Swimming efficiency in a shear-thinning fluid

Micro-organisms expend energy moving through complex media. While propulsion speed is an important property of locomotion, efficiency is another factor that may determine the swimming gait adopted by a micro-organism in order to locomote in an energetically favorable manner. The efficiency of swimming in a Newtonian fluid is well characterized for different biological and artificial swimmers. However, these swimmers often encounter biological fluids displaying shear-thinning viscosities. Little is known about how this nonlinear rheology influences the efficiency of locomotion. Does the shear-thinning rheology render swimming more efficient or less? How does the swimming efficiency depend on the propulsion mechanism of a swimmer and rheological properties of the surrounding shear-thinning fluid? In this work, we address these fundamental questions on the efficiency of locomotion in a shear-thinning fluid by considering the squirmer model as a general locomotion model to represent different types of swimmers. Our analysis reveals how the choice of surface velocity distribution on a squirmer may reduce or enhance the swimming efficiency. We determine optimal shear rates at which the swimming efficiency can be substantially enhanced compared with the Newtonian case. The nontrivial variations of swimming efficiency prompt questions on how micro-organisms may tune their swimming gaits to exploit the shear-thinning rheology. The findings also provide insights into how artificial swimmers should be designed to move through complex media efficiently.

Figure 1

(a) Power dissipation in a shear-thinning fluid $\mathcal{P}$ (scaled by the corresponding Newtonian value ${\mathcal{P}}_N$) for a neutral squirmer ($\alpha =0$, gray ◯), puller ($\alpha =5$, blue $\square$), and pusher ($\alpha =-5$, red $△$) by numerical simulations (symbols) agrees well with the asymptotic expansions in viscosity ratio for a neutral squirmer (solid line) and a pusher or puller (dashed line) at $\varepsilon =1-\beta =0.1$ and $n=0.25$. The variation in power dissipation at low shear rates is also well captured by the asymptotic expansion in Carreau number, Cu, given by Eq. (22) (dash-dotted lines). (b) Numerical computations of the power dissipation at $\varepsilon =0.99$ and $n=0.25$. The insets show the corresponding swimming velocity $U$ compared with the Newtonian value $U_N$.

Figure 2

(a) Enhanced swimming efficiency in a shear-thinning fluid $\eta$ (compared with the Newtonian efficiency ${\eta }_N$) for a neutral squirmer ($\alpha =0$, ◯), puller ($\alpha =5$, $\square$), and pusher ($\alpha =-5$, $△$) by numerical simulations (symbols) is well captured by the asymptotic expansions in viscosity ratio for a neutral squirmer (solid line) and a pusher or puller (dashed line) at $\varepsilon =1-\beta =0.1$ and $n=0.25$. The asymptotic expansion in Carreau number, Cu, given by Eq. (23) (dash-dotted lines) is effective in predicting the swimming efficiency at low shear rates. (b) Numerical computations of the power dissipation at $\varepsilon =0.99$ and $n=0.25$. For a given viscosity ratio, an optimal Cu maximizing the swimming efficiency of a swimmer exists.

Figure 3

Swimming efficiency $\eta$ as a function of $\alpha =B_2/B_1$ at different values of Cu with $\varepsilon =1-\beta =0.99$ and $n=0.25$ in a shear-thinning fluid: $\text{Cu}=0.1$ (black ◯), $\text{Cu}=1$ (gray $\square$), and $\text{Cu}=10$ (light gray $△$). The inset shows the enhancement in efficiency relative to the Newtonian value $\eta /{\eta }_N$ as a function of $\alpha$, where ${\eta }_N=1/(2+{\alpha }^2)$.

Figure 4

Swimming can be more efficient or less in a shear-thinning fluid, depending on the details of the swimming gait. (a) Swimming efficiency at small Carreau number ($\eta \sim {\eta }_N+{\text{Cu}}^2{\eta }_1$) given by Eq. (24) predicts instances of more efficient (${\eta }_1>0$, region in red) and less efficient (${\eta }_1<0$, region in blue) swimming depending on squirming modes represented by $\alpha =B_2/B_1$ and $\zeta =B_3/B_1$. The level set curve (dashed line) separates the two regions. For comparison, the efficiency and swimming speed of a two-mode squirmer ($\alpha =15$ and $\zeta =0$) are shown as black dash-dotted lines in (b) and (c), respectively. With only the first two modes, the swimmer gains efficiency but loses swimming speed. The addition of a third mode can either enhance or degrade the swimming performance depending on the choice of $\zeta$. For illustration, a swimmer chosen above the level set curve ($\alpha =15$ and $\zeta =15$) displays in (b) less efficient swimming at small Cu and in (c) slower swimming speed for all Cu, represented by the blue dashed lines; nevertheless, the swimmer can gain efficiency above the Newtonian limit at large Cu in (b). For the case $\alpha =15$, we determine a threshold value of ${\zeta }_c=-5.4$ below which both the efficiency and swimming speed of the squirmer are enhanced for all values of Cu, represented by thin green solid lines in (b) and (c). To further illustrate, a swimmer below the threshold value ($\alpha =15$ and $\zeta =-15$) is chosen as another example to demonstrate that swimming in a shear-thinning fluid can be both (b) more efficient and (c) faster than the Newtonian case, represented by the red solid lines. The asymptotic expansion in Cu, Eq. (24), is effective in predicting the swimming efficiency at small Cu (dotted lines) in (b).