Asynchronous oscillations of rigid rods drive viscous fluid to swirl

We present a minimal system for generating flow at low Reynolds number by oscillating a pair of rigid rods in silicone oil. Experiments show that oscillating them in phase produces no net flow, but a phase difference alone can generate rich flow fields. Tracer particles follow complex trajectory patterns consisting of small orbital movements every cycle and then drifting or swirling in larger regions after many cycles. Observations are consistent with simulations performed using the method of regularized Stokeslets, which reveal complex three-dimensional flow structures emerging from simple oscillatory actuation. Our findings reveal the basic underlying flow structure around oscillatory protrusions such as hairs and legs as commonly featured on living and nonliving bodies.

Figure 1

Schematics of the experimental setup viewed from (a) the side and (b) the top. Servo-driven horizontal rods (rods 1 and 2) of length $L$ and diameter $\epsilon$ oscillate at distance $H$ below the air–silicone oil interface. The two pivot points are separated by distance $D$. ${\theta }_1$ and ${\theta }_2$ are the angles of rod 1 and rod 2 with respect to the $x$ axis. (c) Three floating tracer particles (top, middle, and bottom) trace the fluid flow in the $x\text{−}y$ plane. (d) Angular variations of the rods within one cycle for the three representative cases.

Figure 2

Experimental trajectories of the three tracer particles for a duration of ten periodic cycles in (a) case 1, (b) case 2, and (c) case 3. The initial and final positions are marked with a circle and square, respectively.

Figure 3

Experimental trajectories of the three tracer particles sampled after each cycle in (a) case 1, (b) case 2, and (c) case 3 for a duration of ten periodic cycles. For each case, three separate runs were repeated. Arrows point in the general direction of flow.

Figure 4

(a) A side-view diagram of the rod submerged beneath an air–silicone oil interface. (b) A side-view diagram of a symmetric pair of rods completely immersed in silicone oil. The flow below the plane of symmetry is a reflection of that above the plane and is equivalent to the flow beneath the interface.

Figure 5

Particle trajectories starting at different positions in case 1 (top panel), case 2 (middle panel), and case 3 (bottom panel). Experimental data (black curves) are compared with simulations (gray curves) within one oscillatory cycle. Open circles represent final positions after one complete cycle. Arrows indicate the sense of rotation.

Figure 6

Net motion of particle tracers in case 2 (top panel) and case 3 (bottom panel). Experimental data of particle positions sampled every periodic cycle are connected by black lines, with circle markers representing the positions every five cycles. Blue arrows indicate simulated displacements after five cycles. The black bars represent the initial configuration of the rods. Note that the tracers are above the plane of the rods and can appear to pass through them.

Figure 7

Vertically upward component of the displacement field after one cycle at the plane of the rods, located at a depth $z=-6$ mm below the plane of symmetry. The vertical component $d_z$ is scaled by (a) $L$ or (b) $\left|d\right|$, the magnitude of the total displacement, in case 2. [(c), (d)] Corresponding results in case 3. The black bars represent the initial configuration of the rods.

Figure 8

Simulated displacements of the top particle (initially located at $x/L=D/2,y/L=1.5)$ after one cycle depending on [(a), (b)] the phase difference $\varphi$, (c) amplitude $A$, and (d) distance $D$ between the two pivot points. In panel (b), the displacement direction is measured by the angle ${\theta }_d$ it makes with the $x$ axis. The angle ${\theta }_d$ switching from zero to $\pi$ corresponds to reversal in the direction of displacement. The circle cursor is for case 2 and the square cursor is for case 3.