Microscale Hydrodynamic Cloaking and Shielding via Electro-Osmosis

We demonstrate theoretically and experimentally that injection of momentum in a region surrounding an object in microscale flow can yield both “cloaking” conditions, where the flow field outside the cloaking region is unaffected by the object, and “shielding” conditions, where the hydrodynamic forces on the object are eliminated. Using field-effect electro-osmosis as a mechanism for injection of momentum, we present a theoretical framework and analytical solutions for a range of geometrical shapes, validate these both numerically and experimentally, and demonstrate the ability to dynamically switch between the different states.

Figure 1

Illustration of the concept of hydrodynamic cloaking and shielding and the configuration used for its modeling and experimental visualization. (a) The setup consists of a microfluidic chamber containing a pillarlike object with arbitrary cross-sectional geometry confined between two parallel plates. On the bottom plate, surrounding the object, is a gate electrode designed to cloak or shield the object. The chamber is filled with a liquid subjected to a pressure gradient resulting in a flow that in the native case curves around the object and exerts a force on it. By activating and controlling the EOF around the object, the velocity field and pressure distribution can be tuned either to cloak the object (b), such that the flow outside the electrode region remains uniform, or to shield the object (c) by creating a uniform pressure in its vicinity and eliminating the force on it.

Figure 2

Schematic illustration of the depth-averaged two-dimensional system used for theoretical modeling. ${\tilde{r}}_{\mathrm{obj}}(\theta )$ and ${\tilde{r}}_{\mathrm{cl},\mathrm{sh}}(\theta )$ represent the parametrization of the object shape and the cloaking and shielding region in which $f(\theta )$ and $g(\theta )$ describe the geometrical deviations from perfectly cylindrical and circular shapes, respectively.

Figure 3

Demonstration of hydrodynamic cloaking and shielding for the case of a cylindrical object. (a)–(c) Theoretical pressure distribution (color map) and streamlines (white lines) corresponding to (a) pressure-driven flow, (b) shielding with ${\tilde{u}}_{\mathrm{ext}}=10.2\mu \mathrm{m/s}$, and (c) cloaking with ${\tilde{u}}_{\mathrm{ext}}=51.0\mu \mathrm{m/s}$, satisfying Eqs. ((8)) and ((7)) when ${\tilde{u}}_{\mathrm{EOF}}=27.2\mu \mathrm{m/s}$. (d)–(f) Experimental flow fields (blue arrows) and resulting streamlines (black lines) corresponding to (d) pressure-driven flow with ${\tilde{u}}_{\mathrm{ext}}=9.38\mu \mathrm{m/s}$, (e) shielding with ${\tilde{u}}_{\mathrm{ext}}=9.33\mu \mathrm{m/s}$, and (f) cloaking with ${\tilde{u}}_{\mathrm{ext}}=53.8\mu \mathrm{m/s}$, for ${\tilde{u}}_{\mathrm{EOF}}=27.2\mu \mathrm{m/s}$. The cloaking and shielding region (purple line) is a circular annulus with ${\tilde{r}}_a=2{\tilde{r}}_0$.

Figure 4

Demonstration of hydrodynamic cloaking and shielding for the case of a flower-shaped object. (a),(b) Theoretical pressure distribution (color map) and streamlines (white lines) corresponding to (a) shielding with ${\tilde{u}}_{\mathrm{ext}}=9.86\mu \mathrm{m/s}$, and (b) cloaking with ${\tilde{u}}_{\mathrm{ext}}=49.3\mu \mathrm{m/s}$, satisfying Eqs. (8) and (7) when ${\tilde{u}}_{\mathrm{EOF}}=26.3\mu \mathrm{m}/\mathrm{s}$. (c),(d) Experimental flow field (blue arrows) and resulting streamlines (black lines) corresponding to (c) shielding with ${\tilde{u}}_{\mathrm{ext}}=10.0\mu \mathrm{m}/\mathrm{s}$, and (d) cloaking with ${\tilde{u}}_{\mathrm{ext}}=48.1\mu \mathrm{m}/\mathrm{s}$, for ${\tilde{u}}_{\mathrm{EOF}}=26.3\mu \mathrm{m}/\mathrm{s}$. The object (white region) is defined by $g(\theta )=-\cos (5\theta )$ and $\beta =0.1$, and the cloaking and shielding region (purple line) is defined by Eqs. (9) and (10) with ${\tilde{r}}_a=2{\tilde{r}}_0$.