Simple swimmer at low Reynolds number: Three linked spheres

We propose a very simple one-dimensional swimmer consisting of three spheres that are linked by rigid rods whose lengths can change between two values. With a periodic motion in a nonreciprocal fashion, which breaks the time-reversal symmetry as well as the translational symmetry, we show that the model device can swim at low Reynolds number. This model system could be used in constructing molecular-sized machines.

Figure 1

Three linked spheres connected by two arms of negligible thickness.

Figure 2

Complete cycle of the proposed nonreciprocal motion of the swimmer, which is composed of four consecutive time-reversal breaking stages: (a) the left arm decreases its length with the constant relative velocity $W$, (b) the right arm decreases its length with the same velocity, (c) the left arm opens up to its original length, and finally, (d) the right arm elongates to its original size. By completing the cycle the whole system is displaced to the right side by an amount $\Delta$.

Figure 3

An auxiliary (fictitious) movement in which the right arm has a constant length $\delta$ while the left arm changes its length from $D$ to $D-ϵ$. During this movement the middle sphere will be displaced by an amount ${\Delta }_f\left(\delta \right)$.

Figure 4

Dimensionless displacement of the swimmer in a complete cycle as a function of the dimensionless relative displacement between neighboring spheres.