Empirical resistive-force theory for slender biological filaments in shear-thinning fluids

Many cells exploit the bending or rotation of flagellar filaments in order to self-propel in viscous fluids. While appropriate theoretical modeling is available to capture flagella locomotion in simple, Newtonian fluids, formidable computations are required to address theoretically their locomotion in complex, nonlinear fluids, e.g., mucus. Based on experimental measurements for the motion of rigid rods in non-Newtonian fluids and on the classical Carreau fluid model, we propose empirical extensions of the classical Newtonian resistive-force theory to model the waving of slender filaments in non-Newtonian fluids. By assuming the flow near the flagellum to be locally Newtonian, we propose a self-consistent way to estimate the typical shear rate in the fluid, which we then use to construct correction factors to the Newtonian local drag coefficients. The resulting non-Newtonian resistive-force theory, while empirical, is consistent with the Newtonian limit, and with the experiments. We then use our models to address waving locomotion in non-Newtonian fluids and show that the resulting swimming speeds are systematically lowered, a result which we are able to capture asymptotically and to interpret physically. An application of the models to recent experimental results on the locomotion of Caenorhabditis elegans in polymeric solutions shows reasonable agreement and thus captures the main physics of swimming in shear-thinning fluids.

Figure 1

A straight filament of length $2\ell$ and cross-sectional radius $a$ may translate in a fluid along its length (velocity ${\mathbf{u}}_{\parallel }$) or perpendicular to it (${\mathbf{u}}_{\perp }$). The axis of the cylinder is along the $\widehat{\mathbf{z}}$ direction while $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ are in the cross section.

Figure 2

Inverse of the drag coefficient correction factor, $R_E$, as obtained experimentally by measuring the sedimentation speed of cylindrical rods under gravity in a variety of shear-thinning fluids [41]. Results from both vertical ($\parallel$) and horizontal ($\perp$) rod orientations are shown and different fluids are marked by different symbols. Here $n$ is the power index of the fluid, ${\mathrm{\Gamma }}^{-1}$ the critical shear rate for transition to shear-thinning behavior, $U$ the velocity of the rod, and $d_s$ a relevant length scale characterizing the rod (see text). Inset: empirical formula, Eq. (18), proposed to fit all data [41]. Adapted and reprinted from Ref. [41] with permission from Elsevier.

Figure 3

Non-Newtonian correction factors as a function of the dimensionless flow shear rate for the experimental results [(a) $R_E]$ and the Carreau fluid model [(b) $R_C]$. Increasing shear-thinning indices $n$ are shown from light gray to dark gray with the Newtonian limit $n=1$ shown in black.

Figure 4

Sinusoidal traveling waveform as a model for the locomotion of flagellated eukaryotic cell. Swimming, with speed $U$, is in the opposite direction to the traveling wave, with wave speed $V$. See text for notation.

Figure 5

The numerical Newtonian results are compared to the analytic Newtonian results, (a) small-amplitude analytical expansion (solid line, $\epsilon \ll 1$) compared to numerical results (red diamonds) and (b) full analytical result of Eq. (35) (dashed line) compared to numerical results (red diamonds).

Figure 6

Ratio between the non-Newtonian and Newtonian swimming speeds, $U_{NN}/U_N$, as a function of the properties of the fluid. (a) Speed ratio with fixed $n=0.3$ for a range of amplitudes $\epsilon$ plotted against the critical fluid time $\mathrm{\Gamma }$. (b) Speed ratio with fixed value of $\mathrm{\Gamma }=\pi /20$ for a range of wave amplitudes $\epsilon$ plotted against the power index of the fluid, $n$. The swimming speed is always reduced in a shear-thinning fluid.

Figure 7

Ratio of the non-Newtonian to Newtonian swimming speeds $U_{NN}/U_N$ for fixed critical time $\mathrm{\Gamma }=\pi /40$, for a range of power indices $n$ as a function of the flagellum amplitude $\epsilon$, (a) experimental model ($R_E$) and (b) Carreau model ($R_C$).

Figure 8

First-order corrections to the Newtonian velocity scaled by the Newtonian speed, $U_1/U_0$, for the Carreau model (solid line) and the experimental model (dashed line) as a function of the wave amplitude $\epsilon$. The correction is always negative indicating a decrease in the swimming speed.

Figure 9

Dimensionless shear rates along a waving flagellum as a function of the dimensionless position $x$ along the flagellum for the drag problem (dashed lines) and the thrust problem (solid lines), shown for a small amplitude ($\epsilon =0.25$) and large amplitude ($\epsilon =1$) swimmer.

Figure 10

Experimental results from Ref. [52] (open symbols with error bars) plotted together with our simulation results (line or filled symbols). Newtonian swimming speeds are shown in (a) and non-Newtonian swimming speeds in (b). The simulation results are represented by line symbols for thin rod results and filled symbols for fat rod results. The lines in each of the different colours are straight lines of best fit to their matching color symbol. Each shear-thinning simulation data point shares fluid and swimmer properties with the red experimental shear-thinning swimming speed at that particular effective viscosity.