Realizing the thinnest hydrodynamic cloak in porous medium flow

Abstract

Transformation mapping theory offers us great versatility to design invisible cloaks for the physical fields whose propagation equations remain invariant under coordinate transformations. Such cloaks are typically designed as a multi-layer shell with anisotropic material properties, which makes no disturbance to the external field. As a result, an observer outside the cloak cannot detect the existence of this object from the field disturbances, leading to the invisible effect in terms of field prorogation. In fact, for many prorogating fields, at a large enough distance, the field distortion caused by an object is negligible anyway; thus, a thin cloak is desirable to achieve near-field invisibility. However, a thin cloak typically requires more challenging material properties, which are difficult to realize due to the huge variation of anisotropic material parameters in a thin cloak region. For a flow field in a porous medium, by applying the bilayer cloak design method and integrating the inner layer with the obstacle, we successfully reduce the anisotropic multi-layer cloak into an isotropic single-layer cloak. By properly tailoring the permeability of the porous medium, we realize the challenging material parameters required by the ultrathin cloak and build the thinnest shell-shaped cloak of all physical fields up to now. The ratio between the cloak's thickness and its shielding region is only 0.003. The design of such an ultrathin cloak may help to achieve the near-field invisibility and concealment of objects inside a fluid environment more effectively.

Graphical abstract

Figure 1

Schematic of the designed cloak in the porous medium domain (A) An impermeable obstacle (r = R₁) in the steady flow and streamlines in the background are distorted. (B) The obstacle with the cloaking shell (R₁ < r < R₂) (marked as the purple circular shell), and streamlines in the background remain parallel and uniform. (C) The curve represents the ratio of permeability in the background and the cloaking shell k_II/k_III versus the thickness ratio R₂/R₁. The orange star indicates the value of the cloak realized in our experiments. The inset is the cloak with uniform pressure contours, denoted as black dashed lines.

Figure 2

The ultrathin cloak realized by the porous medium (A) The structure of the ultrathin cloak composed of a shell of free space and the background medium of isotropic pillars. (B) Detailed view of the cloak structure, in which the yellow color represents the free-space shell (region II) with the thickness of R₂ − R₁. (C) Plot of the normalized numerical velocity changing with x in the case of cloak and only obstacle. All the numerical velocities in the two cases are normalized by the velocity at x = 0 in the case of cloak, and x is the position of the velocity. The inset shows the observation lines in the two cases. (D) The pressure distribution at the observation lines. The inset about the pressure curve indicates the zero-pressure gradient in the cloaking shell. The positions of observation lines in the two cases are shown in the inset at the top right.

Figure 3

Numerical and experimental validation of the ultrathin cloak in hydrodynamics (A–C) Parametric simulations of velocity fields with the required permeability in the bare case (A), only obstacle (B), and the cloak (C). (D–F) Simulated velocity field in the bare case (D), only obstacle (E), and the cloak (F)with the porous medium of isotropic pillars. (G–I) The experimentally captured streamlines in the porous medium for the bare case (G), only obstacle (H), and the cloak case (I). (J) The fluorescence microscopy setup. (K) The PDMS microfluidic device, and the inset shows the thickness of the free-space shell of 60 μm and the pillar-to-pillar spacing of 10 μm. (L) Comparison of the velocity between experiment and simulation. The velocities in simulation are normalized by the numerical value at x = 0 in the case of cloak, and the velocity in experiment is normalized by the experimental value at x = 0 in the case of cloak. The device in the experiment is scaled down twice so that the simulation and experiment data can be collapsed in the same plot.

Figure 4

The numerical simulations of the 3D ultrathin hydrodynamic cloak (here R₁ = 1 mm and R₂ = 1.003 mm) (A–F) The streamlines (A) and the iso-pressure surface (B) are straight with the 3D cloak, while they are significantly distorted near the object without the 3D cloak in (D) and (E). The 3D images with and without the 3D ultrathin cloak are shown in (C) and (F). Since the thickness of the cloak is only 0.3% of the cloaked object, the size difference between (C) and (F) cannot be distinguished by the naked eye.