Air-propelled, herringbone-textured platelets

Solids can be levitated by actively nourishing an air cushion beneath them, using air blown through a porous substrate. It has recently been demonstrated that introducing asymmetry in the airflow close to a hovercraft can lead to its propulsion by viscous entrainment. In this work, we focus on the major consequence of a simple modification in the set-up: instead of engraving the texture on the substrate as in previous works, we directly carve it into the hovercraft. Provided that air-flux is strong enough, propulsion at any Reynolds number is observed. However, motion happens in the direction opposite to the viscous scenario. To understand experimental observations, an analytical model is proposed to account for the force of propulsion and the pressure needed for levitation, emphasizing asymptotic situations at low and high Reynolds numbers, further confirmed by numerical simulations.

Figure 1

Image sequences showing trajectories of levitating plates; time between pictures is 0.5 s. Hovercrafts are 30 mm long and 160 µm thick; their velocity is null in the top picture. A herringbone texture (highlighted by orange lines) is engraved either on the substrate (a) or on the levitating object (b). Patterns are identically oriented, but the motion of the hovercraft takes place in opposite directions. Blue arrows show the movement of the center of mass of the plates. Typical final speeds are around 6 cm/s in both cases.

Figure 2

(a) Side and (b) top view of the set-up. An overpressure $P_2$ above atmospheric pressure $P_0$ is imposed in a Plexiglas box whose porous top has holes with radius $r$, spacing $d$, and length $e$. Air flows through the pores (blue arrows) and interacts with a textured glass plate (length $a$, width $b$, and thickness $c$). The bottom side of the plate is made of channels arranged in a herringbone design (opening angle 2α, height $h$, and thickness λ spaced by $w$), in which the overpressure is $p$. Air velocity at a position $x$ in a channel has horizontal and vertical components, $u$ and $v$. After being redirected by the channels, air escapes in the direction pointed by the red arrows. Propulsion of the slider at a velocity V (yellow arrow) happens in the opposite direction.

Figure 3

Cross-section of the channel domain used in the model. Flow happens below the solid of thickness $c$ in a volume of height $h$ and length $l=b/\left(2\sin \alpha \right)$. The horizontal velocity of air is $u$($x$, $y$) and the vertical velocity is $v$($x$, $y$). The overpressure in the reservoir is $P_2$ and inside the channel is $p$($x$, $y$), larger than the atmospheric pressure $P_0$.

Figure 4

Channel pressure $p$ as a function of the position $x$. High Reynolds asymptotic cases are plotted in red, while low Reynolds cases are in blue. Dashed lines stand for short channels, while solid lines correspond to long channels. The values chosen for this graph are the minimum and maximum numerical Reynolds numbers (0.25–295), corresponding to the minimum and maximum values of K (0.05–927) for our experiments. Only for Re $\ll$ 1 and K $\gg$ 1 (blue dashed line), we used $K=30$ to emphasise the nonzero and nonmonotonic behavior of the pressure.

Figure 5

(a) Ratio of the experimental total force of propulsion F to the force needed to enable levitation, $F_g$, as a function of the scaling factor appearing in the theoretical expression (13). The high Reynolds numbers area is colored in red, while the low Reynolds numbers area is in blue; the short channel limit is shown by the dashed lines while the long channel asymptotic case is displayed by the thin lines. The symbol shape shows the corresponding parameter K: circles (K < 1) and triangles (K > 1). Given our experimental characteristics, it varies from 0.05 to 927. The symbol color qualitatively shows the corresponding numerical Reynolds number of data, from blue (Re $\ll$ 1) to red (Re $\gg$ 1). It is found to range from 0.25 to 295. (b) Experimental pressure $P_2-P_0$ imposed by the operator to observe levitation as function of its theoretical expression from Eq. (15c), shown by the dashed-dotted line. Symbol shapes and colors are the same as in (a). (c) Ratio of the numerical total force of propulsion F to the force needed for levitation $F_g$ as function of the scaling factor appearing in the theoretical Eq. (13). Symbols, colors, and lines are the same as in (a). (d) Numerical pressure $P_2-P_0$ required to induce levitation as a function of Eq. (15c), indicated by the dashed-dotted line. Horizontal lines show the long channels limit (14). Symbols, shape, and color are the same as in (a).

Figure 6

The functions (5/6)$\left\{{\int }_0^1{\left[g^{\prime }\left(y\right)\right]}^{2}dy\right\}$(Re) in (a), $(1/6)\left[g^{″}\left(1\right)\right]$(Re) in (b), and $(1/12)[g^{″}\left(1\right)-g^{″}\left(0\right)]$(Re) in (c). Symbols are numerical values, solid lines represent the analytical approximations, Eqs. (B4, B5, B6).