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Physics of rheologically enhanced propulsion: Different strokes in generalized Stokes
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Figures

Image of FIG. 1.

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FIG. 1.

Swimming techniques in inertialess flows that are examined in this study. Conceptual swimmers may comprise sliding spheres that have simple kinematics, such as (a) the collinear motion of the Najafi-Golestanian swimmer and (b) paddling motion. These swimmers can provide insight into more complex biological systems. (c) Ciliates beat many surface cilia in a coordinated fashion. This is often modeled mathematically with envelope methods, either as a small perturbation to the cell morphology (dashed), or through a surface slip velocity. (d) Sperm, an archetypal “monoflagellate pusher,” propagate a bending wave down a single flagellum, shown here in a time-lapse manner.

Image of FIG. 2.

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FIG. 2.

(a) A schematic of the fluid domain containing a model human sperm ∂ , showing no-slip channel walls ∂ and open boundaries ∂ . The relationship between the lab frame, (, ) and the body frame, ( , ) is also shown, where the body frame origin is a fixed point on the swimmer. Femlets are distributed along the boundary ∂ , shown here as a sperm head and flagellum. (b) The full computational domain used in this study. The domain and swimmer are shown to scale.

Image of FIG. 3.

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FIG. 3.

The envelope function of the force exerted by the flagellum on the fluid. The function is approximately zero in the black regions, and increases as the colors lighten. (a) An example elongated femlet cut-off function, given by a two-dimensional elongated Gaussian, oriented by a coordinate transform to align locally with the swimmer's body. (b) A plot showing the smooth force distribution envelope generated by a sum of such cut-off functions when projected on a finite element mesh; femlet centroids are marked by dots.

Image of FIG. 4.

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FIG. 4.

A complete beat cycle of the Najafi-Golestanian swimmer showing the position of the outer spheres relative to the central sphere, the direction in which the propulsive sphere moves (solid arrow) relative to the payload, and the direction and magnitude of swimming (dashed arrow).

Image of FIG. 5.

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FIG. 5.

The effects of shear-thinning on the Najafi-Golestanian swimmer with the four-stage beat pattern given in Table I . (a) The progress during each effective stroke and (b) the regress during each recovery stroke as functions of the power-law index . Since the decrease in regress is greater for < 1, the overall effect of shear-thinning is an increase in net progress as decreases (c). In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.

Image of FIG. 6.

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FIG. 6.

Simulation results of the position of the Najafi-Golestanian swimmer over five beat cycles, demonstrating how decreasing the instantaneous swimming speed at all times in shear-thinning fluid can lead to an increase in overall progress, provided swimming speed is decreased more during the recovery stroke. The rheological parameters of the Carreau fluid are μ = 2, = 0.5, and Sh = 1.

Image of FIG. 7.

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FIG. 7.

The effective viscosity of Carreau fluid, normalized to μ = 1, surrounding the Najafi-Golestanian swimmer (Table I ) at (a) the start of effective stroke 1 and (b) the start of recovery stroke 2 for μ = 2, = 0.5, and Sh = 1. The fluid around the propulsive sphere is thinner than that around the payload.

Image of FIG. 8.

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FIG. 8.

The velocity relative to the Newtonian case of the Najafi-Golestanian swimmer when initiating an effective stroke (a) and a recovery stroke (b) as a function of the viscosity differential μ. The velocity has been calculated while varying the three rheological parameters of Carreau flow for = 0.5, μ ∈ [1, 2], Sh = 0.5 (dark gray), = 0.5, μ = 2, Sh ∈ [0, 0.5] (medium gray), and ∈ [0.5, 1], μ = 2, Sh = 0.5 (light gray). This figure demonstrates an apparent proportionality between the velocity and the viscosity differential, and that the viscosity differential is enhanced during the recovery stroke.

Image of FIG. 9.

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FIG. 9.

The effects of shear-thinning on the paddler (a) with the two-stage beat pattern given in Table II . During the portions of the beat represented by the dashed black lines, the swimmer does not progress and as such they are not considered here. The dashed arrow shows the swimming direction. (b) The progress during the effective stroke and (c) the regress during the recovery stroke as functions of the power-law index . The greater increase in regress results in a decrease in net progress with shear-thinning rheology, (d). In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.

Image of FIG. 10.

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FIG. 10.

Simulation results of the position of the paddler over five beat cycles, demonstrating how increasing the instantaneous swimming speed at all times in shear-thinning fluid can lead to an decrease in net progress, provided swimming speed is decreased more during the recovery stroke. The observed effect is exactly opposite to that of the Najafi-Golestanian swimmer, summarized in Figure 6 . The rheological parameters of the Carreau fluid are μ = 2, = 0.5, and Sh = 1.

Image of FIG. 11.

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FIG. 11.

The effective viscosity of Carreau fluid, normalized to μ = 1, surrounding the paddler (Table II ) at (a) the start of the effective stroke and (b) the start of the recovery stroke for μ = 2, = 0.5, and Sh = 1. The fluid around the propulsive sphere is thinner than that around the payload.

Image of FIG. 12.

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FIG. 12.

The velocity relative to the Newtonian case of the paddler at the commencement of (a) an effective stroke and (b) a recovery stroke as functions of the viscosity differential μ. The velocity has been calculated while varying the three rheological parameters of Carreau flow for = 0.5, μ ∈ [1, 2], Sh = 0.5 (dark gray), = 0.5, μ = 2, Sh ∈ [0, 0.5] (medium gray), and ∈ [0.5, 1], μ = 2, Sh = 0.5 (light gray). During the recovery stroke (b), spheres are far apart and there is approximate proportionality between the increase in velocity and the viscosity differential. During the effective stroke (a), however, interactions between the viscosity fields of the spheres reduce the effect of the viscosity differential. For low values of Sh (medium gray), more shear is required to thin the flow. Thus, proportionality between velocity increase and viscosity differential is maintained with the same constant for effective and recovery strokes due to decreased viscosity field interactions.

Image of FIG. 13.

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FIG. 13.

A schematic of a ciliated surface. Cilia beat with an effective-recovery stroke pattern, marked with E and R, respectively, pushing fluid locally in the direction shown. The cilia are activated in a coordinated, metachronal fashion. The envelope of this motion is given by the dashed green line.

Image of FIG. 14.

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FIG. 14.

(a) A schematic of a two-dimensional treadmilling squirmer, along with (b) a micrograph of a colony, showing surface cilia that beat in a coordinated fashion to propel the colony forwards. This cell also shows a number of characteristic “daughter” colonies within it. Image taken by Professor Raymond E. Goldstein, University of Cambridge; reprinted with permission.

Image of FIG. 15.

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FIG. 15.

The velocity of the treadmilling squirmer with slip velocity given by Eq. (12) as a function of (a) the viscosity ratio μ with = 0.5 and Sh = 1, (b) the shear index Sh with = 0.5 and μ = 2, and (c) the power-law index with μ = 2 and Sh = 1. In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.

Image of FIG. 16.

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FIG. 16.

The effective viscosity μ of Carreau fluid, normalized to μ = 1, surrounding the treadmilling squirmer for μ = 2, = 0.5, and Sh = 0.5. These parameter values are the extremal values used for the data in Figures 17 and 18 . Away from the swimmer surface, contours of equi-viscosity are approximately circular. On the surface, fluid is relatively thicker surrounding the propulsive portions of the swimmer. The squirmer is aligned to the positive -axis, as in Figure 14(a) , and the direction of travel is indicated by the dashed arrow.

Image of FIG. 17.

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FIG. 17.

The effective viscosity of the fluid envelope surrounding the treadmilling squirmer. (a) Changes in the viscosity field as a function of the radial coordinate for different values of the power-law index . The swimmer surface is given by = 0.5. (b) For fixed values of , the effective viscosity exhibits a near linear dependence upon the power-law index .

Image of FIG. 18.

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FIG. 18.

The velocity relative to the Newtonian case of the treadmilling squirmer as a function of (a) the effective viscosity on the contour = 0.52 and (b) the rate of decay α of the velocity from the surface of the squirmer relative to the Newtonian case α. The velocity has been calculated while varying the three rheological parameters of Carreau flow for = 0.5, μ ∈ [1, 2], Sh = 0.5 (dark gray), = 0.5, μ = 2, Sh ∈ [0, 0.5] (medium gray), and ∈ [0.5, 1], μ = 2, Sh = 0.5 (light gray). This figure demonstrates a striking proportionality between the velocity and the decay rate of the fluid.

Image of FIG. 19.

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FIG. 19.

Swimming parameters for the trajectory (dark gray) of a swimmer moving from right to left over one beat cycle of period . The instantaneous velocity is the derivative of arclength along the path with respect to time.

Image of FIG. 20.

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FIG. 20.

(a) The flagellar waveform generated by shear angle (15) and (b) a micrograph of a human sperm in medium containing 1% methylcellulose, a fluid with comparable viscosity to that of cervical mucus.

Image of FIG. 21.

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FIG. 21.

Trajectories of the body frame origin , given by the head-flagellum junction, of a two-dimensional sperm-like swimmer in Carreau fluid for different values of the viscosity ratio μ, showing an increase in progress and a decrease in ALH as μ increases.

Image of FIG. 22.

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FIG. 22.

(a) Trajectories of the cells with head morphologies given in Table III , swimming in Stokes flow with = 0.5, μ = 4, and Sh = 1. For = 0.5 and Sh = 1, the effect of varying the viscosity ratio μ on (b) the swimmers' progress, (c) the amplitude of the swimmers' lateral head displacement, (d) the path length of the swimmers' trajectories, and (e) the swimmers' path straightness.

Image of FIG. 23.

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FIG. 23.

The impact of varying Sh = λω on the effective viscosity μ of Carreau fluid surrounding a two-dimensional sperm-like swimmer at (a) Sh = 0.2, (b) Sh = 0.8, (c) Sh = 1.5, and (d) Sh = 3 with μ = 2 and = 0.5. In these figures, the area of the cell head is 0.002π, the wavenumber = 2.5 and the maximum shear angle = 0.45π.

Image of FIG. 24.

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FIG. 24.

The magnitude of the force that the flagellum exerts upon the fluid at time = 0 for Newtonian (dark gray) and Carreau (light gray) fluids with μ = 2, = 0.5, and Sh = 0.8, close to the optimal value of Sh found by Montenegro-Johnson

Image of FIG. 25.

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FIG. 25.

The magnitude and direction of swimming of a sperm oriented in the negative direction with wavenumber = 2.5 and maximum shear angle = 0.45π at times = 0, 0.1, 0.2, 0.3, 0.4, for varying viscosity ratio. These times span half a complete beat cycle. This figure demonstrates that shear-thinning results in straighter swimming and increased instantaneous velocity.

Image of FIG. 26.

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FIG. 26.

Reciprocal sliding sphere swimmers that cannot progress through inertialess Newtonian fluid, but may progress through inertialess Carreau fluid. These swimmers are pusher and puller versions of (a) the Najafi-Golestanian swimmer and (b) the paddler, showing the effective and recovery strokes with an indication of the velocity of the propulsive sphere (solid arrow) and the magnitude and direction of progress over each stroke (dashed arrow).

Image of FIG. 27.

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FIG. 27.

(a) The speed of the flow arising from a regularized force of the form (A1) , with ε = 0.1, situated at the origin in a no-slip circular cavity of radius 10 as calculated by the method of femlets and (b) the absolute difference between the flow speed as calculated by the method of femlets and the method of regularized stokeslets.

Image of FIG. 28.

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FIG. 28.

Relative error in the calculated speed of the flow induced by the treadmilling squirmer in Newtonian fluid, compared with the analytical solution of Blake for an infinite fluid. The maximum relative error close to the squirmer is 1.2%, and is approximately 0.2% throughout the majority of the domain.

Tables

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Table I.

The body frame positions of the three spheres of the Najafi-Golestanian swimmer we will model, for = 0.5, = 0.25, over each portion of its beat cycle.

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Table II.

The body frame positions of the three spheres for the paddling swimmer over the effective and recovery stroke, where in our model = 0.5, = 0.75, and = 0.25.

Generic image for table

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Table III.

Elliptical head morphologies of constant eccentricity, but different area scaled with flagellum length, corresponding to the data in Figure 22 . These morphologies, from top to bottom correspond with dark to light plots.

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Table IV.

The body frame positions of the three spheres of the speed-asymmetric collinear pusher over its effective stroke, which lasts for 3/4 of the beat period, and the recovery stroke, which lasts for 1/4 of the beat period.

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Table V.

The velocity of the treadmilling squirmer as calculated with the method of femlets as a function of the regularization parameters σ, σ, showing that the error associated in approximating a moving Dirichlet boundary by femlets decreases as approximately .

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/content/aip/journal/pof2/25/8/10.1063/1.4818640
2013-08-21
2014-04-18

Abstract

Shear-thinning is an important rheological property of many biological fluids, such as mucus, whereby the apparent viscosity of the fluid decreases with shear. Certain microscopic swimmers have been shown to progress more rapidly through shear-thinning fluids, but is this behavior generic to all microscopic swimmers, and what are the physics through which shear-thinning rheology affects a swimmer's propulsion? We examine swimmers employing prescribed stroke kinematics in two-dimensional, inertialess Carreau fluid: shear-thinning “generalized Stokes” flow. Swimmers are modeled, using the method of femlets, by a set of immersed, regularized forces. The equations governing the fluid dynamics are then discretized over a body-fitted mesh and solved with the finite element method. We analyze the locomotion of three distinct classes of microswimmer: (1) conceptual swimmers comprising sliding spheres employing both one- and two-dimensional strokes, (2) slip-velocity envelope models of ciliates commonly referred to as “squirmers,” and (3) monoflagellate pushers, such as sperm. We find that morphologically identical swimmers with different strokes may swim either faster or slower in shear-thinning fluids than in Newtonian fluids. We explain this kinematic sensitivity by considering differences in the viscosity of the fluid surrounding propulsive and payload elements of the swimmer, and using this insight suggest two reciprocal sliding sphere swimmers which violate Purcell's Scallop theorem in shear-thinning fluids. We also show that an increased flow decay rate arising from shear-thinning rheology is associated with a reduction in the swimming speed of slip-velocity squirmers. For sperm-like swimmers, a gradient of thick to thin fluid along the flagellum alters the force it exerts upon the fluid, flattening trajectories and increasing instantaneous swimming speed.

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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Physics of rheologically enhanced propulsion: Different strokes in generalized Stokes
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/8/10.1063/1.4818640
10.1063/1.4818640
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