Axisymmetric squirmers in Stokes fluid with nonuniform viscosity

The ciliary locomotion and feeding of an axisymmetric microswimmer in a complex fluid whose viscosity depends on a surrounding nutrient field are investigated numerically in order to extend previous asymptotic results for weak nutrient-viscosity coupling. Numerical simulations capture nonlinearities inherent in the full system that are missed using perturbation-based linearization methods. The microswimmer's ciliary beating is modeled by a slip velocity, i.e., the squirmer model, and body geometry is modeled by spheroids. It is found that swimming speed and feeding are most affected by a nonuniform viscosity environment when the ratio of advection forces to diffusion transport, characterized by the nondimensional Peclét number, is moderate, i.e., $\text{Pe}=O\left(5\right)$. These changes are correlated to significant increases in the pressure force on the surface of the squirmer, as well as differences in power expenditure and hydrodynamic efficiency compared to the constant-viscosity case. Additionally, the swimming and feeding changes are found to be more significant in oblate spheroids than prolate spheroids, although the shape has a smaller effect on performance than Peclét number or surface stroke. These results suggest that nonlocal effects from viscosity variation are caused by a modification to the pressure force, as opposed to the strain rate. These results should be useful in interpreting experiments where a microswimmer affects a fluid's local rheology.

Figure 1

The nutrient-viscosity relationship for (a) various $k_{\mu }$, fixed $\xi =1$, and (b) various $\xi$, fixed $k_{\mu }>0$, and (c) fit to two real materials, using Eq. (5). Fitting the data for viscosity vs concentration of mucin, a protein, gives $k_{\mu }=-0.76$ and $\xi =0.70$ [65]; fitting the data for viscosity vs water temperature gives $k_{\mu }=1.45$ and $\xi =1.35$ [66]. The power-law coefficient $\xi$ determines the nonlinearity of relationship, while the magnitude and sign of $k_{\mu }$ determines the strength of the relationship and whether the viscosity increases or decreases near the surface of the squirmer ($C=0, c=1$), respectively. Note that $C$ is the dimensional nutrient variable and $c$ is its dimensionless counterpart.

Figure 2

Spheroidal microswimmer where $b=c$. The shape is prolate if $a>b$, spherical if $a=b$, and oblate if $a<b$.

Figure 3

Asymptotic validation and extension to strong coupling. (a) The first-order velocity improvement $U_1$ for $k_{\mu }=0.001$ is plotted, where $U=U_0+k_{\mu }{U}_1$ ($k_{\kappa }=0$). The dash-dotted line represents the low-Pe asymptotic results from Ref. [53]. (b) Extension as $k_{\mu }$ increases to strongly nonlinear regime shows $\xi$ and $k_{\mu }$ dependence of $V$, as defined in Eq. (16) to represent modification to swimming speed.

Figure 4

Effect of Pe and $\xi$ on speed ($V$) and flux ($J$) modification. [(a), (b)] $V$ for $k_{\mu }=-0.5$ and 0.5 and [(c), (d)] $J$ for $k_{\mu }=-0.5$ and 0.5, respectively. Both $V$ and $J$ monotonically increase in $k_{\mu }$. It is found that the nonlinear relationship between viscosity and nutrient concentration does not change the qualitative contours of $V$ or $J$.

Figure 5

Effect of Pe and $\xi$ on modification to power $\left({\mathcal{P}}^{*}\right)$ and efficiency $\left({\eta }^{*}\right)$. [(a), (b)] Relative power use for $k_{\mu }=\mp 0.5$ and [(c), (d)] relative low-Re swimming efficiency, as defined in Eq. (17), for $k_{\mu }\mp 0.5$.

Figure 6

Effect of Pe on a treadmill squirmer with $\xi =1.0$ and $k_{\mu }=0.5$; nutrient field with streamlines on top and pressure field on bottom. (a) $\text{Pe}=0.01$, both nutrient and pressure are symmetric due to the diffusion-dominated regime. (b) $\text{Pe}=10$, advection-dominated regime causes an asymmetry in pressure and nutrient fields which results in larger modifications to swimming speed and nutrient flux.

Figure 7

Viscosity fields for two $\xi$ ($k>0$): (a) $\xi =0.5$, (b) $\xi =2.0$. The nutrient boundary layer is similar in both regimes ($\mathrm{Pe}=10$); changing $\xi$ affects the viscosity field which determines the thickness of the viscosity boundary layer, which in turn affects the swimming and feeding performances.

Figure 8

Effect of $k_{\text{ratio}}$ and $\xi$ on modification to swimming speed $V$. [(a), (b)] $k_{\mu }=\mp 0.5$ at $\text{Pe}=0.1$ and [(c), (d)] $k_{\mu }=\mp 0.5$ at $\text{Pe}=10$. Velocity changes, while quantitatively different as Pe increases, also depends on the value of $k_{\mu }$.

Figure 9

Effect of $k_{\text{ratio}}$ and $\xi$ on modification to nutrient flux $J$. [(a), (b)] $k_{\mu }=\mp 0.5$ at $\text{Pe}=0.1$ and [(c), (d)] $k_{\mu }=\mp 0.5$ at $\text{Pe}=10$. While for $k_{\text{ratio}}=0$ nutrient flux modification is minimal, the modification is drastic as the diffusivity is allowed to vary proportionally to the viscosity variation.

Figure 10

Effect of $k_{\text{ratio}}$ on nutrient boundary layer. Parameters match those of Figs. 8 and 9. For each parameter regime, the nutrient field for $k_{\text{ratio}}=-1$ (top) or $k_{\text{ratio}}=1$ (bottom) and $\xi =2$ is shown. Changing $k_{\text{ratio}}$ affects the nutrient boundary layer, which has a significant effect on the nutrient flux.

Figure 11

Effect of ${\beta }_2/{\beta }_1$ ratio on $V$ and $J$ for diffusion-dominated ($\text{Pe}=0.01$, dots), balanced ($\text{Pe}=1$, dash), and advection-dominated ($\text{Pe}=30$, solid) regimes. Speed modification ($V$) is asymmetric with respect to stroke mode, while $J$ is approximately symmetric in ${\beta }_2/{\beta }_1$ and increases monotonically with Pe.

Figure 12

Effect of stroke mode on nutrient and pressure fields. Top (bottom) half of panels corresponds to nutrient $c$ (pressure $p$). Left panels [(a), (c)] correspond to balanced regimes ($\text{Pe}=1$) whereas right panels [(b), (d)] correspond to advection-dominated regimes ($\text{Pe}=20$). Top panels [(a), (b)] are pusher-like squirmers (${\beta }_2/{\beta }_1=-3$); bottom panels [(c), (d)] are puller-like (${\beta }_2/{\beta }_1=3$). As all squirmers are swimming in the same direction, the placement of the stagnation point either helps or hinders locomotion, but does not affect the nutrient flux modification.

Figure 13

Effect of shape. Oblate spheroids experience larger increases in speed and flux modifications than prolate spheroids, although the stroke mode dependence largely mirrors the results for spheres.

Figure 14

Effect of shape on nutrient field. Top panels show pusher-like squirmers (${\beta }_2/{\beta }_1=-5$), and the bottom panels show puller-like squirmers (${\beta }_2/{\beta }_1=5$) for various aspect ratios: (a) $\ell =0.6$, (b) $\ell =1.0$, and (c) $\ell =1.7$. The effect of stroke mode for different geometries is qualitatively similar to a sphere, suggesting shape is only a minor effect.