Ke
Qin
^{a},
Zhiwei
Peng
^{b},
Ye
Chen
^{a},
Herve
Nganguia
^{c},
Lailai
Zhu
^{d} and
On Shun
Pak
*^{a}
^{a}Department of Mechanical Engineering, Santa Clara University, Santa Clara, California, 95053, USA. E-mail: opak@scu.edu
^{b}Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
^{c}Department of Mathematical and Computer Sciences, Indiana University of Pennsylvania, Indiana, Pennsylvania 15705, USA
^{d}Department of Mechanical Engineering, National University of Singapore, 117575, Singapore. E-mail: lailai_zhu@nus.edu.sg

Received
1st December 2020
, Accepted 10th February 2021

First published on 12th February 2021

Some micro-organisms and artificial micro-swimmers propel at low Reynolds numbers (Re) via the interaction of their flexible appendages with the surrounding fluid. While their locomotion has been extensively studied with a Newtonian fluid assumption, in realistic biological environments these micro-swimmers invariably encounter rheologically complex fluids. In particular, many biological fluids such as blood and different types of mucus have shear-thinning viscosities. The influence of this ubiquitous non-Newtonian rheology on the performance of flexible swimmers remains largely unknown. Here, we present a first study to examine how shear-thinning rheology alters the fluid-structure interaction and hence the propulsion performance of elastic swimmers at low Re. Via a simple elastic swimmer actuated magnetically, we demonstrate that shear-thinning rheology can either enhance or hinder elastohydrodynamic propulsion, depending on the intricate interplay between elastic and viscous forces as well as the magnetic actuation. We also use a reduced-order model to elucidate the mechanisms underlying the enhanced and hindered propulsion observed in different physical regimes. These results and improved understanding could guide the design of flexible micro-swimmers in non-Newtonian fluids.

While low-Reynolds-number locomotion is relatively well studied with a Newtonian fluid assumption, biological and artificial micro-swimmers invariably encounter complex (non-Newtonian) fluids in their natural habitats and operating environments. These biological fluids often display complex rheological properties such as viscoelasticity and shear-thinning viscosity.^{40} While locomotion in viscoelastic fluids has been extensively studied,^{41,42} including the effect of viscoelasticity on flexible swimmers,^{32,43–47} the effect of shear-thinning rheology has been largely overlooked until more recently. A shear-thinning fluid loses its viscosity with increased shear rates due to changes in the fluid microstructure. Various theoretical and experimental models, including waving sheets,^{48–50} squirmers,^{51,52} rotating helices,^{53,54} and nematodes,^{55–57} among others,^{51,58–60} have revealed scenarios where the swimming speed can increase, decrease, or remain unchanged in a shear-thinning fluid relative to that in a Newtonian fluid. Although a wide variety of swimmer models considered in previous studies demonstrate the profound effects of shear-thinning rheology on locomotion, the shape and swimming gaits of these swimmers are prescribed and fixed. How shear-thinning rheology affects the performance of flexible swimmers, whose shapes and gaits are not known a priori but emerge as a result of fluid–structure interactions, remains largely unknown. A list of questions of both fundamental importance and practical significance remain unanswered: how does shear-thinning rheology affect the shape and gait of an elastic swimmer? Do they swim faster or slower in a shear-thinning fluid? What are the mechanisms underlying any enhancement or hindrance of propulsion? How should artificial flexible propellers be designed to maximize their propulsion performance in a shear-thinning fluid? An improved understanding of elastohydrodynamic propulsion in shear-thinning fluids will not only guide the design of this major class of artificial micro-swimmers, but also shed light on how microorganisms may adapt to rheologically complex fluids by better exploiting the fluid–structure interaction for locomotion.

In this work, we present a first study on the effect of shear-thinning rheology on elastohydrodynamic propulsion via a simple yet representative elastic swimmer actuated by an external magnetic field. We note that shear-thinning rheology can induce both local and non-local effects on locomotion.^{61} The local effect corresponds to the reduction of fluid viscosity due to the increased local shear rates, whereas the non-local effect is concerned with a change in the flow field around the swimmer.^{50,53,61} As a first step, we focus only on the local effect in this work by adopting a local drag model recently proposed by Riley and Lauga,^{61} which is effective in capturing the main physical features of swimming in a shear-thinning fluid. The local drag model is based on the Carreau constitutive equation,^{40} which was shown to describe well the rheological measurements of various biological fluids such as blood,^{62,63} bile,^{64} and lung and cervical mucus.^{49} We will utilize this framework to fill in the gap of missing knowledge on elastohydrodynamic propulsion in a shear-thinning fluid.

The paper is organized as follows. In Section 2 we introduce the model elastic swimmer and formulate the equations governing its elastohydrodynamics in a shear-thinning fluid. In Section 3, we contrast the propulsion performance in a Newtonian fluid (Section 3.1) with that in a shear-thinning fluid (Section 3.2). We also use a reduced-order model in Section 3.3 to further elucidate the essential physics underlying the observed propulsion characteristics in different physical regimes. Finally, we conclude this work with remarks on the limitations and future directions in Section 4.

We model the elastic filament as an Euler–Bernoulli beam with an energy functional,^{65,66}

(1) |

f^{e} = −δ/δx = −∂_{s}[Aκ_{s}n − τt], | (2) |

f^{h} = −R_{C}(ξ_{⊥}nn + ξ_{‖}tt)·u, | (3) |

R_{C} = [1 + (λ_{C}_{avg})^{2}]^{(n−1)/2}, | (4) |

(5) |

In general, the tangential (u_{‖} = u·t) and normal (u_{⊥} = u·n) velocity components and hence the local shear rate vary along a deforming filament. The hydrodynamic force density f^{h} is therefore modified by the spatially and temporally varying correction factor R_{C}. Riley and Lauga^{61} showed that such modifications cause undulatory swimmers with a prescribed shape to swim slower in a shear-thinning fluid than in a Newtonian fluid. Here we adopt the modified RFT to examine the effect of shear-thinning rheology on the propulsion of a flexible filament, whose shapes are not known a priori but emerge as a result of the interaction between the deforming filament and its surrounding fluid.

(6) |

Differentiating eqn (6) with respect to the arc-length s, together with the local inextensibility condition, the normal and tangential components of the resulting equation are, respectively, given by

(7) |

(8) |

(9) |

F_{ext}(L,t) = τ(L,t)t − Aψ_{ss}(L,t)n = 0, | (10) |

T_{ext}(L,t) = Aψ_{s}(L,t) = T^{m}. | (11) |

At the other end (s = 0), the filament is free of force and torque:

F_{ext}(0,t) = −τ(0,t)t + Aψ_{ss}(0,t)n = 0, T_{ext}(0,t) = −Aψ_{s}(0,t) = 0. | (12) |

(13) |

(14) |

(15) |

At the actuated end ( = 1), the dimensionless boundary conditions for ψ and are given by

ψ_{}(1,) = MSp^{4}[λcosψ(1,)sin − sinψ(1,)], ψ_{}(1,) = 0, (1,) = 0, | (16) |

The dimensionless boundary conditions at the free end ( = 0) are given by

ψ_{}(0,) = 0, ψ_{}(0,) = 0, (0,) = 0. | (17) |

We solve the coupled system of nonlinear partial differential equations, eqn (13)–(15), subject to the boundary conditions, eqn (16) and (17) numerically. The numerical simulations are conducted by a finite element method (FEM) based on COMSOL Multiphysics. A backward differentiation formulation is used for time marching the equations. We use 50 to 100 seventh-order Hermite elements to discretize the filament depending on the value of Sp, and a direct solver for solving the linear systems. The computational model is cross-validated against results in the Newtonian limit (Cu = 0) based on a multi-link framework (see Section 3.1 and Appendix for details).

The dynamics of a flexible filament in a Newtonian fluid in different regimes of Sp has been characterized in previous studies.^{17,19,24,29,76} Despite differences in various configurations, the elastohydrodynamic propulsion mechanisms display similar general characteristics as a function of Sp. We illustrate these characteristics with our model swimmer: at low Sp (e.g., Sp = 0.5), the filament is relatively too stiff to undergo significant deformation along the filament; the filament thus behaves largely like a rigid rod performing reciprocal motion [inset, Fig. 2(a)], which leads to ineffective propulsion as constrained by the scallop theorem. As Sp increases, the deformation of the filament enhances its propulsion speed, which reaches a maximum at Sp ≈ 2.4 [see the corresponding filament deformations in Fig. 2(a) inset]. At exceedingly large Sp (e.g., Sp = 10), the filament becomes too soft and hence the deformation is largely localized around the actuated end, as shown in the inset [Fig. 2(a)]; here a large portion of the filament remains horizontal throughout the actuation, which leads to minimal propulsion. A typical swimming trajectory of the filament at Sp = 2 is shown in Fig. 2(c) for the Newtonian case (Cu = 0, black solid line). The filament follows an oscillatory trajectory with net translation in the x-direction.

It is noteworthy that the filament becomes effectively more flexible as Sp increases, which could lead to large deformations that may not be captured quantitatively by the Euler–Bernoulli beam model. Although previous predictions based on the beam model show quantitative agreement with experimental measurements over the experimentally relevant range of Sp,^{17,19,76} future investigations based on geometrically nonlinear rod theories can be considered to address these limitations.^{67,68}

Under a relatively strong magnetic torque (e.g., M = 1), the propulsion speed generally decreases as the fluid becomes shear-thinning (increasing value of Cu) at most Sp, as shown in Fig. 2(a). This reduction is more substantial at lower Sp (e.g., Sp = 2). Although shear-thinning rheology was shown to reduce the swimming speed for undulatory swimmers of prescribed shapes previously,^{61} the mechanism underlying the observed reduction here is more complex, because shear-thinning rheology also alters the shape and gait of the swimmer. We visualize the shape of the filament at different time instances in Fig. 2(b), contrasting the deformation in a Newtonian fluid (Cu = 0) with that in a shear-thinning fluid (Cu = 0.1). At Sp = 2, it is apparent that the filament displays less deformation in a shear-thinning fluid than in a Newtonian fluid. In Fig. 2(c), we also compare the swimming trajectories of the filament in a Newtonian (solid black line) and shear-thinning (solid green line) fluids at Sp = 2. While the oscillatory trajectories display similar amplitudes in the y-direction, the net translation in the x-direction is substantially reduced at this Sp.

Although the elastic filament generally propels slower in a shear-thinning fluid under a relatively strong magnetic torque in Fig. 2(a)–(c), we demonstrate in Fig. 2(d)–(f) that shear-thinning rheology can also enhance propulsion under weaker magnetic torques (e.g., M = 0.02). As shown in Fig. 2(d), whether shear-thinning rheology enhances or hinders propulsion also depends on the value of Sp. At lower Sp (e.g., Sp ≲ 3), enhanced propulsion in a shear-thinning fluid is observed, whereas hindered propulsion occurs at higher Sp. Similar to the results in Fig. 2(b), we visualize the shape of the filament at different time instants in a Newtonian and shear-thinning fluids in Fig. 2(e). At Sp = 2, it is apparent that the filament displays a greater range of angular movements in a shear-thinning fluid than in a Newtonian fluid. We also contrast the swimming trajectory at Sp = 2 in a Newtonian fluid (solid black line) with that in a shear-thinning fluid (solid green line) in Fig. 2(f). Unlike the trajectories under a relatively strong magnetic torque shown in in Fig. 2(c), which display similar oscillatory amplitudes, the trajectory of the filament in a shear-thinning fluid has a substantially larger amplitude compared with that in a Newtonian fluid under a relatively weak magnetic torque as shown in in Fig. 2(f).

Taken together, shear-thinning rheology can either increase or decrease the propulsion speed of a flexible swimmer, depending on the specific values of M and Sp. This is unlike the case of undulatory swimmers with prescribed gaits in a shear-thinning fluid,^{61} where the swimming speeds are systematically lowered. Our observations here thus highlight the effect of gait changes of a flexible swimmer in a shear-thinning fluid (i.e., the second mechanism referred to above) on the propulsion performance. We will further elucidate these results with the use of a reduced-order model in Section 3.3.

Fig. 3 Reduced-order modeling of elastohydrodynamic propulsion. (a) The elastic filament in Fig. 1 is represented by a minimal model consisting of two rigid links connected by a torsional spring. (b) The amplitude of the angle of the actuated link (link 2), _{2}, as a function of the relative strength of the magnetic torque, M, at different values of λ = B_{y}/B_{x} in a Newtonian fluid with K = 0.1. The amplitude _{2} increases with M before leveling off to the maximum value tan^{−1}λ set by the external magnetic field at larger values of M. (c) and (d) The shape of the two-link swimmer over one actuation period T = 2π at equal time intervals (T/4) in a Newtonian fluid; the intensity of color increases as time advances. The dotted lines in (c) and (d) represent the external magnetic field B and the angle spanned by its oscillation for λ = 1. At low M [e.g., M = 0.02 in (c)], the actuated link spans an angle smaller than that spanned by the external magnetic field, _{2} < tan^{−1}λ = π/4. At a large M [e.g., M = 1 in (d)], the actuated link follows closely the external magnetic field and attains the same angle spanned by the external magnetic field, _{2} < tan^{−1}λ = π/4. |

We consistently use the same non-dimensionalizations described in Section 2.5 to scale lengths, time, and forces in this reduced-order model. Hereafter we shall work with dimensionless variables only while adopting the same notations for their dimensionless counterparts for convenience. Instead of Sp for a continuous elastic filament, a dimensionless spring constant K = k/(L^{3}ξ_{⊥}ω), which compares the elastic to viscous torques, emerges in this two-link model. Here we note that the dimensionless spring constant K plays a physically similar (but inverse) role to Sp in a continuous filament as shown in Fig. 2. The dynamics of the two-link swimmer is governed by the balance of forces,

F^{h}_{1} + F^{h}_{2} = 0, | (18) |

T^{h}_{1,1} + T^{h}_{2,1} + T^{m} = 0, | (19) |

T^{h}_{2,2} + T^{e} + T^{m} = 0. | (20) |

The distinct effects of shear-thinning rheology on the propulsion performance under strong and weak magnetic torques revealed in Fig. 2 can be better understood by examining the response of the two-link swimmer to the magnetic actuation. In this two-link model [Fig. 3(a)], the actuation comes from link 2 attempting to follow the external magnetic field (characterized by the angle θ_{2}), whereas link 1 responds elastically to the actuation via the torsional spring (characterized by the relative angle Δθ = θ_{2} − θ_{1}). The propulsion behavior of the two-link swimmer can be described in terms of the amplitude of the angles _{2} = max(θ_{2}) and . Therefore, understanding the effect of shear-thinning rheology on elastohydrodynamic propulsion can be reduced to elucidating how _{2} and are modified in a shear-thinning fluid in various scenarios.

The maximum angle _{2} spanned by the actuated link (link 2) under the oscillating magnetic field largely depends on the relative strength of the magnetic torque, M. We first examine this dependence in a Newtonian fluid in Fig. 3(b). At low M, the amplitude _{2} increases with M before leveling off to the maximum angle (tan^{−1}λ) set by the external magnetic field (λ = B_{y}/B_{x}) at large M ≥ 1. For instance, Fig. 3(c) displays the time evolution of a two-link swimmer under a weak magnetic torque (M = 0.02 and λ = 1), where the actuated link spans an angle much smaller than tan^{−1}λ = π/4; essentially, the magnetic torque is relatively too weak to actuate the link fast enough to follow closely the oscillatory magnetic field. On the other hand, when the magnetic torque is relatively strong (e.g., M ≥ 1), the actuated link follows the magnetic field almost synchronously, spanning the maximum angle _{2} ≈ tan^{−1}λ = π/4 as shown in Fig. 3(d); in this synchronous regime, a further increase in the strength of the magnetic torque M has little effect on the dynamics of the actuated link and hence that of the swimmer. The difference in the dynamics of the actuated link under relatively weak and strong magnetic torques is crucial in understanding the hindered and enhanced propulsion observed in a shear-thinning fluid. We next discuss how shear-thinning rheology modifies the propulsion of the two-link swimmer under relatively weak and strong magnetic torques in terms of _{2} and and compare the results with that for a continuous filament in Fig. 2.

3.3.1 Propulsion at large M.
Under a relatively strong magnetic torque (e.g., M = 1), shear-thinning rheology generally reduces the propulsion speed of the two-link swimmer at most K as shown in Fig. 4(a), similar to the results for a continuous filament in Fig. 2(a). First, we examine how shear-thinning rheology alters the gaits (characterized by _{2} and ) and hence the propulsion speed of a two-link swimmer. At large M, due to the dominance of the magnetic actuation on the dynamics of the actuated link, reduced viscous torques in a shear-thinning fluid has little influence on the actuation angle _{2} [Fig. 4(b)]. The actuated link spans approximately the same maximum angle _{2} = tan^{−1}λ = π/4 dictated by the external magnetic field (λ = B_{y}/B_{x} = 1) at different values of Cu. Modifications on the swimming gait in this large-M regime therefore stem only from changes in the relative angle in a shear-thinning fluid [inset, Fig. 4(b)]. As Cu increases, the shear-thinning viscosity reduces the viscous torque; a smaller elastic torque is thus required to balance the viscous torque, leading to a reduced amplitude of the relative angle . The effect is analogous to a further increase in the spring constant K. The two-link swimmer hence behaves increasingly more like a rigid rod in a shear-thinning fluid as Cu increases [Fig. 4(c), K = 0.1], which acts to hinder propulsion. The same mechanism contributes to the decrease in propulsion speed for a continuous filament at high M [Fig. 2(a)], where shear-thinning rheology reduces the deformation along the filament owing to the reduced viscous forces [see Fig. 2(a) and (b)]. As a remark, even without any shape changes, an undulatory swimmer with the same gaits can swim slower in a shear-thinning fluid, because thrust is reduced to a larger extent than drag for undulatory swimmers.^{61} This latter effect combines with the effect due to gait changes to significantly reduce the propulsion performance at larger values of K in Fig. 4(a) [or smaller values of Sp in Fig. 2(a)]. On the other hand, at smaller values of K (e.g., K = 0.01), propulsion is relatively ineffective in the Newtonian limit due to the dominance of the viscous effect over the elastic effect on the dynamics of link 1. In this regime, the reduction in the viscous effect caused by shear-thinning rheology indeed allows more effective swimming gaits to emerge (the effect is analogous to an increased K). The two mechanisms, which act in tandem to hinder propulsion at large K, now counter-act and lead to less significant changes in the overall propulsion performance in the small K (or large Sp) regime.

Fig. 4 Propulsion of a magnetically-driven two-link swimmer in a shear-thinning fluid under relatively strong [M = 1, panels (a)–(c)] and weak [M = 0.02, panels (d)–(f)] magnetic torques with λ = 1. (a) and (d) Average propulsion speed 〈V〉 of the swimmer as a function of the dimensionless spring constant (K) for varying Carreau number (Cu). Insets in (a) display the shape of the swimmer over one actuation period T = 2π at equal time intervals (T/4) in the Newtonian limit at various K. We note that K of a two-link swimmer plays a physically similar (but inverse) role to Sp of a continuous filament as shown in Fig. 2. Similar to the results for a continuous filament, while the two-link swimmer generally propels slower in a shear-thinning fluid under a relatively strong magnetic torque [M = 1, panel (a)], enhanced propulsion can also occur under a relatively weak magnetic torque [M = 0.02, panel (d)]. (b) and (e) The amplitude of the actuated link's angle (_{2}) and that of the relative angle ( insets) as a function of K at varying Cu. (c) and (f) The shape of the swimmer over one actuation period T = 2π at equal time intervals (T/4) in Newtonian and shear-thinning fluids at different K; the intensity of the color increases as time advances. |

3.3.2 Propulsion at small M.
The propulsion performance of the two-link swimmer is modified by shear-thinning rheology in a qualitatively different manner at small M [Fig. 4(d)], where enhanced propulsion can occur. We attribute the difference to the distinct ways shear-thinning rheology alters the swimming gaits at small and large M [compare Fig. 4(b) and (e)]. While _{2} at large M always attains the maximum angle tan^{−1}λ = π/4 allowed by the external magnetic field, the actuated link spans angles that are considerably smaller (_{2} ≪ π/4) under a relatively weak magnetic torque (e.g., M = 0.02). The viscous effect dominates the dynamics of the actuated link in this regime and limits the amplitude of its angular movements, _{2}. When the fluid becomes shear-thinning, the reduced viscous effect on the actuated link allows the link to span larger angles _{2} as Cu increases [Fig. 4(e)]. We argue that this increased angle of actuation [apparent in the visualizations shown in Fig. 4(f) at K = 0.1], induced by shear-thinning rheology at small M, is responsible for the enhanced propulsion observed in Fig. 4(d). A similar effect is at play for a continuous filament actuated with a small M [Fig. 2(e), Sp = 2], where the filament's actuated (right) end displays an increased amplitude and hence the propulsion speed in a shear-thinning fluid [Fig. 2(d)]. It is noteworthy that the increase in the actuation amplitude becomes less significant at lower values of K (or higher Sp); in this regime, the other shear-thinning effect unrelated to gait changes but a greater reduction in thrust than drag for undulatory swimmers,^{61} which acts to hinder propulsion, may become more important. A reduction in the propulsion speed is apparent for a continuous filament at high Sp as shown in Fig. 2(d).

We summarize the essential physical pictures at large (Section 3.3.1) and small (Section 3.3.2) M as follows. Via the two-link model, we reduce the description of a flexible swimmer into two portions: the portion responsible for actuation (link 2) and the portion responsible for the elastic response (link 1). Under a relatively small magnetic torque (M ≪ 1), the amplitude of actuation is suppressed by excessively large viscous effects on the actuated portion. Here shear-thinning rheology enables a larger amplitude of actuation (_{2}) by reducing the viscous effect, which leads to enhanced propulsion in this regime. In contrast, under a relatively strong magnetic torque (M ≥ 1), the actuated portion already maximizes the amplitude of actuation allowed by the magnetic field; shear-thinning rheology hence alters the swimming gait only via changes in the relative angle . Shear-thinning rheology generally reduces because a smaller elastic torque is required to balance the reduced viscous torque in a shear-thinning fluid. At higher values of K, a reduced makes the two-link swimmer behave more like a rigid rod with hindered propulsion performance. On the other hand, at smaller values of K, a reduced could act to enhance propulsion by allowing the two-link swimmer to deviate from relatively ineffective gaits in a Newtonian fluid in this regime. Overall, while these gait changes induced by shear-thinning rheology affect propulsion, we also note that, even without inducing any gait changes, shear-thinning rheology can also hinder the propulsion of undulatory swimmers by reducing the thrust more than drag.^{61} This later effect can act in tandem or counter-act with the effect due to gait changes to enhance or hinder propulsion by varying extents in different physical regimes. Taken together, the physical picture presented here captures qualitatively the behaviors observed for a continuous filament.

We discussed several limitations in this work, which provide directions for subsequent studies. First, as a first step we considered a local drag model here and therefore only accounted for the local shear-thinning effect. This also confines the validity of our results to the small Carreau number regime to be consistent with the local nature of the model.^{61} The change in the flow field due to non-local shear-thinning effects and non-local hydrodynamic interactions remains to be investigated. Second, for simplicity we prescribed a magnetic moment at one of the filament's end as a minimal model of a magnetic swimmer in this work. We have therefore ignored the hydrodynamic effect of the magnetic head geometry, which is another design parameter for optimizing the propulsion performance of magnetic swimmers.^{39} Finally, we note that magnetic actuation is considered here for its common use as an actuation mechanism for artificial micro-swimmers.^{80,81} The same framework can be employed to consider other types of boundary or distributed actuation mechanisms that are more relevant to biological swimmers.^{25,82–84} We believe that the essential physical pictures discussed in this work could still be generally useful in interpreting the effect of shear-thinning rheology in other swimmer configurations.

(21) |

(22) |

Fig. 5 Schematic diagram illustrating a multi-link discretization of an elastic filament and notations. |

The dynamics of the multi-link model is governed by the overall balance of force

(23) |

(24) |

(25) |

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