) 23 3.0 65 0.9 10.0 1.8 0.09 0.52 (29 ◦ ) 0.60 56.8R2 ( □ , (cid:4) ) 23 3.0 65 0.9 10.0 3.2 0.16 0.79 (45 ◦ ) 1.07 45.9R3 ( ▽ , ▼ ) 23 3.0 65 0.9 10.0 4.6 0.23 0.96 (55 ◦ ) 1.53 37.3R4 ( ◊ , ⧫ ) 23 3.0 65 0.9 10.0 11.8 0.59 1.31 (75 ◦ ) 3.93 16.8A1 ( ◁ , ◀ ) 17.3 4.1 76 1.0 9.5 3.0 0.16 0.75 (45 ◦ ) 0.73 54.3A2 ( ▷ , ▶ ) 17.3 4.1 80 1.0 9.5 7.0 0.37 1.13 (67 ◦ ) 1.71 39.36A3 ( △ , ▲ ) 17.3 4.1 83 1.0 9.5 15.0 0.79 1.34 (77 ◦ ) 3.66 23.2A4 ( (cid:7) , (cid:8) ) 17.3 4.1 80 1.0 5.0 3.5 0.35 1.13 (65 ◦ ) 0.85 37.6A5 ( □ , (cid:4) ) 17.3 4.1 80 1.0 13.0 9.0 0.35 1.13 (65 ◦ ) 2.20 37.2TABLE I. Dimensions of the ten helical swimmers used in this study. All length are reported inmillimeters. Symbols are defined in Fig. 1 while L is the total contour length of the tail and θ is thepitch angle in radians (degrees). The empty and solid symbols represent experiments conductedin Newtonian and Boger fluids, respectively. Two types of fluids were used, a Newtonian and a viscoelastic Boger fluid, and inboth cases we have two test fluids. The viscoelastic fluid was prepared with glucose,water and a small amount of polyacrylamide (PAA, molecular weight 5 × g / molfrom Sigma-Aldrich). The Boger fluids were was fabricated by slowly dissolving thepolyacrylamide in non-ionic water for 24 hours. Afterwards, the polymeric solution wasadded to the glucose and the mixture was mixed slowly for four days. We show inTable II the properties of the two pairs of tests fluids used in this study. For the first testpair, we used industrial grade glucose and for the second commercial corn syrup (Karobrand). While the properties of the industrial grade glucose varied from batch to batch,commercial corn syrup was consistently the same. In both cases, the Newtonian referenceliquid was fabricated by adding water to glucose until the fluid had similar viscosity tothat of the viscoelastic fluid.The rheological properties of the fluids were determined using a rheometer with par-allel plates with 40 mm diameter and 1 mm gap (TA Instruments, ARES-G2). Both steadyand oscillatory tests were conducted to measure the dynamic viscosity, µ , the storage and6oss moduli, G ′ and G ′′ , respectively. The mean relaxation time is calculated by fitting G ′ and G ′′ to a generalized Maxwell model [9]. The density of the fluids are obtained usinga 25 ml pycnometer.The motion of the swimmer in both Newtonian and a viscoelastic (Boger) fluid wasfilmed with a digital camera at 60 frames per second. The images were processed digitallywith the software Tracker . Each experiment was repeated at least three times to ensurerepeatability. The temperature in the experiment ranged between 23 ◦ C and 24 ◦ C. III. EXPERIMENTAL RESULTS
Each swimmer was tested in a fluid pair and its swimming speed was measured asa function of rotational frequency allowing direct comparison between Newtonian andviscoelastic results. We show in Fig. 2 shows three typical experimental results chosento illustrate the three possible qualitative results. The swimming speed is plotted as afunction of rotational frequency for swimmers F1, R1 and R4 from Table I for the first fluidpair N1 and B1, from Table II. Clearly, for a helical swimmer, three di ff erent behaviors arepossible: the swimmer can swim faster in a viscoelastic fluid compared to the Newtoniancase (green rhombus), is can go slower (red circles) or with approximately the same speed(grey triangles). The three swimmers, despite the changes in their geometrical parameters,are propelled by the same helical action and the most notable di ff erence between themare the value of their pitch angle, θ , and tail-to-head size ratios, D ∗ = R / D H . The angles Fluid G / W / PAAM ρ µ n τ (%) kg / m Pa s (-) sN1 89 / / / / / / / / ρ ), dynamic viscosity ( µ ),power index ( n ), and mean relaxation time ( τ ). FIG. 2. Measured swimming speed, U , as a function of rotational frequency, ω / π , for threerepresentative swimmers (F1, R1 and R4 from Table I where symbols are defined). Empty andfilled symbols show the results for Newtonian and viscoelastic fluids, respectively (fluids N1 andB1 from Table II). range from 29 ◦ (slower swimming) to 52 ◦ (same speed) to 75 ◦ (faster swimming) while thesize ratios are D ∗ = . U NN / U N , where U NN and U N are the measured mean speedsin the viscoelastic and Newtonian fluids, respectively. To assess the relative importanceof viscoelastic e ff ects, we calculate the Deborah number as De = ωτ , where τ is the fluidrelaxation time (from Table II). The ratio U NN / U N is then plotted in Fig. 3 (left) as a functionof De for all the swimmers studied here (from Table I). Despite the large range of Deborahnumbers in our experiments (from below 1 to above 20), a clear trend is not apparent inthe data.Instead of the Deborah number, one could argue that the relevant parameter to interpretthe data is the Weissenberg number, Wi, which, instead of comparing the relaxation timeof the fluid with the rotation rate of the swimmer, compares it to the relative rate ofdeformation in the flow. Hence, we can define this number as Wi = ˙ γτ , where ˙ γ is thecharacteristic shear rate. For a rotating helix, the shear rate scales as R ω / λ ; therefore,we have Wi ∼ ( R / λ ) De. We plot in Fig. 3 (right) the normalized mean speed, U NN / U N as a function of the Weissenberg number, for all experiments. Similar to the previous8 FIG. 3. Ratio of viscoelastic to Newtonian swimming speeds, U NN / U N , as a function of theDeborah number (left) and Weissenberg numbers (right). For symbols, see Table I. case, the data shows an unidentifiable dependence on Wi. These dimensionless numberscan therefore not be used alone to characterize the changes in swimming speed whenviscoelastic e ff ects are present.Contrasting our data with the experimental results from Ref. [6], we notice that in thiswork also the dependence of the swimming speeds with De for helices with di ff erent pitchangles did not collapse into a single curve. The follow-up numerical study in Ref. [11]showed also that the ratio U NN / U N was a ff ected by both the Deborah number and thehelix pitch angle. Guided by these studies, we re-plot our data in Fig. 4 (left) with theswimming speed increase now shown as a function of R / λ = tan θ /( π ) . Displayed inthis manner, we see a remarkably consistent increase of swimming enhancement with R / λ (i.e. with the helix angle, θ ) regardless of the value of the Deborah number. A valueof R / λ ≈ . θ ≈ . ◦ , appears to mark the transitionfrom a decrease to an increase in swimming speed. We have also included the data fromRef. [6] in Fig. 4 (left) ( ∗ and × symbols); the small number of data points in that studyappear to fit within the uncertainty of our experiments. Note, however, that the increasein U NN / U N found by these authors was very modest in comparison to the present datawhere we obtain increases of up to a factor of five.One important aspects of the geometry of the swimmers shown in Table I is that thesize of the head, D H , remains relatively constant for all swimmers; however, to achievedi ff erent pitch angles, the size of the helix, 2 R , varies significantly. Therefore, the helix-9 -1 FIG. 4. (left) Ratio of viscoelastic to Newtonian swimming speeds, U NN / U N , as a function of thehelix aspect ratio, R / λ ; the ( ∗ ) and ( × ) symbols show the data from Ref. [6] for R / λ = .
40 and R / λ = .
18, respectively. (right) Ratio of viscoelastic to Newtonian swimming speeds, U NN / U N ,as a function of helix to head diameter ratio D ∗ ≡ R / D H . All filled experimental symbols followTable I. to-head size ratio, D ∗ = R / D H , varies form 0.6 to 3.9. The helix diameter can thereforebe smaller, similar or larger that the head diameter. To explore the way in which thischange in geometry a ff ects the swimming speed, we show in Fig. 4 (right) the normalizedswimming speed, U NN / U N , as a function of the size ratio D ∗ , for all the experimentsconducted in this investigation. Clearly, and similarly to the results in Fig. 4 (left), acorrelation can be be identified; when the head is smaller than the helix, the swimmingspeed in the viscoelastic fluid is larger that the Newtonian one, and when the head islarger then the opposite happens. IV. PHYSICAL INTERPRETATION
How can we explain theoretically the influence of viscoelasticity on the swimmingspeed ratio, U NN / U N ? While viscoelastic e ff ects are undoubtedly important, the valuesof the Deborah or Weissenberg numbers alone are not able to quantify the impact ofelastic stresses on the swimming speed. As shown above, both the helix angle, and thehelix-to-head size ratio, appear to play a role in the balance between thrust and drag onthe swimmer. We consider them both separately in what follows.10 . Local resistive model Using the observation, shown in Fig. 2, that the swimming speed increases approxi-mately linearly with the rotational frequency in all cases, we can first attempt to rationalisethe impact of the helical slope using resistive-force theory for low-Reynolds number swim-mers [26]. This is known to be valid in the Newtonian case for slender swimmers, andthus should remain approximately valid at small Deborah numbers in the viscoelasticcase. To address the role of the helix angle, we consider the limit of the small swimmerhead so that both propulsion and thrust are dominated by the rotating helical tail.Neglecting the viscous drag on the head of the swimmer, the swimming speed of aforce-free helix is predicted by the resistive-force theory framework to be given by U ω R = ( ξ − ) tan θ + ξ tan θ , (2)where ξ = c ⊥ / c ∥ is the ratio between the drag coe ffi cient for local portions of the slenderhelix moving perpendicularly and parallel to the local tangent [26] and tan θ is the tangentof the helix angle. Assuming that a similar local hydrodynamic analysis can be conductedfor a viscoelastic Boger fluid at small De, the helix swimming speed would then be ( U ω R ) NN = ( ξ NN − ) tan θ + ξ NN tan θ , (3)where ξ NN = c NN ⊥ / c NN ∥ is the drag coe ffi cient ratio for a viscoelastic flow. Hence, assumingthat ξ ≈ U NN U N = ( ξ NN − ) ( + θ + ξ NN tan θ ) , (4)which can, theoretically, be smaller or larger that one depending on the the value of tan θ and on ξ NN .Our experimental results from Fig. 4 (left) show that U NN / U N < R / λ ). This would be consistent with the model in Eq. (4) in this limit if 1 < ξ NN < R / λ ), the experiments showthat U NN / U N >
1. This would be consistent with the model in Eq. (4) only if the drag ratiosatisfied ξ NN >
2. 11here is therefore a contradiction. Of course, such a local resistive-force theory ap-proach could very well not be valid in a viscoelastic fluid, for example if nonlocal e ff ects(hydrodynamic interactions) played an important role. Alternatively, if the local theorywas valid, the ratio of drag coe ffi cients ξ NN would have to depend on the value of theangle θ , i.e. the local orientation of the helix relative to the fluid in which it moves. Whilerecent numerical work reported that the elastic stresses in the wake of rigid cylindersdepend on the orientation of the cylinder relative to is velocity [19], the dependance ofthe drag coe ffi cient ratio for di ff erent angles in viscoelastic flows have not been reportedto date. This resistive-force theory approach does not appear, therefore, to explain theresults from Fig. 4 (left) in a physically-intuitive way. B. The snowman e ff ect We can, however, provide a physical mechanism for the change in swimming plottedas in Fig. 4 (right) by turning to past work that addressed the e ff ect of asymmetry forrotating swimmers in viscoleastic fluids. These theoretical [23] and experimental stud-ies [24] showed that a snowman, i.e. a dumbbell composed of two spheres of di ff erentdiameters, could swim in a viscoelastic fluid when rotating about its symmetry axis. Thephysical origin of the propulsion lies in the secondary flows generated in elastic fluidsby normal-stress di ff erence that, for a rotating sphere, lead to fluid flows directed awayfrom the sphere along its rotation axis. A dumbbell made of two spheres of di ff erent sizesexperiences therefore an imbalance of drag due to these two elastic flows, resulting inswimming. This viscoelastic propulsion force is directed in the direction from the largestto the smallest sphere [23, 24].Our data in Fig. 4 (right) clearly indicate that the front-back asymmetry of the helicalswimmers does control the normalized swimming speed. We propose therefore that it isthe size asymmetry between the head and the tail that leads to an additional snowman-like viscoelastic force a ff ecting the swimming speed. If this mechanism is correct, and forlocomotion that takes place head-first (the case in our experiments), a swimmer with ahead smaller than the helix should swim faster due to this viscoelastic snowman e ff ect;conversely, if the head is larger than the tail the swimming speed should decrease. Thisis indeed what we see in our experiments.12n order to be more quantitative, we consider the theoretical expression derived inRef. [23], and estimate the additional viscoelastic force resulting from the di ff erence insize between the head and helix. Assuming as a first approximation that the additionalviscoelastic force is generated regardless of the detailed shape of the head or helix, andidentifying the diameters of the spheres in Ref. [23] to the diameters of the head and helixin our experiment, the snowman propulsive force predicted theoretically is given by P S = c S ω ( D H ) De D ∗ ( D ∗ − )( + D ∗ ) , (5)where c S is a viscous drag coe ffi cient ( c S = k µ where k is a dimensionless shape factor), D ∗ = R / D H is the rise ratio and De = ωτ is the Deborah number. Next, we assume forsimplicity that the propulsion, P helix , and viscous drag on the helix, D helix , are not far fromthose given by the Newtonian resistive-force theory, and similarly for the drag force onthe head of the swimmer, D head , the steady force balance on the swimmer in a viscoelasticfluid is now given by P helix + P S = D helix + D head . (6)Using the classical expressions for P helix , D helix and D head from Ref. [4], and combining themwith Eq. (5), we obtain the result U NN = U N + U S , (7)where U N is the Newtonian swimming velocity given by U N = ω R ( ( ξ − ) tan θ + ξ tan θ + ξ o L ∗ sec θ ) , (8)with L ∗ = L H / L and ξ o = c H / c ∥ is the normalized head drag coe ffi cient ( c H is the headdrag coe ffi cient). The additional snowman speed U S is given by U S = ω R ⎛⎜⎜⎜⎝ ξ S D ∗ H θ De D ∗ ( D ∗ − )( + D ∗ ) + ξ tan θ + ξ o L ∗ sec θ ⎞⎟⎟⎟⎠ , (9)with ξ S = c S / c ∥ and D ∗ H = D H / L .Using this model, the additional viscoelastic thrust resulting from the front-back asym-metry leads to the normalized swimming speed written as a sum U NN U N = + U S U N , (10)13 FIG. 5. Modified extra swimming speed, U ∗ S (defined in Eq. 12), as a function of helix to headdiameter ratio D ∗ ≡ R / D H , for De ≈ .
8. The symbols are the experimental values while thelines show the theoretical predictions of the model in Eq. (12) with values ξ s =
20 (solid line), 40(dashed line) and 80 (dash-dotted line). where U S U N = ξ s De2 ( ξ − ) D ∗ H sin θ D ∗ ( D ∗ − )( + D ∗ ) . (11)This final expression indicates that the viscoelastic contribution due to the asymmetry ofthe swimmer depends on many factors, including the Deborah number and the size ratio D ∗ . Importantly, the ratio U S / U N can be positive or negative depending on the valueof D ∗ relative to one. Since ξ >
1, swimmers with D ∗ > D ∗ < U S / U N , termed U ∗ S , as U ∗ S = U S U N sin θ D ∗ H = ξ s De2 ( ξ − ) D ∗ ( D ∗ − )( + D ∗ ) , (12)where both sin θ and D ∗ H are known quantities in our experiments. The value of U ∗ S canthen be plotted as a function of D ∗ for given values of De and ξ s . To do so, we extractdata from Fig. 3 (left) for an approximately constant value of De ≈ . U ∗ S can be calculated. We show in Fig. 5 the comparisonbetween the model, Eq. (12) and the experimental values using three possible values forthe dimensionless factor ξ s . The model is able to reproduce the experimental trend and14hows a clear transition for U S / U N from negative to positive values, thus explaining thetransition from slower to faster than Newtonian when the helix to tail size ratio goes fromsmaller to larger than unity. V. CONCLUSION
In this work we have carried out experiments on the locomotion of free-swimmingmagnetically-driven rigid helices in Newtonian and viscoelastic (Boger) fluids. We var-ied the sizes of the swimmer body and its helical tail and showed that the impact ofviscoelasticity depends critically on the geometry of the swimmer: it can lead to a largeincrease of the swimming speed, a decrease or it can have approximately no impact.We proposed that the influence of viscoelasticity on helical propulsion is controlled by asnowman-like viscoelastic e ff ect, previously reported for dumbbell swimmers, whereinthe front-back asymmetry of the swimmer generates a non-Newtonian elastic propulsionforce that can either favor or hinder locomotion.The obvious next step in this investigation would be to address a similar question forbiological swimmers propelled by helical flagellar filaments. Swimming bacteria such as E. coli have a cell body whose width is approximately D H ≈ . µ m while the diametersof the helical flagella is approximately 2 R ≈ . µ m. The dimensionless ratio in that caseis therefore given by D ∗ = R / D H ≈ .
45. Since this is less than one, our results suggesttherefore that bacteria self-propeling in similar fluids would have their swimming speeddecreased by elastic stresses.
ACKNOWLEDGEMENTS
V. Angeles is grateful to Conacyt-Mexico for a graduate student scholarship and sup-port. This project has received funding from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (grant agree-15ent 682754 to EL). [1] E. Lauga and T. R. Powers. The hydrodynamics of swimming microorganisms.
Rep. Prog.Phys. , 72:096601–096637, 2009.[2] E. M. Purcell. The e ffi ciency of propulsion by rotating flagellum. Proc. Natl. Acad. Sci. USA ,94:11307–11311, 1997.[3] E. Lauga. Bacterial hydrodynamics.
Annu. Rev. Fluid Mech. , 48:105–130, 2016.[4] B. Rodenborn, C.-H. Chen, H. L Swinney, B. Lui, and H. P. Zhang. Propulsion of microor-ganisms by a helical flagellum.
Proc. Natl. Acad. Sci. USA , 110:E338–E347, 2013.[5] E. Lauga. Propulsion in a viscoelastic fluid.
Phys. Fluids , 19:083104, 2007.[6] B. Liu, T. R. Powers, and K. S. Breuer. Force-free swimming of a model helical flagellum inviscoelastic fluids.
Proc. Natl. Acad. Sci. USA , 108:19516–19520, 2011.[7] X. Shen and P. E. Arratia. Undulatory swimming in viscoelastic fluids.
Phys. Rev. Lett. ,106:208101, 2011.[8] D. A. Gagnon, X. N. Shen, and P. E. Arratia. Undulatory swimming in fluids with polymernetworks.
Europhys. Lett. , 104:14004, 2013.[9] J. Espinosa-Garcia, E. Lauga, and R. Zenit. Fluid elasticity increases the locomotion of flexibleswimmers.
Phys. Fluids , 25:031701, 2013.[10] M. Dasgupta, B. Liu, H. C. Fu, M. Berhanu, K. S. Breuer, T. R. Powers, and A. Kudrolli. Speedof a swimming sheet in Newtonian and viscoelastic fluids.
Phys. Rev. E , 87:013015, 2013.[11] S. E. Spagnolie, B. Liu, and T. R. Powers. Locomotion of helical bodies in viscoelastic fluids:Enhanced swimming at large helical amplitudes.
Phys. Rev. Lett. , 111:068101, 2013.[12] B. Thomases and R. D. Guy. Mechanisms of elastic enhancement and hindrance for finite-length undulatory swimmers in viscoelastic fluids.
Phys. Rev. Lett. , 113:098102, 2014.[13] F. A. God´ınez, L. Koens aand T. D. Montenegro-Johnson, R. Zenit, and E. Lauga. Complexfluids a ff ect low-reynolds number locomotion in a kinematic-dependent manner. Exp. Fluids ,56:97, 2015.[14] J. Teran, L. Fauci, and M. Shelley. Viscoelastic fluid response can increase the speed ande ffi ciency of a free swimmer. Phys. Rev. Lett. , 104:038101, 2010.
15] G. J. Elfring and G. Goyal. The e ff ect of gait on swimming in viscoelastic fluids. J. Non-Newtonian Fluid Mech. , 234:8 – 14, 2016.[16] H. C. Fu, T. R. Powers, and C. W. Wolgemuth. Theory of swimming filaments in viscoelasticmedia.
Phys. Rev. Lett. , 99:258101, 2007.[17] H. C. Fu, C. W. Wolgemuth, and T. R. Powers. Swimming speeds of filaments in nonlinearlyviscoelastic fluids.
Phys. Fluids , 21:033102, 2009.[18] C. Li, B. Qin, A. Gopinath, P. E. Arratia, B. Thomases, and R. D. Guy. Flagellar swimming inviscoelastic fluids: role of fluid elastic stress revealed by simulations based on experimentaldata.
J. Roy. Soc. Interface , 14:20170289, 2017.[19] C. Li, B. Thomases, and R. D. Guy. Orientation dependent elastic stress concentration at tipsof slender objects translating in viscoelastic fluids.
Phys. Rev. Fluids , 4:031301(R), 2019.[20] Z. Qu and K. S. Breuer. E ff ects of shear-thinning viscosity and viscoelastic stresses on flagel-lated bacteria motility. Phys. Rev. Fluids , 5:073103, 2020.[21] G. Li and A. M. Ardekani. Undulatory swimming in non-Newtonian fluids.
J. Fluid Mech. ,784:R4, 2015.[22] S. Gomez, F. Godinez, E. Lauga, and R. Zenit. Helical propulsion in shear-thinning fluids .
J.Fluid Mech. , 812:R3, 2016.[23] O. S. Pak, L. Zhu, L. Brandt, and E. Lauga. Micropropulsion and microrheology in complexfluids via symmetry breaking.
Phys. Fluids , 24:103102, 2012.[24] J. A. Puente-Vel´azquez, F. A. God´ınez, E. Lauga, and R. Zenit. Viscoelastic propulsion of arotating dumbbell.
Microfluid. Nanofluid. , 23:108, 2019.[25] F. A. God´ınez, O. Ch´avez, and R. Zenit. Note: Design of a novel rotating magnetic fielddevice.
Rev. Sci. Instrum. , 83:066109, 2012.[26] C. Brennen and H. Winet. Fluid mechanics of propulsion by cilia and flagella.
Ann. Rev. Fluid.Mech. , 9:339–398, 1977., 9:339–398, 1977.