Magnetic Levitation Stabilized by Streaming Fluid Flows

We demonstrate that the ubiquitous laboratory magnetic stirrer provides a simple passive method of magnetic levitation, in which the so-called “flea” levitates indefinitely. We study the onset of levitation and quantify the flea’s motion (a combination of vertical oscillation, spinning and “waggling”), finding excellent agreement with a mechanical analytical model. The waggling motion drives recirculating flow, producing a centripetal reaction force that stabilized the flea. Our findings have implications for the locomotion of artificial swimmers and the development of bidirectional microfluidic pumps, and they provide an alternative to sophisticated commercial levitators.

Figure 1

(a) Experimental setup showing the flea levitating at $z>z_b$. (b) Measured flea angle $\theta$ versus drive magnet angle ${\omega }_{d}t$, for a range of ${\omega }_d$. The dotted line corresponds to the synchronously spinning flea when ${\omega }_d<{\omega }_{\uparrow }$. (c) Overhead 2D images of the flea have been stacked to form three-dimensional space-time spirals, visualizing the waggling and rotational motion of the flea over a 2 s period, at ${\omega }_d=73.6$, 133, and $201\mathrm{rad}{\mathrm{s}}^{-1}$ (from top to bottom). Colors correspond to the relevant curves in panel (b). (d) Waggle speed (${\omega }_w$), spin speed (${\omega }_s$), and waggle amplitude ($A$) versus ${\omega }_d$. Dashed down arrows indicate that the threshold drive speed ${\omega }_d={\omega }_{\uparrow }$ is reached, as ${\omega }_d$ is increased from 0, when the flea jumps up from its initial position on the base $z=z_b$ to a stable levitation point at $z>z_b$. The colors of data points and arrows correspond to $z_b=22$ (black), 26 (red), 30 (green), and 34 (blue) mm. Black lines are analytic solutions using the experimental value ${\omega }_{\downarrow }=63\mathrm{rad}{\mathrm{s}}^{-1}$.

Figure 2

(a) Flea levitating in castor oil (drive speed ${\omega }_d$ labeled). (i) A still photograph of the levitating flea. (ii)–(v) Projections of the central pixel column [red line in (i)] over 0.5 s. (b) Mean vertical position $⟨z⟩$ of the flea versus ${\omega }_d$ for a range of $z_b$. The dashed down arrow shows where ${\omega }_d={\omega }_{\downarrow }$ for all experiments. The solid black line gives the mean vertical position calculated by numerical integration of Eqs. (2) and (3). Plotted symbols show experimental results: synchronous spinning (hollow); asynchronous, levitating (colored solid); asynchronous, nonlevitating (gray solid). The blue shaded region denotes synchronous spinning. The boundary is calculated analytically using ${\omega }_{\uparrow }=M(z_b)/D$. The gray shaded region denotes asynchronous, nonlevitating spinning.

Figure 3

(a) The solid line shows the experimentally determined path of the flea following displacement of the drive magnet (${\omega }_d=73\mathrm{rad}{\mathrm{s}}^{-1}$) 14 mm from the origin (averaged over 7 frames). The images are projections of the waggle motion as the flea spirals towards the rotation axis of the drive magnet, taken at 15 points, at intervals of 0.14 s (34 frames), corresponding to the positions indicated by circles on the spiral. Red dots indicate the axis of the drive magnet relative to the flea. The inset diagram indicates the “head” and “tail” of the asymmetrically waggling flea when the rotation axes of the two magnets are displaced horizontally. (b) 1 s projections of suspended particle paths (directionality given by black lines with white arrowheads), driven by our “artificial waggler” (the ends of the rod are oscillating in and out of the page). The left and right sides of the image are taken from separate experiments at ${\mathrm{Re}}_s=11.7\pm 0.4$ (left) and ${\mathrm{Re}}_s=400\pm 12$ (right). An image of the brass bar used in the experiment has been superimposed for clarity. (c) Plots of the simulated fluid flows: the left and right plots represent the same ${\mathrm{Re}}_s$ as in experimental flows shown in (b). The color map and arrows indicate the vorticity and direction of flows, respectively.