Intangible Hydrodynamic Cloaks for Convective Flows

Although studies of hydrodynamic cloaking have attracted considerable attention, most pioneering studies have so far focused on creeping flows, in which nonlinear convective terms are neglected and cloaks are fabricated with inhomogeneous or anisotropic metamaterials. Here we discover that intangible uniform-viscosity hydrodynamic cloaks, which are capable of hiding obstacles immersed in narrow-gap Navier-Stokes flows with non-negligible convection, can be possibly designed. Via the convection-diffusion-balance method and the realizable thermally controlled tactic, the diffusion effect exerted by the obstacle surface can be balanced by the convection counterpart exerted by the manipulation of temperature-dependent viscosity inside the intangible cloak. This analysis has been validated by finite-element computational simulations that tackle cloaked-cylinder flows for Reynolds numbers in the laminar regime. The proposed study may lay the foundation for attempts to further explore hydrodynamic cloaks and possibly other metamaterial-related devices.

Figure 1

Schematic models of a hydrodynamic cloaks. Geometry with height ($H$) $\times$ width ($W$) $\times$ depth ($D$) = $1\mathrm{cm}\times 1\mathrm{cm}\times 50\mu \mathrm{m}$, $R_1=2\mathrm{mm}$, and $R_2=4\mathrm{mm}$. The radius of the obstacle and the outside radius of the cloak are $R_1$ and $R_2$, respectively. Water with the dynamic viscosity $\mu =1$ mPa s and the density $\rho =997.1\mathrm{kg}/{\mathrm{m}}^3$ at room temperature are chosen as the working fluid. (a) Three-dimensional (3D) model with parabolic velocity profiles on the $y$-$z$ plane. (b) Two-dimensional (2D) model.

Figure 2

Velocity distributions superimposed with streamlines (black color) and isobars (white color) on the $z=0$ plane at $\mathrm{Re}=1.03$. (a) Obstacle absent, (b) obstacle existent, (c) cloaking.

Figure 3

Velocity distributions superimposed with streamlines (black lines) and isobars (white lines) of the hydrodynamic cloak on the $z=0$ plane at various laminar large $\mathrm{Re}$’s.

Figure 4

Comparing $y$-component velocities near the outside cloak at $a(1.1R_2,0)$, $b(0,1.1R_2)$, and $c(0,-1.1R_2)$ with inlet velocity ($v_{\mathrm{in}}$) on the $z=0$ plane versus $\mathrm{Re}$. Geometry with $H\times L\times D=1\mathrm{cm}\times 1\mathrm{cm}\times 50\mu \mathrm{m}$, $R_1=2\mathrm{mm}$, and $R_2=4\mathrm{mm}$.

Figure 5

Cloaking at $\mathrm{Re}=4.51$. Geometry with $H\times L\times D=1\mathrm{cm}\times 1\mathrm{cm}\times 50\mu \mathrm{m}$, $R_1=2\mathrm{mm}$, $R_2=3.3\mathrm{mm}$, and $R_3=4\mathrm{mm}$. (a)–(c) Viscosity distributions of water on (a) $z=\pm 0.4D$ plane, (b) $z=\pm 0.2D$ plane, and (c) $z=0$ plane. (d)–(f) Velocity distributions of water on (d) $z=\pm 0.4D$ plane, (e) $z=\pm 0.2D$ plane, and (f) $z=0$ plane. (g) Velocity distributions at five different $y$ positions on the $z=0$ plane. Five red lines in the inset-figure model correspond to five legend-caption positions, respectively. (h) Thermally manipulating the viscosity of the cloak prescribed at $z=\pm 0.5D$.

Figure 6

Cloaking at $\mathrm{Re}=27.97$. Geometry with $H\times L\times D=5\mathrm{cm}\times 3\mathrm{cm}\times 50\mu \mathrm{m}$, $R_1=2\mathrm{mm}$, $R_2=3.3\mathrm{mm}$, and $R_3=9.6\mathrm{mm}$. (a)–(c) Temperature distributions of water on (a) $z=\pm 0.4D$ plane, (b) $z=\pm 0.2D$ plane, and (c) $z=0$ plane. (d)–(f) Velocity distributions of water on (d) $z=\pm 0.4D$ plane, (e) $z=\pm 0.2D$ plane, and (f) $z=0$ plane.

Figure 7

Vorticity distributions ($\mathrm{\Omega }=|\mathrm{\nabla }\times \mathbf{u}|$) at $\mathrm{Re}=4.51$ on various planes for the obstacle-existent case.

Figure 8

(a),(b) Pressure distributions on the cylinder surface parameterized in several Reynolds numbers on the $z=0$ plane for the obstacle-existent case. (c),(d) Velocity and vorticity distributions on the $z=0$ plane at $\mathrm{Re}=41.28$. (e),(f) Velocity and vorticity distributions on the $z=0$ plane at $\mathrm{Re}=159.38$. Geometry with $H\times L\times D=1\mathrm{cm}\times 1\mathrm{cm}\times 50\mu \mathrm{m}$, $R_1=2\mathrm{mm}$.