^{1,a)}, Eric Lauga

^{2}and Luca Brandt

^{1}

### Abstract

We use numerical simulations to address locomotion at zero Reynolds number in viscoelastic (Giesekus) fluids. The swimmers are assumed to be spherical, to self-propel using tangential surface deformation, and the computations are implemented using a finite element method. The emphasis of the study is on the change of the swimming kinematics, energetics, and flow disturbance from Newtonian to viscoelastic, and on the distinction between pusher and puller swimmers. In all cases, the viscoelastic swimming speed is below the Newtonian one, with a minimum obtained for intermediate values of the Weissenberg number, *We*. An analysis of the flow field places the origin of this swimming degradation in non-Newtonian elongational stresses. The power required for swimming is also systematically below the Newtonian power, and always a decreasing function of *We*. A detail energetic balance of the swimming problem points at the polymeric part of the stress as the primary *We*-decreasing energetic contribution, while the contributions of the work done by the swimmer from the solvent remain essentially *We*-independent. In addition, we observe negative values of the polymeric power density in some flow regions, indicating positive elastic work by the polymers on the fluid. The hydrodynamic efficiency, defined as the ratio of the useful to total rate of work, is always above the Newtonian case, with a maximum relative value obtained at intermediate Weissenberg numbers. Finally, the presence of polymeric stresses leads to an increase of the rate of decay of the flow velocity in the fluid, and a decrease of the magnitude of the stresslet governing the magnitude of the effective bulk stress in the fluid.

Funding by VR (the Swedish Research Council) to L.B., by the Linné Flow Centre at KTH, and the US National Science Foundation (Grant No. CBET-0746285 to E.L.) is gratefully acknowledged. Computer time provided by SNIC (Swedish National Infrastructure for Computing) is also acknowledged.

I. INTRODUCTION

II. MATHEMATICAL MODEL

A. Squirmer model

B. Polymeric dynamics

III. NUMERICAL METHOD

IV. SWIMMING SPEED IN VISCOELASTIC FLUIDS

V. SWIMMING POWER IN VISCOELASTIC FLUIDS

VI. SWIMMING EFFICIENCY IN VISCOELASTIC FLUIDS

VII. VELOCITY DECAY IN VISCOELASTIC FLUIDS

VIII. STRESSLET IN VISCOELASTIC FLUIDS

IX. CONCLUSION

### Key Topics

- Polymers
- 68.0
- Viscoelasticity
- 45.0
- Tensor methods
- 16.0
- Polymer flows
- 15.0
- Elasticity
- 13.0

## Figures

Vector plot of flow fields generated by pusher and puller squirmers in a Newtonian fluid. Left: co-moving frame. Right: laboratory frame. On each panel, the data for the pusher are displayed on the left, and those for the puller on the right. The large arrow (blue) indicates the swimming direction and the background color scheme indicates the velocity magnitudes.

Vector plot of flow fields generated by pusher and puller squirmers in a Newtonian fluid. Left: co-moving frame. Right: laboratory frame. On each panel, the data for the pusher are displayed on the left, and those for the puller on the right. The large arrow (blue) indicates the swimming direction and the background color scheme indicates the velocity magnitudes.

Nondimensional swimming speed, *U*, of the swimmer as a function of the Weissenberg number, *We*, for pusher α = −5 (squares, green), puller α = 5 (triangles, blue) and neutral squirmer α = 0 (circles, red). The swimming speed is scaled by its corresponding value in the Newtonian fluid, *U* _{New}.

Nondimensional swimming speed, *U*, of the swimmer as a function of the Weissenberg number, *We*, for pusher α = −5 (squares, green), puller α = 5 (triangles, blue) and neutral squirmer α = 0 (circles, red). The swimming speed is scaled by its corresponding value in the Newtonian fluid, *U* _{New}.

Distribution of the component of the polymeric stress for the neutral squirmer and three values of the Weissenberg number, *We* = 0.1, 2, and 6. (Left: *We* = 0.1 and 2. Right: *We* = 6 and 2.)

Distribution of the component of the polymeric stress for the neutral squirmer and three values of the Weissenberg number, *We* = 0.1, 2, and 6. (Left: *We* = 0.1 and 2. Right: *We* = 6 and 2.)

Distribution of the component of the polymeric stress for the pusher (left) and puller (right) swimming in the viscoelastic fluid. For both gaits, *We* = 0.1 and *We* = 2 are chosen for comparison. The inset plots show regions with high value of .

Distribution of the component of the polymeric stress for the pusher (left) and puller (right) swimming in the viscoelastic fluid. For both gaits, *We* = 0.1 and *We* = 2 are chosen for comparison. The inset plots show regions with high value of .

Distribution of the component of the polymeric stress along the symmetry axis. Shaded circles and arrows indicate the squirmer and its swimming direction. Data for the pusher are shown in the top whereas the neutral squirmer and the puller are reported at the bottom. The solid and short-dashed lines correspond to *We* = 0.1, the long-dashed and dotted-dashed lines to *We* = 2.

Distribution of the component of the polymeric stress along the symmetry axis. Shaded circles and arrows indicate the squirmer and its swimming direction. Data for the pusher are shown in the top whereas the neutral squirmer and the puller are reported at the bottom. The solid and short-dashed lines correspond to *We* = 0.1, the long-dashed and dotted-dashed lines to *We* = 2.

Distribution of the and components of the polymeric stress for pullers in the viscoelastic fluid with *We* = 0.5. Both the and components lead to a net polymeric (elastic) force indicated by the gray arrows, hindering locomotion. The location of this elastic forces is right above the vortex ring, implying that they are generated by the elongational flow.

Distribution of the and components of the polymeric stress for pullers in the viscoelastic fluid with *We* = 0.5. Both the and components lead to a net polymeric (elastic) force indicated by the gray arrows, hindering locomotion. The location of this elastic forces is right above the vortex ring, implying that they are generated by the elongational flow.

Swimming power, , for pusher (squares, green), puller (triangles, blue) and neutral (circles, red) squirmers, nondimensionalized by the swimming power in the Newtonian case, . The power consumption of swimming in the Newtonian fluid are equal to (neutral squirmer) and (both pusher and puller).

Swimming power, , for pusher (squares, green), puller (triangles, blue) and neutral (circles, red) squirmers, nondimensionalized by the swimming power in the Newtonian case, . The power consumption of swimming in the Newtonian fluid are equal to (neutral squirmer) and (both pusher and puller).

Contributions to the total power expended by swimming, , versus Weissenberg number *We*, β = 0.5, for neutral squirmer (left) and puller (right). The three contributions are defined as: , , and .

Contributions to the total power expended by swimming, , versus Weissenberg number *We*, β = 0.5, for neutral squirmer (left) and puller (right). The three contributions are defined as: , , and .

Polymeric power density, *D* _{ P }, for pushers in fluids with *We* = 0.1 (left) and *We* = 2 (right). Dotted elliptical circles mark areas with negative values of *D* _{ P }. The values of *D* _{ P } along the swimmer surface are displayed in Fig. 10.

Polymeric power density, *D* _{ P }, for pushers in fluids with *We* = 0.1 (left) and *We* = 2 (right). Dotted elliptical circles mark areas with negative values of *D* _{ P }. The values of *D* _{ P } along the swimmer surface are displayed in Fig. 10.

Spatial distribution of polymeric power density, *D* _{ P }, along the surface of pushers in viscoelastic fluids with *We* = 0.1 (solid line, red) and 2 (dashed line, blue); θ is defined in Sec. II, as the angle between the swimming direction and the position vector of each point.

Spatial distribution of polymeric power density, *D* _{ P }, along the surface of pushers in viscoelastic fluids with *We* = 0.1 (solid line, red) and 2 (dashed line, blue); θ is defined in Sec. II, as the angle between the swimming direction and the position vector of each point.

Swimming speed , scaled to ensure locomotion at constant power, versus Weissenberg number, *We*, for β = 0.5. Nondimensional form is plotted for the three swimming gaits: neutral (circles, red), puller (triangles, blue), and pusher (squares, green).

Swimming speed , scaled to ensure locomotion at constant power, versus Weissenberg number, *We*, for β = 0.5. Nondimensional form is plotted for the three swimming gaits: neutral (circles, red), puller (triangles, blue), and pusher (squares, green).

Swimming efficiency, η, normalized by the corresponding value for Newtonian swimming, η_{New}, versus Weissenberg number, *We*. The efficiencies of swimming in the Newtonian fluid are equal to η_{New} = 0.5 (neutral squirmer) and η_{New} = 0.037 (both pusher and puller).

Swimming efficiency, η, normalized by the corresponding value for Newtonian swimming, η_{New}, versus Weissenberg number, *We*. The efficiencies of swimming in the Newtonian fluid are equal to η_{New} = 0.5 (neutral squirmer) and η_{New} = 0.037 (both pusher and puller).

Power law exponent, γ, for spatial decay of the axial velocity along the axis of the domain (*r* = 0) (|*u*| ∼ *r* ^{−γ}, see text), as a function of the Weissenberg number, *We*. The inset plot highlights the difference between pusher and puller.

Power law exponent, γ, for spatial decay of the axial velocity along the axis of the domain (*r* = 0) (|*u*| ∼ *r* ^{−γ}, see text), as a function of the Weissenberg number, *We*. The inset plot highlights the difference between pusher and puller.

Stresslets induced by swimmers in viscoelastic fluids: *rr* and *zz* components of the tensor **S**, Eq. (21), for different swimming gaits. Left: pusher (α = −5) and puller (α = 5). Right: neutral squirmer (α = 0).

Stresslets induced by swimmers in viscoelastic fluids: *rr* and *zz* components of the tensor **S**, Eq. (21), for different swimming gaits. Left: pusher (α = −5) and puller (α = 5). Right: neutral squirmer (α = 0).

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