Three-sphere swimmer in a nonlinear viscoelastic medium

A simple model for a swimmer consisting of three colinearly linked spheres attached by rods and oscillating out of phase to break reciprocal motion is analyzed. With a prescribed forcing of the rods acting on the three spheres, the swimming dynamics are determined analytically in both a Newtonian Stokes fluid and a zero Reynolds number, nonlinear, Oldroyd-B viscoelastic fluid with Deborah numbers of order one (or less), highlighting the effects of viscoelasticity on the net displacement of swimmer. For instance, the model predicts that the three-sphere swimmer with a sinusoidal, but nonreciprocal, forcing cycle within an Oldroyd-B representation of a polymeric Boger fluid moves a greater distance with enhanced efficiency in comparison with its motility in a Newtonian fluid of the same viscosity. Furthermore, the nonlinear contributions to the viscoelastic constitutive relation, while dynamically nontrivial, are predicted a posteriori to have no effect on swimmer motility at leading order, given a prescribed forcing between spheres.

Figure 1

The Najafi-Golestanian three-sphere swimmer. Three colinear, identical, neutrally buoyant spheres are connected by rods of mean length $D$ with periodic, but, in general, nonreciprocal, oscillations given by ${\xi }^R\left(t\right)$ and ${\xi }^F\left(t\right)$. The force acting on each sphere is denoted ${\mathbf{f}}^{\left(\alpha \right)}\left(t\right)$, where $\alpha \in \{1,2,3\}$, and the net displacement over one beat cycle is denoted $\Delta$ for the Newtonian problem and ${\Delta }_v$ for the viscoelastic analog.

Figure 2

The dimensionless net displacement over one beat cycle, $\Delta$ for a Newtonian fluid and ${\Delta }_v$ for the viscoelastic medium, is plotted as a function of rod phase difference, $\varphi$, where the internal forcing of sphere 1 is $\sin \left(t\right)$ and of sphere 3 is $\sin (t-\varphi )$. Here, $\epsilon =0.1$, $\delta =0.01$, $\Gamma =0$, as for an upper convected Maxwell fluid, and $\Lambda$ ranges from 0, i.e., the Newtonian case (given by the dashed line) to 2 in steps of $0.5$, where each step increases the amplitude of the curve.