A metamaterial-free fluid-flow cloak

The model of ideal fluid flow around a cylindrical obstacle exhibits a long-established physical picture where originally straight streamlines will be deflected over the whole space by the obstacle. As inspired by transformation optics and metamaterials, recent theories have proposed the concept of fluid cloaking able to recover the straight streamlines as if the obstacle does not exist. However, such a cloak, similar to all previous transformation-optics-based devices, relies on complex metamaterials, being difficult to implement. Here we deploy the theory of scattering cancellation and report on the experimental realization of a fluid-flow cloak without metamaterials. This cloak is realized by engineering the geometry of the fluid channel, which effectively cancels the dipole-like scattering of the obstacle. The cloaking effect is demonstrated via direct observation of the recovered straight streamlines in the fluid flow with injected dyes. Our work sheds new light on conventional fluid control and may find applications in microfluidic devices.

Ideal fluid flow around a cylinder is a fundamental problem discussed in many textbooks of fluid mechanics [1][2][3][4]. Being inviscid and incompressible, an ideal fluid satisfies the mass continuity equation, which can be simplified into Laplace's equation at steady states. When encountering a circular cylinder, the ideal fluid no longer follows straight streamlines, but flows around the cylinder with deflected streamlines described by a conformal mapping [1,4]. This model provides a physical picture to understand general fluid flow in fluid mechanics when a complex-shaped obstacle or fluid viscosity is involved.
In this Letter, we extend the previous scattering cancellation approach to fluid control, and construct a fluid-flow cloak to hide a cylindrical obstacle without disturbing the straight external streamlines (see the movie in Supplementary Material demonstrating the effectiveness of such a fluid cloak [36]). In particular, the use of scattering cancellation in fluid flow has gained an unprecedented unique feature, i.e., being "metamaterial-free": our fluid cloak is realized by changing the geometry of the fluid channel, rather than employing any complex metamaterial design. By injecting dye particles into the fluid flow, we have observed the successful recovery of straight streamlines passing through the obstacle, as if the obstacle does not exist.
We shall firstly point out that the ideal fluid with zero viscosity in fact does not exist in nature (it is "dry water" as quoted from John von Neumann [37]). The flow of a real fluid with finite viscosity is governed by the Navier-Stokes equation, which has nearly no analytical solution due to its nonlinear viscosity term. Nevertheless, a viscous fluid flow in a narrow gap between two parallel plates, known as Hele-Shaw flow, can be described by a scalar potential function, exhibiting similar features of two-dimensional (2D) ideal fluid flow [3,4,38,39]. Moreover, Hele-Shaw flow plays a significant role in microfluidic devices and plastic-forming manufacturing operations. A realistic cloak for Hele-Shaw flow may find useful applications in these related fields.
Let us start with Fig. 1(a), which shows the ideal fluid flow around a cylinder with radius in a 2D geometry [1][2][3][4]. By denoting the stream function as and the velocity potential as (here which has the dipole strength vector 2 . Therefore, in the language of electromagnetics, the cylinder induces a "dipole field" of fluid flux, disturbing the flow at every location.
The analogy to electromagnetics implies that it might be straightforward to construct a fluid cloak by applying the scattering cancellation. Indeed, we can consider a cloak as shown in Fig. 1 which consists of a shell with an outer radius and an inner radius that encloses the cylindrical obstacle completely. This cloak can guide the fluid flux smoothly around the obstacle leaving the external fluid flux undisturbed. In the previous magnetic/thermal cloaks [31,33,34], the 2D calculation from Laplace's equation requires their magnetic permeability/thermal conductivity to take the relative value of / . By the same token, we can simply write down the mathematical condition of the fluid cloak as [36] . # 1 Here is the fluid density in the background, and is that inside the cloak shell.
However, we still have two problems to tackle. To tackle the problems mentioned above, we consider the creeping flow with low Reynold number (Re 1) in a narrow gap between two plates, which is known as Hele-Shaw flow. In this case, a viscous flow can be simplified into an ideal fluid flow satisfying Laplace's equation [3,4,38,39]. Hence, the problem of viscosity can be solved. As illustrated in Fig. 1(c), we consider fluid with a density that flows into a narrow channel with a height of . A solid cylinder with radius that penetrates through the channel serves as a cylindrical obstacle. So, it can be expected that the viscous flow in the narrow channel in the presence of the cylindrical obstacle will behave like the picture in Fig. 1(a) (there will be some discrepancies in thin layers close to the boundary of the obstacle; to be discussed later).
Now we design the cloak. As mentioned above, it is impractical to compress the fluid density to fulfill the cloaking condition in Eq. (1). However, from the 2D perspective, we can emulate a higher local fluid density by extending the height of the channel at the cloak shell region, as illustrated in Fig. 1(c). According to mass conservation, the extended height required to construct a fluid flow cloak satisfies the formula below,

# 2
It is worth mentioning that the value of in Eq. (2) is only an estimation. The no-slip condition (i.e., only zero velocity is allowed at the boundary) gives rise to distortion of streamlines in the Hele-Shaw flow in the vicinity of boundaries of the obstacle, being different from the ideal fluid flow. Hence, the optimal height for the fluid flow cloak should be slightly shifted from directly calculated from Eq.
(2). The method we used to optimize will be discussed in a later section. Here we introduce our optimization method for the height of the cloak shell. The initial value of = 9.85 mm is obtained from Eq. (2). Therefore, the simulation using the same setup with the experiment was repeated with a range of close to 9.85 mm in COMSOL Multiphysics 5.2 in order to obtain the optimized . The creeping flow that is governed by Stokes equations was used in the simulation. It is worth noting that the optimized should show straight streamlines in the background region, as shown in Fig. 1(b). Hence, we first extracted the -coordinates of 40 streamlines, which were spaced equally in the -direction, from each simulation. Next, we removed those -coordinates that were far away from the obstacle or located inside the cloak shell region (see   Fig. 1(a). It is worthwhile noting that the distortion of streamlines arose even in places away from the obstacle.
Finally, the results for the obstacle sample with the cloak, which demonstrates the realization of fluid cloaking, are as shown in Fig. 4(e) and (f). The cloaking shell region was represented by a pink dotted circle. As anticipated, the distortion of streamlines outside the shell region was cancelled and the streamlines were straight. The background region in Fig. 4(f) exhibited the same pattern as shown in Fig. 4(b) and 1(b). The cylindrical obstacle was therefore "hidden" from the external fluid fluxes.
In summary, our work demonstrates an innovative approach to realize a fluid flow cloak that can hide a cylindrical obstacle in a narrow fluid channel. In contrast to previous designs employing complex metamaterials, our fluid cloak adopts the scattering cancellation approach and is realized by merely adjusting the geometry of the fluid channel. Such a simple solution provides a new venue for fluid control by transplanting novel concepts of metamaterials (rather than using complex metamaterials themselves) to fluid engineering.
Note: In the final stage of submission, we found two publications reporting hydrodynamic cloaks [41,42], and one of them implements a fluid flow cloak with multilayer metamaterials [42].
The fundamental difference of our work lies in the scattering cancellation in fluid control and the absence of complex metamaterial design.  The glycerin that dyed by white paints was pumped into a sink first through a wide rubber tube and then flowed into a rectangular channel uniformly. The indicators, glycerin dyed by black dyes, were injected into the channel manually with injection syringes through the injection tubes at steady state. The flowing channel has a width of 50 mm and a length of 146 mm. A piece of glass block was placed on top of the samples to enclose the channel. The shell region was extended downwards, like a trough surrounding the obstacle. The height of the background and effective shell region are = 5 mm and = 10 mm respectively. The entire processes were recorded by a camera from top.