Self-propulsion of a freely suspended swimmer by a swirling tail in a viscoelastic fluid

Microscopic organisms must frequently swim through complex biofluids, including bacterial biofilms and the mucus lining of the upper respiratory tract. Recently, there has been great interest in understanding how the non-Newtonian behavior of the fluids, in particular fluid elasticity, can enable methods of self-propulsion that otherwise would not work in a Newtonian fluid. We present such a swimmer that consists of two spheres of unequal size that rotate in opposite directions. The swimmer is force- and torque-free and placed in an elastic fluid. Using a combination of analytical theory and numerical simulations, we show that this model swimmer, which has zero propulsion in a Newtonian fluid under Stokes flow, swims in the direction of the larger of the two spheres in an elastic fluid. We show that the speed of swimming increases nearly linearly with the Deborah number De or primary normal stress coefficient ${\mathrm{\Psi }}_1$, which is an appropriate measure of the elasticity of the fluid and for $\text{De}\le 1$ is also nearly linear in the concentration of polymer in the fluid. The dependence of the swim speed on the relative size of the two spheres is nonmonotonic, exhibiting a maximum at a size ratio of about 0.75. By analyzing the forces acting on the swimmer and the surrounding flow field, we find that propulsion is driven by thrust due to pressure applied along the swimmer. This thrust originates from flow advection driven by hoop stresses surrounding the faster-spinning smaller sphere. We compare our predictions to experimental measurements of swimming speeds for the bacterium E. coli, which swims via a rotating flagellar bundle and counterrotating body, and find that the speeds predicted by our analysis are remarkably close to the speed increase E. coli experiences in viscoelastic fluids. Finally, we conclude our work by showing how our analysis can be extended to different swimmer configurations and gaits, as long as the propulsion is driven by swirl alone.

Figure 1

(a) Depiction of a swimming E. coli cell (left) and the schematic of a coarse-grained model swimmer (right) that can be tuned to reproduce the azimuthal flow field created by the E. coli rotating flagellar bundle and counterrotating body. (b) Dimensions and coordinate system for the model two-sphere swimmer.

Figure 2

(a) Swimming speed $U$ as a function of the Deborah number De and the viscosity ratio $\beta$ for $h^{*}=0.05$ and $r^{*}=0.5$. Markers connected by solid lines denote simulation data, while the dashed lines denote the low-De theory, i.e., Eqs. (12) and (13). As indicated by the black arrow, as one moves down the set of curves, the value of $\beta$ increases from 0.25 to 0.75, i.e., an increasing contribution from the Newtonian solvent. The theory and simulations show that the speed of the two-sphere swimmer increases with increasing fluid elasticity (De) and increasing polymer concentration (decreasing $\beta$). The inset shows the same set of data but with the speed $U$ now normalized by $1-\beta$ to show that all data collapses onto a single curve for low De, as predicted by the theory. (b) Swimming speed $U$ as a function of time for $h^{*}=0.05$, $r^{*}=0.5$, and $\beta =0.5$.

Figure 3

(a) Swimming speed $U$ as a function of the size ratio $r^{*}$ for $h^{*}=0.05$, $\text{De}=1$, and $\beta =0.5$. Both theory and simulations predict a nonmonotonic dependence of $U$ on $r^{*}$, with $U$ vanishing in the case of $r^{*}=0$ and 1, while reaching a maximum at $r^{*}\approx 0.75$. (b) Theoretical prediction for the swimming speed $U$ as a function of the size ratio $r^{*}$ for $\text{De}=1$, $\beta =0.5$, and a range of values of $h^{*}$. As $h^{*}$ is increased, $U$ decreases in magnitude for all $r^{*}$. However, the functional dependence of $U$ with respect to $r^{*}$ is not strongly affected by the value of $h^{*}$.

Figure 4

Normalized swimming speed $U/\text{De}(1-\beta )$ as a function of the gap size $h^{*}$ for a size ratio of $r^{*}=0.75$. The numerical simulations were conducted at $\text{De}=1$ and $\beta =0.5$. While the theory predicts a slight increase in the swimming speed before a monotonic decrease with increasing gap size, the simulation results predict a purely decreasing function of gap size. As expected, the two predict the same result for large values of $h^{*}$ since the theory becomes more accurate as the separation increases.

Figure 5

Contributions to the net force acting on the swimmer as a function of De for $h^{*}=0.05$, $r^{*}=0.5$, and $\beta =0.5$. (a) Total integrated force contributions acting on each sphere. Solid lines denote forces on the smaller sphere, while dash-dotted lines denote force on the larger sphere. The plot indicates that pressure acting on the larger sphere creates a positive (i.e., propulsive) contribution to the net hydrodynamic force, while contributions from pressure, viscous, and elastic stresses acting on the smaller sphere are negative and thus oppose the swimming motion. (b) Total integrated force contributions acting over the entire swimmer (i.e., summing individual forces acting on the large and the small sphere). This plot indicates that the net effect of pressure is propulsive, while those of viscous and elastic stresses are resistive.

Figure 6

Contour plots in the $x\text{−}z$ plane at $y=0$ of (a) the pressure and (b) the polymer stress energy ${\tau }_{ii}^p$ for the simulation at $\text{De}=1$, $\beta =0.5$, $h^{*}=0.05$, and $r^{*}=0.5$. Streamlines indicate the direction of the radial flow field in the comoving frame of reference; the gray arrow in each panel indicates the swimming direction in the laboratory frame. Given the results shown in Fig. 5, propulsion appears to created by a buildup of pressure towards the rear of the swimmer that originates from flow pulled inward from the faster-spinning smaller sphere. In (b) we see the development of hoop stresses from the rotation of the smaller sphere that otherwise act to impede motion according to Fig. 5.

Figure 7

Comparison of the pressure $p$ predicted by lubrication theory [i.e., Eq. (18)] to that from our numerical simulation as a function of the radial coordinate $r$ in cylindrical coordinates for the case of $r^{*}=0.5$, $h^{*}=0.05$, $\beta =0.5$, and $\text{De}=1$.

Figure 8

Illustration of one particular example of an axisymmetric swimmer consisting of two counterrotating bodies of revolution.

Figure 9

(a) Schematic of an axisymmetric swimmer with a spherical head and prolate spheroidal tail oriented along the vertical $z$ direction. (b) Swimming velocity $U_z$ as a function of the ratio $D_L{l}_L^4/D_S{l}_S^4$ for the case of $c/R_L=0.75$, De = 1, $\beta =0.5$, and $h^{*}=0.05$. From far-field hydrodynamic interactions, we predict that the swimmer shown in (a) should self-propel in the direction of the spherical head if $D_L{l}_L^4/D_S{l}_S^4>1$. From this set of simulations, we find that $U_z$ changes sign at a value of approximately 1.3; we expect this value to converge to unity as $h^{*}$ increases and the far-field approximation becomes more accurate.

Figure 10

Illustration of a proposed three-sphere swimmer that translates in a viscoelastic fluid by rotating the two smaller spheres located at its rear.

Figure 11

Velocity $U_z$ of the three-sphere swimmer illustrated in Fig. 10 at De = 1, $\beta =0.5$, and $h^{*}=1$. In (a) the velocity is plotted as a function of the angle $\alpha$ and the size ratio $r^{*}$ and indicates that the speed of the swimmer $U=|U_z|$ is maximized by selecting the smallest angle and largest size ratio possible. Note that the range of $\alpha$ values considered for each curve is given by $\alpha ={\alpha }_{\min }=1/(1+h^{*}+r^{*})$ to $\alpha =\pi /2$, where ${\alpha }_{\min }$ is the smallest achievable angle for a given size ratio (i.e., taking into account the finite size of the spheres). In (b) the velocity is plotted as a function of $r^{*}$, assuming $\alpha ={\alpha }_{\min }$.