2.1. Swimming in viscoelastic and shear-thinning fluids

To model the elasticity and shear-thinning properties of biological fluids, we use the Giesekus constitutive relation (Giesekus Reference Giesekus1982), in which ${\bf\tau}$ can be split into solvent and polymer contributions as ${\bf\tau}={\bf\tau}^{s}+{\bf\tau}^{p}$ , where ${\bf\tau}^{s}={\it\beta}_{s}\dot{{\bf\gamma}}$ ,

and ${\it\beta}_{s}$ is the ratio of the solvent viscosity to the zero-shear-rate viscosity of the polymeric solution. This ratio is equal to the ratio of the infinite-shear-rate viscosity and the zero-shear-rate viscosity. The Deborah number $\mathit{De}={\it\lambda}/t_{c}$ is the ratio of the polymer relaxation time ${\it\lambda}$ and the characteristic flow time scale $t_{c}=1/{\it\Omega}$ . Another important dimensionless number for swimming in viscoelastic fluids is the Weissenberg number $\mathit{Wi}={\it\lambda}\dot{{\it\gamma}}_{c}$ , where the characteristic shear rate in this study is $\dot{{\it\gamma}}_{c}={\it\Omega}A$ . The mobility factor ${\it\alpha}$ , which is in the range of $0$ to $1/2$ , represents the anisotropic hydrodynamic drag exerted on the polymer molecules by the surrounding solute molecules and affects the viscosity of the polymeric solution. At ${\it\alpha}=0$ , the Giesekus constitutive equation recovers to the Oldroyd-B model and has a constant viscosity. The notation $\overset{{\bigtriangledown}}{}$ represents the upper-convected derivative.

We use the Carreau constitutive model (Carreau, Dekee & Chhabra Reference Carreau, Dekee and Chhabra1997) to investigate the motion in a shear-thinning inelastic fluid,

where ${\it\beta}_{e}$ is the normalized effective viscosity and $|\dot{{\it\gamma}}|=\sqrt{\dot{{\bf\gamma}}\boldsymbol{ : }\dot{{\bf\gamma}}/2}$ is the effective shear rate. The Carreau number $\mathit{Cu}={\it\lambda}_{c}{\it\Omega}$ (similar to $\mathit{De}$ ) is the ratio between the characteristic time scale ${\it\lambda}_{c}$ of the solution and the characteristic flow time scale $1/{\it\Omega}$ , where ${\it\lambda}_{c}$ is the inverse of the shear rate at which the fluid transitions from Newtonian-like to power-law behaviour. Similar to the Weissenberg number, the shear Carreau number is related to the shear rate $\mathit{Cr}=\mathit{Cu}\,A$ . The power-law index $n$ determines how fast the viscosity decreases with increasing shear rate. The larger $n$ is, the more slowly the viscosity decreases. At ${\it\beta}_{s}=1$ , $\mathit{Cu}=0$ or $n=1$ , the model recovers to the Newtonian fluid. In biological materials, such as biofilm and mucus, ${\it\lambda}$ and ${\it\lambda}_{c}$ vary from $O(10)$ to $O(10^{3})$  s, $n$ from 0.1 to 0.9 and ${\it\beta}_{s}$ from $O(10^{-3})$ to $O(10^{-1})$ . The typical beating frequency of the flagellum ranges from 2 to 30 Hz. Therefore, for micro-organisms, $\mathit{De},\mathit{Wi},\mathit{Cu}$ and $\mathit{Cr}$ vary in a very wide range, from $O(1)$ to $O(10^{4})$ .

3.1. Verification of the numerical code

The swimming speed $U$ and the hydrodynamic efficiency ${\it\eta}$ for an infinitely long waving sheet in a Newtonian fluid are shown in figure 1(a). The numerical results agree well with the second- and fourth-order analytical results for a small-amplitude sheet (Taylor Reference Taylor1951) up to $A=0.1{\rm\pi}$ . Within this range, the maximum difference between our simulations and the fourth-order analytical results is less than $0.5\,\%$ . The swimming speed of the sheet reaches a plateau for $A>0.4{\rm\pi}$ . Interestingly, the efficiency of the sheet peaks at approximately $A=0.3{\rm\pi}$ , which lies inside the range of typical biological observations, such as C. elegans (Gagnon et al. Reference Gagnon, Keim and Arratia2014) and sperm (Brokaw Reference Brokaw1965).

Figure 1.

(a) Swimming speed $U$ and efficiency ${\it\eta}$ for an infinitely long sheet of different amplitudes in a Newtonian fluid. (b) Shear-dependent viscosity of Giesekus and Carreau models in a simple shear flow.

Figure 2.

The normalized (a) swimming speed and (b) hydrodynamic power as a function of $\mathit{De}$ (or $\mathit{Cu}$ ) for an infinitely long sheet of small ( $A=0.01{\rm\pi}$ ) and large amplitude ( $A=0.4{\rm\pi}$ ). The dashed and dash–dot lines correspond to (4.1) and (4.2) respectively.

In figure 1(b), we compare the shear-thinning behaviour of the Giesekus and Carreau models (2.3b ) in a simple shear flow. The shear rate corresponding to the transition from Newtonian-like to power-law behaviour in a Giesekus fluid is determined by both ${\it\lambda}$ and ${\it\alpha}$ , whereas the shear-thinning rate remains the same (Bird et al. Reference Bird, Armstrong, Hassager and Curtiss1977). We can match the effective viscosity of the Carreau fluid to the Giesekus fluid to unravel the effect of both elasticity and shear-thinning behaviour (see figure 1 b). Verification of numerical results for a Taylor’s swimming sheet with small amplitude in non-Newtonian fluids is given in figure 2. Our results recover the analytical results for a small-amplitude sheet in both viscoelastic and shear-thinning fluids.

4.1. Scaling law in shear-thinning fluids

To fully understand the shear-thinning effects on a swimming sheet, we conduct a series of simulations for various oscillation amplitudes $A$ and viscosity ratios ${\it\beta}_{s}$ . In figure 4, we plot the normalized swimming speed and power as a function of $\mathit{Cr}$ . It is clear that using $\mathit{Cr}$ instead of $\mathit{Cu}$ leads to a much better collapse of the data. The swimming speed is less affected at small values of $\mathit{Cr}$ $({<}0.1)$ , since the fluid viscosity does not vary at such a low shear rate. The analysis of Vélez-Cordero & Lauga (Reference Vélez-Cordero and Lauga2013) is valid in this range. The rheological effects begin to matter when $\mathit{Cr}\sim O(0.1)-O(1)$ . The transition occurs at a smaller $\mathit{Cr}$ as the undulation amplitude increases and ${\it\beta}_{s}$ decreases. There is a maximum speed for cases with non-zero ${\it\beta}_{s}$ . In the limits of zero and infinite $\mathit{Cr}$ , the swimming speed of the sheet is the same as its Newtonian value because the normalized viscosity is constant in both cases, equal to 1 and ${\it\beta}_{s}$ respectively. The maximum speed and the corresponding Cr number, $\mathit{Cr}_{max}$ , depend on the oscillation amplitude $A$ , as well as the viscosity ratio ${\it\beta}_{s}$ . Larger speed enhancement and $\mathit{Cr}_{max}$ are observed for larger $A$ and smaller ${\it\beta}_{s}$ . The value of $\mathit{Cr}_{max}$ ranges from the order of $O(1)$ to $O(10^{2})$ , which corresponds to the range where the fluid viscosity is effectively thinned but before it reaches the infinite-shear-rate plateau (see figure 1 b). Interestingly, experiments show that the circulation generated by C. elegans first increases with $\mathit{Cr}$ and then decreases (Gagnon et al. Reference Gagnon, Keim and Arratia2014). In the limit of zero ${\it\beta}_{s}$ , the swimming speed monotonically increases due to the absence of infinite-shear-rate plateau.

The power consumption of an undulating sheet follows a universal scaling law. As shown in figure 4(b), all of the data collapse onto solid curves defined as

in which the constant $3/8$ is chosen so that the first term of the Taylor expansion of (4.3) matches the power consumption calculated for small-amplitude oscillations, i.e. (4.2). The power consumption of an undulating sheet is equal to the energy dissipation integral $P=\int _{S}{\it\beta}^{e}|\dot{{\it\gamma}}|^{2}\,\text{d}S$ , where ${\it\beta}^{e}=1$ in a Newtonian fluid. Assuming that the magnitude of the shear rate is roughly unchanged and using (2.3b ), we get the following scaling for power consumption $(1-P/P_{N})/(1-{\it\beta}_{s})\sim 1-(1+\mathit{Cu}^{2}|\dot{{\it\gamma}}_{t}|^{2})^{(n-1)/2}$ , where $|\dot{{\it\gamma}}_{t}|$ is the typical shear rate. This equation can be simplified to (4.3) if the effective shear rate scales with the oscillation amplitude ( $|\dot{{\it\gamma}}_{t}|\propto A$ ). Later, we will confirm both these assumptions.

Figure 4.

Normalized (a) swimming speed and (b) hydrodynamic power as a function of $\mathit{Cr}$ .

The present numerical results extend the analytical results by Vélez-Cordero & Lauga (Reference Vélez-Cordero and Lauga2013) for a small-amplitude waving sheet. Our results are also consistent with the simulations of Montenegro-Johnson et al. (Reference Montenegro-Johnson, Smith, Smith, Loghin and Blake2012, Reference Montenegro-Johnson, Smith and Loghin2013) for a finite sperm-like swimmer, in which the velocity peaks at a certain $\mathit{Cr}$ . Montenegro-Johnson et al. (Reference Montenegro-Johnson, Smith and Loghin2013) related this velocity peak with the maximal viscosity gradient along the sheet, but their explanation does not hold for an infinitely long sheet. Here, we propose another explanation for the speed enhancement caused by shear-thinning viscosity. The fluid viscosity around the sheet is reduced due to the high values of the shear rate and it gradually recovers to the zero-shear-rate viscosity away from the sheet (figure 5 a– $c$ ). As a result, the sheet swims inside a corridor of low-viscosity fluid surrounded by a higher-viscosity fluid. If ${\it\beta}_{s}=0$ , the fluid viscosity can be reduced to the solvent viscosity, which is infinitesimal compared with the unstirred fluid. The sheet swims as if it is inside a confined channel and gains an enhanced speed due to the confinement (Katz Reference Katz1974). The differences in the streamlines of two cases with different $\mathit{Cr}$ are shown in figure 5(d,e). The effect of this corridor of low-viscosity fluid is weakened with increasing ${\it\beta}_{s}$ due to the increased viscosity ratio inside and outside the channel, but the same mechanism still exists. The further decrease in the fluid viscosity inside the channel at larger $\mathit{Cr}$ strengthens the confinement effect. On the other hand, the enlarged shear-thinning region weakens the confinement effect by increasing the width of the low-viscosity layer. These two competing effects lead to a peak in the swimming speed. For a finite-length waving sheet, this mechanism as well as the mechanism proposed by Montenegro-Johnson et al. (Reference Montenegro-Johnson, Smith and Loghin2013) exist and may work in tandem to enhance the swimming speed.

To further test the role of corridors of low-viscosity fluids, we have conducted simulations of a waving sheet swimming in a layer of low-viscosity Newtonian fluid surrounded by an outer layer of high-viscosity Newtonian fluid. The ratio of inner fluid viscosity to outer fluid viscosity is ${\it\beta}_{s}$ . As shown in figure 6(a), the swimming speed of the waving sheet increases as ${\it\beta}_{s}$ decreases (for ${\it\beta}_{s}<1$ ), and it approaches the swimming speed between two solid walls as ${\it\beta}_{s}$ approaches zero. We should note that the swimming enhancement of a waving sheet in shear-thinning fluids is weaker than that swimming between two solid walls due to continuous change in the fluid viscosity as well as the finite ratio of the fluid viscosity near the swimmer to the zero-shear-rate viscosity.

Figure 5.

( $a$ – $c$ ) Distribution of effective viscosity around a waving sheet at ${\it\beta}_{s}=0.5$ and (d–e) the velocity field and streamlines around a sheet at ${\it\beta}_{s}=0$ ; in all of the cases, $A=0.4{\rm\pi}$ ; ( $a$ )  $Cu=1.2{\rm\pi}$ , ( $b$ )  $Cu=6{\rm\pi}$ , ( $c$ )  $Cu=20{\rm\pi}$ , ( $d$ )  $Cu=0.02{\rm\pi}$ , ( $e$ )  $Cu=200{\rm\pi}$ .

In figure 6(b,c), we plot the distribution of $|\dot{{\it\gamma}}|,u,v$ and ${\it\beta}^{e}$ along the $y$ -direction at $x={\rm\pi}/2$ . Independent of the fluid rheology and the oscillation amplitude, the magnitude of the shear rate roughly follows the same relation (light grey line),

which confirms the assumption we used for the derivation of (4.3). This scaling is the same as for the small-amplitude analytical results, in which the first- and second-order velocities are not affected by shear-thinning effects. In this case, the shear dependence of the viscosity affects the third and higher orders of the shear rate (Vélez-Cordero & Lauga Reference Vélez-Cordero and Lauga2013). The $u$ velocity component for the case with maximum speed changes its direction earlier than other cases, and the $v$ component is the lowest, indicating the strongest confinement effect in this case. The effective viscosity ${\it\beta}^{e}$ of Carreau fluid can be derived as a function of $\mathit{Cr}$ and $y$ using (2.3b ) and (4.4) (light grey lines in figure 6 c). The effective viscosity obtained from (2.3b ) and (4.4) has a minimum at $y=1$ , which agrees well with our numerical results for a small-amplitude sheet. For a large-amplitude sheet at low $\mathit{Cr}$ , the viscosity near the sheet is lower than the abovementioned prediction on one side, while it is less thinned on the other side. At high $\mathit{Cr}$ , where the viscosity reaches its minimum value near the sheet, the prediction holds well.

Figure 6.

(a) The normalized swimming speed of a sheet inside a channel of low-viscosity fluid. Distribution of (b) $|\dot{{\it\gamma}}|,u$ and $v$ and ( $c$ )  ${\it\beta}^{e}$ along the $y$ -direction at $x={\rm\pi}/2$ . The light grey lines correspond to (4.4) in (b) and (2.3b ) in ( $c$ ) after substituting (4.4) for the effective shear rate. The vertical dashed lines represent the sheet of $A=0.01{\rm\pi}$ and $0.4{\rm\pi}$ respectively; ${\it\beta}_{s}=0.5$ in all cases in (b,c).

Using this prediction, we can estimate $\mathit{Cr}_{max}$ by evaluating the strength of the shear-thinning effect, $(1-{\it\beta}_{min}^{e})/d$ . Here, ${\it\beta}_{min}^{e}$ is the minimum effective viscosity and $d$ is the thickness of the shear-thinned layer, which is defined as the location where the viscosity recovers to the zero-shear-rate viscosity. Figure 7(a) shows $(1-{\it\beta}_{min}^{e})/d$ as a function of $\mathit{Cr}$ . The $\mathit{Cr}$ corresponding to the maximum value of $(1-{\it\beta}_{min}^{e})/d$ is close to the $\mathit{Cr}_{max}$ obtained from our numerical simulations. At $n=0.3$ , $\mathit{Cr}_{max}$ is approximately $10$ and increases with decreasing ${\it\beta}_{s}$ and increasing $n$ . In typical biological fluids, where the viscosity ratio ${\it\beta}_{s}$ is usually low, we expect the microswimmers to benefit from the speed enhancement. In figure 7(b), we plot the normalized speed as a function of the amplitude at $\mathit{Cu}=200{\rm\pi}$ and ${\it\beta}_{s}=0$ . This is approximately the maximum speed enhancement of microswimmers for $n=0.3$ due to shear-thinning effects. For a small-amplitude sheet, approximately $20\%$ speed enhancement is achieved. For a large-amplitude sheet, the speed is more than doubled. Finally, we check whether the thickness, the finite length and the swimming stroke affect the results. The thickness of the thick waving sheet is $0.16{\rm\pi}$ , similar to the measurements of C. elegans (Gagnon et al. Reference Gagnon, Keim and Arratia2014). For the finite-length swimmer, the sheet has one complete waving form. Linearly increasing/decreasing amplitude towards the tail is chosen for kicker/burrower swimmers. As shown in figure 7( $c$ ), swimming enhancement occurs for all of the cases. The thickness does not affect the speed; the finite length of the waving sheet leads to larger $\mathit{Cr}_{max}$ and weaker speed enhancement. Kicker and burrower swimmers have the same swimming speeds.

Figure 7.

(a) The strength of the shear-thinning effect, $(1-{\it\beta}_{min}^{e})/d$ , as a function of $\mathit{Cr}$ for different $n$ and ${\it\beta}_{s}$ . The symbols show the maximum value of $(1-{\it\beta}_{min}^{e})/d$ , with $d$ being the distance where the effective viscosity is $95\,\%$ of the zero-shear-rate viscosity. The error bars span the range of $d$ corresponding to $91\,\%$ and $99\,\%$ of the zero-shear-rate viscosity. (b) The normalized swimming speed as a function of the amplitude $A$ at $\mathit{Cu}=200{\rm\pi}$ and ${\it\beta}_{s}=0$ . ( $c$ ) The normalized swimming speed for various waving sheets as a function of $\mathit{Cr}$ at $A=0.4{\rm\pi}$ and ${\it\beta}_{s}=0.5$ .

There are some differences between the simulation results and the experimental measurements of C. elegans by Gagnon et al. (Reference Gagnon, Keim and Arratia2014). Even though experiments show that the circulation generated by C. elegans first increases with $\mathit{Cr}$ and then decreases, the swimming speed is reported to be the same as that in a Newtonian fluid. The difference may come from the fact that the swimming speed of C. elegans depends on the fluid viscosity, even in Newtonian fluids, while in the simulations, the swimming speed is the same. The difference between these numerical results and the experiments of Gagnon et al. (Reference Gagnon, Keim and Arratia2014) could be due to the three-dimensional nature of the experiments and the subtle viscosity-dependent changes in the swimming kinematics of C. elegans. Additionally, in simulations, the kinematics of the waving sheet is imposed without any limit on the power consumption, whereas in experiments there will be a limit on the power consumption. If we compared the experimentally measured swimming speeds in Newtonian and shear-thinning fluids at the same zero-shear-rate viscosity, we would see a higher speed in shear-thinning fluids, which is consistent with our study. An area of future research could be to evaluate the optimal efficiency in non-Newtonian fluids. The optimal efficiency of a sheet in a Newtonian fluid has been previously studied by Montenegro-Johnson & Lauga (Reference Montenegro-Johnson and Lauga2014).