Two-sphere swimmers in viscoelastic fluids

We examine swimmers comprising of two rigid spheres which oscillate periodically along their axis of symmetry, considering when the spheres oscillate both in phase and in antiphase, and study the effects of fluid viscoelasticity on the swimmers' motion. These swimmers display reciprocal motion in Newtonian fluids and consequently, no net swimming is achieved over one cycle in such fluids. Conversely, in viscoelastic fluids, we find that the effect of viscoelasticity acts to propel the swimmers forward in the direction of the smaller sphere when the two spheres are of different sizes. Finally, we compare the motion of rigid spheres oscillating in viscoelastic fluids with elastic spheres in Newtonian fluids where we find similar results.

Figure 1

Schematic of the two-sphere swimmer. The spheres, labeled ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$, have radii $a\alpha$ and $a$, respectively $\left(\alpha >1\right)$. The spheres are (on average) a distance $d_0$ apart and ${\mathbf{e}}_{\parallel }$ is the unit vector pointing from ${\mathcal{B}}_1$ to ${\mathcal{B}}_2$.

Figure 2

Schematic showing one complete cycle for the two swimmers: (a) The in-phase swimmer maintains the distance between the spheres as it moves forward. (b) In the antiphase swimmer, the spheres converge and diverge. The steps in gray show the transition from one half cycle to the next. The red dot marks the position of the swimmer.

Figure 3

The swimming speed coefficient $\mathcal{U}$ is plotted with variation in the clearance ${\mathrm{\Delta }}_c$ between the two spheres for different size ratios $\alpha$. The square symbols (connected by dashed lines) represent the antiphase swimmer and the circles (connected by solid lines) represent the in-phase swimmer. All quantities are dimensionless.