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Abstract
Shearthinning 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 shearthinning fluids, but is this behavior generic to all microscopic swimmers, and what are the physics through which shearthinning rheology affects a swimmer's propulsion? We examine swimmers employing prescribed stroke kinematics in twodimensional, inertialess Carreau fluid: shearthinning “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 bodyfitted 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 twodimensional strokes, (2) slipvelocity 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 shearthinning 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 shearthinning fluids. We also show that an increased flow decay rate arising from shearthinning rheology is associated with a reduction in the swimming speed of slipvelocity squirmers. For spermlike 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.
T.D.M.J. is funded by Engineering and Physical Sciences Research Council (U.K.) (EPSRC(GB)) First Grant No. EP/K007637/1 to D.J.S. A portion of this work was completed while T.D.M.J. was funded by an EPSRC Doctoral Training Studentship and D.J.S. by a Birmingham Science City Fellowship. Micrograph 11(b) was taken in collaboration with Dr. Hermes Gadêlha and Dr. Jackson KirkmanBrown, and micrograph 8(b) was taken by Professor Raymond E. Goldstein, University of Cambridge. The authors would like to thank Professor John Blake for discussions and mentorship. We also acknowledge the anonymous referees for their valuable suggestions.
I. INTRODUCTION
II. MATHEMATICAL MODELING
A. Fluid mechanics of microscopic swimming
B. The method of femlets
III. RESULTS AND ANALYSIS
A. Sliding sphere swimmers
B. A threesphere “paddler”
C. Slip velocity squirmers
D. Monoflagellate pushers
IV. DISCUSSION
V. CONCLUSIONS
Key Topics
 Shear thinning
 55.0
 Viscosity
 50.0
 Kinematics
 22.0
 Shear rate dependent viscosity
 21.0
 Stokes flows
 11.0
Figures
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 NajafiGolestanian swimmer and (b) paddling motion. These swimmers can provide insight into more complex biological systems. 9 (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 timelapse manner.
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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 NajafiGolestanian swimmer and (b) paddling motion. These swimmers can provide insight into more complex biological systems. 9 (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 timelapse manner.
(a) A schematic of the fluid domain D containing a model human sperm ∂D swim, showing noslip channel walls ∂D dir and open boundaries ∂D neu. The relationship between the lab frame, (x, y) and the body frame, (x ′, y ′) is also shown, where the body frame origin x 0 is a fixed point on the swimmer. Femlets are distributed along the boundary ∂D swim, 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.
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(a) A schematic of the fluid domain D containing a model human sperm ∂D swim, showing noslip channel walls ∂D dir and open boundaries ∂D neu. The relationship between the lab frame, (x, y) and the body frame, (x ′, y ′) is also shown, where the body frame origin x 0 is a fixed point on the swimmer. Femlets are distributed along the boundary ∂D swim, 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.
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 cutoff function, given by a twodimensional 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 cutoff functions when projected on a finite element mesh; femlet centroids are marked by dots.
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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 cutoff function, given by a twodimensional 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 cutoff functions when projected on a finite element mesh; femlet centroids are marked by dots.
A complete beat cycle of the NajafiGolestanian 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).
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A complete beat cycle of the NajafiGolestanian 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).
The effects of shearthinning on the NajafiGolestanian swimmer with the fourstage 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 powerlaw index n. Since the decrease in regress is greater for n < 1, the overall effect of shearthinning is an increase in net progress as n decreases (c). In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.
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The effects of shearthinning on the NajafiGolestanian swimmer with the fourstage 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 powerlaw index n. Since the decrease in regress is greater for n < 1, the overall effect of shearthinning is an increase in net progress as n decreases (c). In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.
Simulation results of the position of the NajafiGolestanian swimmer over five beat cycles, demonstrating how decreasing the instantaneous swimming speed at all times in shearthinning 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 μ0/μ∞ = 2, n = 0.5, and Sh = 1.
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Simulation results of the position of the NajafiGolestanian swimmer over five beat cycles, demonstrating how decreasing the instantaneous swimming speed at all times in shearthinning 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 μ0/μ∞ = 2, n = 0.5, and Sh = 1.
The effective viscosity of Carreau fluid, normalized to μ0 = 1, surrounding the NajafiGolestanian swimmer (Table I ) at (a) the start of effective stroke 1 and (b) the start of recovery stroke 2 for μ0/μ∞ = 2, n = 0.5, and Sh = 1. The fluid around the propulsive sphere is thinner than that around the payload.
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The effective viscosity of Carreau fluid, normalized to μ0 = 1, surrounding the NajafiGolestanian swimmer (Table I ) at (a) the start of effective stroke 1 and (b) the start of recovery stroke 2 for μ0/μ∞ = 2, n = 0.5, and Sh = 1. The fluid around the propulsive sphere is thinner than that around the payload.
The velocity relative to the Newtonian case of the NajafiGolestanian swimmer when initiating an effective stroke (a) and a recovery stroke (b) as a function of the viscosity differential μdiff. The velocity has been calculated while varying the three rheological parameters of Carreau flow for n = 0.5, μ0/μ∞ ∈ [1, 2], Sh = 0.5 (dark gray), n = 0.5, μ0/μ∞ = 2, Sh ∈ [0, 0.5] (medium gray), and n ∈ [0.5, 1], μ0/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.
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The velocity relative to the Newtonian case of the NajafiGolestanian swimmer when initiating an effective stroke (a) and a recovery stroke (b) as a function of the viscosity differential μdiff. The velocity has been calculated while varying the three rheological parameters of Carreau flow for n = 0.5, μ0/μ∞ ∈ [1, 2], Sh = 0.5 (dark gray), n = 0.5, μ0/μ∞ = 2, Sh ∈ [0, 0.5] (medium gray), and n ∈ [0.5, 1], μ0/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.
The effects of shearthinning on the paddler (a) with the twostage 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 powerlaw index n. The greater increase in regress results in a decrease in net progress with shearthinning rheology, (d). In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.
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The effects of shearthinning on the paddler (a) with the twostage 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 powerlaw index n. The greater increase in regress results in a decrease in net progress with shearthinning rheology, (d). In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.
Simulation results of the position of the paddler over five beat cycles, demonstrating how increasing the instantaneous swimming speed at all times in shearthinning 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 NajafiGolestanian swimmer, summarized in Figure 6 . The rheological parameters of the Carreau fluid are μ0/μ∞ = 2, n = 0.5, and Sh = 1.
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Simulation results of the position of the paddler over five beat cycles, demonstrating how increasing the instantaneous swimming speed at all times in shearthinning 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 NajafiGolestanian swimmer, summarized in Figure 6 . The rheological parameters of the Carreau fluid are μ0/μ∞ = 2, n = 0.5, and Sh = 1.
The effective viscosity of Carreau fluid, normalized to μ0 = 1, surrounding the paddler (Table II ) at (a) the start of the effective stroke and (b) the start of the recovery stroke for μ0/μ∞ = 2, n = 0.5, and Sh = 1. The fluid around the propulsive sphere is thinner than that around the payload.
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The effective viscosity of Carreau fluid, normalized to μ0 = 1, surrounding the paddler (Table II ) at (a) the start of the effective stroke and (b) the start of the recovery stroke for μ0/μ∞ = 2, n = 0.5, and Sh = 1. The fluid around the propulsive sphere is thinner than that around the payload.
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 μdiff. The velocity has been calculated while varying the three rheological parameters of Carreau flow for n = 0.5, μ0/μ∞ ∈ [1, 2], Sh = 0.5 (dark gray), n = 0.5, μ0/μ∞ = 2, Sh ∈ [0, 0.5] (medium gray), and n ∈ [0.5, 1], μ0/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.
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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 μdiff. The velocity has been calculated while varying the three rheological parameters of Carreau flow for n = 0.5, μ0/μ∞ ∈ [1, 2], Sh = 0.5 (dark gray), n = 0.5, μ0/μ∞ = 2, Sh ∈ [0, 0.5] (medium gray), and n ∈ [0.5, 1], μ0/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.
A schematic of a ciliated surface. Cilia beat with an effectiverecovery 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.
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A schematic of a ciliated surface. Cilia beat with an effectiverecovery 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.
(a) A schematic of a twodimensional treadmilling squirmer, along with (b) a micrograph of a Volvox carteri 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.
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(a) A schematic of a twodimensional treadmilling squirmer, along with (b) a micrograph of a Volvox carteri 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.
The velocity of the treadmilling squirmer with slip velocity given by Eq. (12) as a function of (a) the viscosity ratio μ0/μ∞ with n = 0.5 and Sh = 1, (b) the shear index Sh with n = 0.5 and μ0/μ∞ = 2, and (c) the powerlaw index n with μ0/μ∞ = 2 and Sh = 1. In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.
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The velocity of the treadmilling squirmer with slip velocity given by Eq. (12) as a function of (a) the viscosity ratio μ0/μ∞ with n = 0.5 and Sh = 1, (b) the shear index Sh with n = 0.5 and μ0/μ∞ = 2, and (c) the powerlaw index n with μ0/μ∞ = 2 and Sh = 1. In each panel, the case corresponding to Newtonian fluid is marked in lighter gray.
The effective viscosity μeff of Carreau fluid, normalized to μ0 = 1, surrounding the treadmilling squirmer for μ0/μ∞ = 2, n = 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 equiviscosity are approximately circular. On the surface, fluid is relatively thicker surrounding the propulsive portions of the swimmer. The squirmer is aligned to the positive xaxis, as in Figure 14(a) , and the direction of travel is indicated by the dashed arrow.
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The effective viscosity μeff of Carreau fluid, normalized to μ0 = 1, surrounding the treadmilling squirmer for μ0/μ∞ = 2, n = 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 equiviscosity are approximately circular. On the surface, fluid is relatively thicker surrounding the propulsive portions of the swimmer. The squirmer is aligned to the positive xaxis, as in Figure 14(a) , and the direction of travel is indicated by the dashed arrow.
The effective viscosity of the fluid envelope surrounding the treadmilling squirmer. (a) Changes in the viscosity field as a function of the radial coordinate r for different values of the powerlaw index n. The swimmer surface is given by r = 0.5. (b) For fixed values of r, the effective viscosity exhibits a near linear dependence upon the powerlaw index n.
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The effective viscosity of the fluid envelope surrounding the treadmilling squirmer. (a) Changes in the viscosity field as a function of the radial coordinate r for different values of the powerlaw index n. The swimmer surface is given by r = 0.5. (b) For fixed values of r, the effective viscosity exhibits a near linear dependence upon the powerlaw index n.
The velocity relative to the Newtonian case of the treadmilling squirmer as a function of (a) the effective viscosity on the contour r = 0.52 and (b) the rate of decay α of the velocity from the surface of the squirmer relative to the Newtonian case αnewt. The velocity has been calculated while varying the three rheological parameters of Carreau flow for n = 0.5, μ0/μ∞ ∈ [1, 2], Sh = 0.5 (dark gray), n = 0.5, μ0/μ∞ = 2, Sh ∈ [0, 0.5] (medium gray), and n ∈ [0.5, 1], μ0/μ∞ = 2, Sh = 0.5 (light gray). This figure demonstrates a striking proportionality between the velocity and the decay rate of the fluid.
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The velocity relative to the Newtonian case of the treadmilling squirmer as a function of (a) the effective viscosity on the contour r = 0.52 and (b) the rate of decay α of the velocity from the surface of the squirmer relative to the Newtonian case αnewt. The velocity has been calculated while varying the three rheological parameters of Carreau flow for n = 0.5, μ0/μ∞ ∈ [1, 2], Sh = 0.5 (dark gray), n = 0.5, μ0/μ∞ = 2, Sh ∈ [0, 0.5] (medium gray), and n ∈ [0.5, 1], μ0/μ∞ = 2, Sh = 0.5 (light gray). This figure demonstrates a striking proportionality between the velocity and the decay rate of the fluid.
Swimming parameters for the trajectory (dark gray) of a swimmer moving from right to left over one beat cycle of period T. The instantaneous velocity is the derivative of arclength s along the path with respect to time.
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Swimming parameters for the trajectory (dark gray) of a swimmer moving from right to left over one beat cycle of period T. The instantaneous velocity is the derivative of arclength s along the path with respect to time.
(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.
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(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.
Trajectories of the body frame origin x 0, given by the headflagellum junction, of a twodimensional spermlike swimmer in Carreau fluid for different values of the viscosity ratio μ0/μ∞, showing an increase in progress and a decrease in ALH as μ0/μ∞ increases.
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Trajectories of the body frame origin x 0, given by the headflagellum junction, of a twodimensional spermlike swimmer in Carreau fluid for different values of the viscosity ratio μ0/μ∞, showing an increase in progress and a decrease in ALH as μ0/μ∞ increases.
(a) Trajectories of the cells with head morphologies given in Table III , swimming in Stokes flow with n = 0.5, μ0/μ∞ = 4, and Sh = 1. For n = 0.5 and Sh = 1, the effect of varying the viscosity ratio μ0/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.
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(a) Trajectories of the cells with head morphologies given in Table III , swimming in Stokes flow with n = 0.5, μ0/μ∞ = 4, and Sh = 1. For n = 0.5 and Sh = 1, the effect of varying the viscosity ratio μ0/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.
The impact of varying Sh = λω on the effective viscosity μeff of Carreau fluid surrounding a twodimensional spermlike swimmer at (a) Sh = 0.2, (b) Sh = 0.8, (c) Sh = 1.5, and (d) Sh = 3 with μ0/μ∞ = 2 and n = 0.5. In these figures, the area of the cell head is 0.002π, the wavenumber k = 2.5 and the maximum shear angle A = 0.45π.
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The impact of varying Sh = λω on the effective viscosity μeff of Carreau fluid surrounding a twodimensional spermlike swimmer at (a) Sh = 0.2, (b) Sh = 0.8, (c) Sh = 1.5, and (d) Sh = 3 with μ0/μ∞ = 2 and n = 0.5. In these figures, the area of the cell head is 0.002π, the wavenumber k = 2.5 and the maximum shear angle A = 0.45π.
The magnitude of the force that the flagellum exerts upon the fluid at time t = 0 for Newtonian (dark gray) and Carreau (light gray) fluids with μ0/μ∞ = 2, n = 0.5, and Sh = 0.8, close to the optimal value of Sh found by MontenegroJohnson et al. 28
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The magnitude of the force that the flagellum exerts upon the fluid at time t = 0 for Newtonian (dark gray) and Carreau (light gray) fluids with μ0/μ∞ = 2, n = 0.5, and Sh = 0.8, close to the optimal value of Sh found by MontenegroJohnson et al. 28
The magnitude and direction of swimming of a sperm oriented in the negative x direction with wavenumber k = 2.5 and maximum shear angle A = 0.45π at times t = 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 shearthinning results in straighter swimming and increased instantaneous velocity.
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The magnitude and direction of swimming of a sperm oriented in the negative x direction with wavenumber k = 2.5 and maximum shear angle A = 0.45π at times t = 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 shearthinning results in straighter swimming and increased instantaneous velocity.
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 NajafiGolestanian 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).
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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 NajafiGolestanian 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).
(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 noslip 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.
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(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 noslip 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.
Relative error in the calculated speed of the flow induced by the treadmilling squirmer in Newtonian fluid, compared with the analytical solution of Blake 52 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.
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Relative error in the calculated speed of the flow induced by the treadmilling squirmer in Newtonian fluid, compared with the analytical solution of Blake 52 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
The body frame positions of the three spheres of the NajafiGolestanian swimmer we will model, for d = 0.5, a = 0.25, over each portion of its beat cycle.
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The body frame positions of the three spheres of the NajafiGolestanian swimmer we will model, for d = 0.5, a = 0.25, over each portion of its beat cycle.
The body frame positions of the three spheres for the paddling swimmer over the effective and recovery stroke, where in our model d = 0.5, y rec = 0.75, and y eff = 0.25.
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The body frame positions of the three spheres for the paddling swimmer over the effective and recovery stroke, where in our model d = 0.5, y rec = 0.75, and y eff = 0.25.
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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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.
The body frame positions of the three spheres of the speedasymmetric 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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The body frame positions of the three spheres of the speedasymmetric 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.
The velocity of the treadmilling squirmer as calculated with the method of femlets as a function of the regularization parameters σ x , σ y , showing that the error associated in approximating a moving Dirichlet boundary by femlets decreases as approximately .
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The velocity of the treadmilling squirmer as calculated with the method of femlets as a function of the regularization parameters σ x , σ y , showing that the error associated in approximating a moving Dirichlet boundary by femlets decreases as approximately .
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Abstract
Shearthinning 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 shearthinning fluids, but is this behavior generic to all microscopic swimmers, and what are the physics through which shearthinning rheology affects a swimmer's propulsion? We examine swimmers employing prescribed stroke kinematics in twodimensional, inertialess Carreau fluid: shearthinning “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 bodyfitted 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 twodimensional strokes, (2) slipvelocity 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 shearthinning 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 shearthinning fluids. We also show that an increased flow decay rate arising from shearthinning rheology is associated with a reduction in the swimming speed of slipvelocity squirmers. For spermlike 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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