WO2019132552A1

WO2019132552A1 - Hydrodynamic cloaking metamaterial and designing method thereof

Info

Publication number: WO2019132552A1
Application number: PCT/KR2018/016790
Authority: WO
Inventors: Jae Ryoun Youn; Young Seok Song; Juhyuk PARK
Original assignee: Seoul National University R&Db Foundation; Industry-Academic Cooperation Foundation, Dankook University
Priority date: 2017-12-28
Filing date: 2018-12-27
Publication date: 2019-07-04
Also published as: KR102150541B1; KR20190080796A

Abstract

The present disclosure relates to a method for designing a hydrodynamic cloaking metamaterial to exclude a fluidic force from a specific space, and a hydrodynamic cloaking metamaterial fabricated by the designing the method.

Description

HYDRODYNAMIC CLOAKING METAMATERIAL AND DESIGNING METHOD THEREOF

The present disclosure relates to a method for designing a hydrodynamic cloaking metamaterial to exclude a fluidic force from a specific space, and a hydrodynamic cloaking metamaterial fabricated by the designing method.

Drag is a frictional fluid flow that acts on an object in the opposite direction to the fluid flow. Strategic control of such a drag is critical for not only engineering industrial applications such as vehicles and pipe flow but also confronting natural disasters such as tsunamis and hurricanes. Indeed, drag-free technology able to preclude frictional resistance to an object in flowing fluids can change our lives completely.

Meanwhile, transformation optics has opened a way for manipulating various physical fields in a domain. It provides mathematical background for designing a cloak that makes an object inside it invisible (transparent). Many studies have been carried out to design and implement electromagnetic metamaterial cloaks by introducing spatially varying material parameters. Further, this concept can be applied to other areas, such as acoustics, quantum mechanics, thermodynamics, solid-mechanics, and recently even time. However, metamaterial cloaks for controlling fluid flow have yet to be reported.

[Prior Art]

Korean Patent No. 10-1498656

The present disclosure relates to a method for designing a hydrodynamic cloaking metamaterial to preclude a fluidic force from a specific space, and a hydrodynamic cloaking metamaterial fabricated by the designing method.

However, problems to be solved by the present disclosure are not limited to the above-described problems. Although not described herein, other problems to be solved by the present disclosure can be clearly understood by a person with ordinary skill in the art from the following description.

A first aspect of the present disclosure provides a method for designing a hydrodynamic cloaking metamaterial, including providing a metamaterial including at least one unit cell in a second region surrounding a first region including a target object or target space present in a fluid flow, wherein each of the at least one unit cell included in the metamaterial has a pre-determined effective viscosity so as to form a drag-free space in the first region for hydrodynamically cloaking the first region including the target object or target space.

A second aspect of the present disclosure provides a hydrodynamic cloaking metamaterial including at least one unit cell configured in a second region surrounding a first region including a target object or target space present in a fluid flow, wherein each of the at least one unit cell included in the metamaterial has a pre-determined effective viscosity so as to form a drag-free space in the first region for hydrodynamically cloaking the first region including the target object or target space.

Fluidic drag is an inevitable phenomenon in aerodynamics and hydrodynamics when a solid object moves relatively with respect to a surrounding fluid. In this sense, exclusion of such a drag force is a great challenge to achieve for scientific and engineering applications. In hydrodynamics, Navier-Stokes equations describe the motion of Newtonian fluids with assumption that stresses in a fluid encompass a diffusing viscous term and a pressure term. Fluid viscosity, a key material parameter of fluid, determines the hydrodynamic behavior of liquid governed by the partial differential equations of elliptic type. Hence, according to the embodiments of the present disclosure, a fluidic space defined by a coordinate-transformed viscosity tensor can provide a method for designing a hydrodynamic cloaking metamaterial that realizes an artificial fictitious fluidic space in it and a hydrodynamic cloaking metamaterial designed by the method.

According to the embodiments of the present disclosure, transformation hydrodynamics is introduced in order to create a desired virtual space and to control fluidic energy in the space and thus to provide a strategic design for a cloaking metamaterial with a drag-free space, and the resulting structure can be implemented with the use of a microfluidic device.

According to the embodiments of the present disclosure, the metamaterial fabricated by theoretical mapping of effective viscosity tensors enables the cloaking of hydrodynamics.

FIG. 1 is a schematic illustration of a hydrodynamic cloak with a drag-free space according to an example of the present disclosure and shows a flow image for the bare space (FIG. 1A), a flow image when an obstacle is placed in the flow (i.e., for space with an obstacle) (FIG. 1B), a flow image when the obstacle is encircled by a hydrodynamic cloak (i.e., for cloak space with an obstacle) (FIG. 1C), and a flow image after removing the obstacle (i.e., for only cloak space without an obstacle) (FIG. 1D). The drag-free space is generated in the center region and the yellow lines indicate the pressure contour.

FIG. 2 shows a concept of fluidic space distortion by transforming coordinate grids from virtual to physical spaces for cloaking a space (FIG. 2A), viscosity profiles of the cloaking shell in the radial and azimuthal directions (FIG. 2B), and pressure fields (upper), pressure gradient fields (middle), and velocity fields (lower) with streamlines simulated for the case of bare space (FIG. 2C), with obstacle (FIG. 2D), with obstacle and cloak (FIG. 2E), and with cloak only (FIG. 2F) according to an example of the present disclosure.

FIG. 3 shows drawings of the microfluidic metamaterials (FIG. 3A), photographs of the patterned Cr masks (FIG. 3B), photographs of the fabricated Si masters (FIG. 3C), photographs of the manufactured microfluidic devices for the cases of the bare space, with obstacle, with obstacle and cloak, and with cloak only (FIG. 3D), and the experimental set-up for observing the streamlines in the microfluidic device (FIG. 3E) according to an example of the present disclosure.

FIG. 4 shows a viscosity tensor as a continuous function of the radial distance (FIG. 4A), simulated pressure fields for the cases with obstacle and cloak (left) and with cloak only (right) (FIG. 4B), pressure gradient fields (FIG. 4C), and velocity fields for the above-described cases (FIG. 4D) according to an example of the present disclosure.

FIG. 5 shows a 10-layer structure and assigned viscosities for cloaking (FIG. 5A), discretized viscosity tensor in each layer along radial and azimuthal axes (FIG. 5B), simulated pressure fields for the cases with obstacle and cloak (left) and with cloak only (right) (FIG. 5C), and pressure gradient fields (FIG. 5D), and velocity fields for the above-described cases (FIG. 5E) according to an example of the present disclosure.

FIG. 6 shows velocity fields and flow streamlines calculated in a designed unit cell when a pressure gradient is imposed in the radial (left) and azimuthal (right) directions (FIG. 6A), fffective viscosity in the radial (left) and azimuthal (right) directions of the unit cell in the cloaking shell (FIG. 6B), and a configuration of the hydrodynamic cloak (FIG. 6C): an microscopic image of the entire fabricated microfluidic device, an image of enlarged microstructure of the device, and a picture of the device according to an example of the present disclosure.

FIG. 7 shows a velocity field in the bare unit cell (FIG. 7A), a velocity field in the background unit cell (FIG. 7B), and velocity fields in the unit cell for each layer (FIG. 7C) according to an example of the present disclosure. The white arrow means r-direction and the yellow arrow for θ-direction. l is the length of micropillar in the unit cell.

FIG. 8 shows calculated pressure fields (FIG. 8A), velocity fields (FIG. 8B), and flow streamlines (FIG. 8C) for the cases of bare space, with obstacle, with obstacle and cloak, and with cloak only, and flow streamlines observed experimentally in the same cases (FIG. 8D) according to an example of the present disclosure.

FIG. 9 shows the comparison of the pressure fields simulated in the cases of continuous viscosity distribution (FIG. 9A), discretized viscosity distribution (FIG. 9B), and pillar array structure (FIG. 9C) according to an example of the present disclosure.

FIG. 10 shows the comparison of the velocity fields simulated in the cases of continuous viscosity distribution (FIG. 10A), discretized viscosity distribution (FIG. 10B), and pillar array structure (FIG. 10C) according to an example of the present disclosure.

FIG. 11 shows the comparison of the streamlines simulated in the cases of continuous viscosity distribution (FIG. 11A), discretized viscosity distribution (FIG. 11B), and pillar array structure (FIG. 11C) according to an example of the present disclosure.

Hereafter, examples will be described in detail with reference to the accompanying drawings so that the present disclosure may be readily implemented by those skilled in the art. However, it is to be noted that the present disclosure is not limited to the examples but can be embodied in various other ways. In the drawings, parts irrelevant to the description are omitted for the simplicity of explanation, and like reference numerals denote like parts through the whole document.

The term "hydrodynamic cloaking", "fluidic cloaking", "hydrodynamically invisible", or "hydrodynamically transparent" used herein refers to a phenomenon or state in which when an object, an obstacle, or a space is present in the flow of fluid, the fluid flows as if the object, the obstacle, or the space were not present. For example, when the object, the obstacle, or the space is situated in a drag-free space, it can be cloaked without experiencing any drag force.

The term "metamaterial" used herein refers to a structure which is specially engineered to remove or avoid an unwanted property of a bulk material or structure or specific phenomena of nature or a material formed by regularly or irregularly repeating composites of one or more identical or different structures. The "metamaterial" is engineered to artificially have a certain physical property that is not found in naturally occurring materials, and a metamaterial or metamaterial cloak used herein is a material showing hydrodynamic cloaking by using a unit cell with effective viscosity that induces fluidic space distortion to form a hydrodynamic cloak or a microfluidic device.

Throughout this document, the term “connected to” may be used to designate a connection or coupling of one element to another element and includes both an element being “directly connected” another element and an element being “electronically connected” to another element via another element.

Through the whole document, the term “on” that is used to designate a position of one element with respect to another element includes both a case that the one element is adjacent to the other element and a case that any other element exists between these two elements.

Further, through the whole document, the term “comprises or includes” and/or “comprising or including” used in the document means that one or more other components, steps, operation and/or existence or addition of elements are not excluded in addition to the described components, steps, operation and/or elements unless context dictates otherwise. Through the whole document, the term “about or approximately” or “substantially” is intended to have meanings close to numerical values or ranges specified with an allowable error and intended to prevent accurate or absolute numerical values disclosed for understanding of the present disclosure from being illegally or unfairly used by any unconscionable third party. Through the whole document, the term “step of” does not mean “step for”.

Through the whole document, the term “combination(s) of” included in Markush type description means mixture or combination of one or more components, steps, operations and/or elements selected from a group consisting of components, steps, operation and/or elements described in Markush type and thereby means that the disclosure includes one or more components, steps, operations and/or elements selected from the Markush group.

Through the whole document, a phrase in the form “A and/or B” means “A or B, or A and B”.

Hereinafter, embodiments and examples of the present disclosure will be described in detail with reference to the accompanying drawings. However, the present disclosure may not be limited to the following embodiments, examples, and drawings.

In an embodiment of the present disclosure, the pre-determined effective viscosity of each of the at least one unit cell included in the metamaterial may be calculated by using a coordinate-transformed viscosity tensor obtained via transformation of a fluidic coordinate system to distort a space surrounding the target object or target space so as to impart zero drag force in the first region for forming a hydrodynamically cloaking shell in the second region, but may not be limited thereto.

In an embodiment of the present disclosure, the pre-determined effective viscosity may be set by numerical simulation for a fluid viscosity tensor represented by the following equation 1, but may not be limited thereto:

[Equation 1]

wherein a is an inner radius of the cloaking shell, and r' is a radial axis in the coordinate transformed physical space. For reference, when the coordinate in the left side in FIG. 2A is transformed into the coordinate in the right side in FIG. 2A, the above r' indicates a radial axis in the coordinate in the right side in FIG. 2A.

In this regard, a basic concept of the hydrodynamic cloak will be described by the expected pressure contours in the fluid with reference to FIG. 1. FIG. 1 is a schematic illustration of a hydrodynamic cloak with a drag-free space. Compared with the bare case shown in FIG. 1A, the obstacle placed in the fluid is subject to a frictional drag (FIG. 1B). As a result, the pressure field developed around it is distorted, and the surrounding fluid can be splashed. Herein, the strategy of the inventors of the present disclosure is to create a drag-free space and to locate the obstacle within the hydrodynamic cloak. The fluidic space is compressed from the cylindrical region (0<r<b) into the annular region (a<r'<b) and then the empty space (0<r'<a) is generated in the coordinate system (FIG. 1C and FIG. 1D), wherein a and b are an inner radius and an outer radius of the cloaking shell, respectively, and r' is a radial axis in the coordinate transformed physical space.

Surprisingly, the pressure field and flow pattern of the engineered fluidic space are completely changed. The flow and pressure distributions outside the annular region are the same as those of the bare space case. Regardless of the imposed pressure and velocity boundary values of fluid flow, hydrodynamic momentum cannot reach the empty space created within the hydrodynamic cloak. The fluid flows as if no objects were present within the space. Consequently, the space becomes "hydrodynamically" invisible. Once an obstacle is situated in the drag-free space, it can be cloaked without experiencing any drag force (FIG. 1C). This implies that the flow field is not affected by the removal of the obstacle (Fig. 1d). The fluidic cloaking demonstrated in FIG. 1 is designed and implemented in the present disclosure.

Theoretical formulation via coordinate transformation needs to be carried out in a bid to prove a form-invariance of the Navier-Stokes equations. FIG. 2 shows theoretical modeling for hydrodynamic cloak. FIG. 2A shows the coordinate transformation from a virtual space (r, θ) to a physical space (r', θ') for fluidic cloaking. Assuming an incompressible Newtonian fluid at steady state with a low Reynolds number, the transformed Navier-Stokes equations can be expressed as below:

(1)

(2)

(3)

Herein, ∇´,

, and

are the nabla operator, the viscous stress tensor, and the velocity field in the new physical space, respectively. The transformed second-order viscosity tensor,

is defined as

, where

indicates the transformation Jacobian matrix between the two coordinate system. It is to be noted that the fluid viscosity in Equation (3) is no longer a scalar constant but a tensor with an anisotropic spatial dependency both in radial and azimuthal axes. The fluid viscosity tensor, a product of the spatial distortion, can be employed to create a virtual fluidic space.

The transformed viscosity tensor,

, for the ideal cloak is given by diag[r'/(r'-a), (r'-a)/r']μ where a is an inner radius of the cloaking shell, and r' is a radial axis in the coordinate transformed physical space (see FIG. 4). FIG. 4 shows modeling of ideal hydrodynamic cloak.

In an embodiment of the present disclosure, a value of the fluid viscosity tensor of the metamaterial is set to change in the second region in a radial direction from the target object or target space such that a velocity contour and and/or a pressure contour of the fluid outside the second region becomes uniform to form the drag-free space in the first region, but may not be limited thereto.

In an embodiment of the present disclosure, the hydrodynamic cloaking metamaterial may form a hydrodynamically cloaking shell in the second region if the target object or target space is present in the fluid flow, but may not be limited thereto.

In an embodiment of the present disclosure, a flow rate in a region of the hydrodynamic cloaking shell formed in the second region may be set to abruptly increase such that a uniform flow rate outside the second region maintains for hydrodynamically cloaking the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, the at least one unit cell may be configured in a spaced apart array to each other, but may not be limited thereto.

In an embodiment of the present disclosure, the at least one unit cell may be arrayed in a monolayer or multilayer in the second region around the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, the method may include simulation which includes setting the at least one unit cell in a multilayer and assigning a different value of the effective viscosity of each layer to decrease in a radial direction from the target object or target space such that a velocity field and and/or a pressure field of the fluid outside the second region becomes uniform, but may not be limited thereto. In this case, the value of the effective viscosity of each layer may be assigned constant in an azimuthal direction from the target object or target space.

In an embodiment of the present disclosure, each of the at least one unit cell may include a microstructure therein, but may not be limited thereto.

In an embodiment of the present disclosure, the effective viscosity of the each unit cell may be represented by the following equation 2, but may not be limited thereto:

[Equation 2]

wherein,

and

are the viscosity fields with and without the microstructure, respectively,

is a superficially averaged quantity, and

is an intrinsic viscosity of the fluid.

In an embodiment of the present disclosure, the microstructure may have a cylindrical, spherical or elliptical shape , or various polyhedral shapes, but may not be limited thereto. For example, the polyhedral shapes may include a rectangular parallelepiped, a tetrahedron, a cube, an octahedron, a dodecahedron, an icosahedron, a triangular pillar, a square pillar, a pentagonal pillar, a tetragonal pyramid, or a pentagonal pyramid, but may not be limited thereto. For example, the cylindrical shapes may include a circular cylinder or elliptic cylinder , but may not be limited thereto.

In an embodiment of the present disclosure, the at least one unit cell in the metamaterial may be arrayed in a multilayer in the second region around the target object or target space, the microstructure included in each of the at least one unit cell may have the same thickness and height to another, and a length of the microstructure may gradually decrease in a radial direction from the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, a difference in length between the microstructures of the respective layers in a radial direction from the target object or target space may gradually decrease in the range of from about 1 mm to about 10 mm, but may not be limited thereto. For example, the difference in length between the microstructures of the respective layers may be in the range of from about 1 mm to about 10 mm, from about 2 mm to about 10 mm, from about 4 mm to about 10 mm, from about 6 mm to about 10 mm, from about 8 mm to about 10 mm, from about 1 mm to about 8 mm, from about 2 mm to about 8 mm, from about 4 mm to about 8 mm, from about 6 mm to about 8 mm, from about 1 mm to about 6 mm, from about 2 mm to about 6 mm, from about 4 mm to about 6 mm, from about 1 mm to about 4 mm, or from about 2 mm to about 4 mm, but may not be limited thereto.

In an embodiment of the present disclosure, a thickness of the microstructure included in each of the at least one unit cell may be from about 1 mm to about 100 mm, and the length of the microstructure of each layer in a radial direction from the target object or target space may gradually decrease from about 200 mm to about 100 mm, but may not be limited thereto. For example, the thickness of the microstructure may be from about 1 mm to about 100 mm, from about 10 mm to about 100 mm, from about 20 mm to about 100 mm, from about 30 mm to about 100 mm, from about 40 mm to about 100 mm, from about 50 mm to about 100 mm, from about 60 mm to about 100 mm, from about 70 mm to about 100 mm, from about 80 mm to about 100 mm, from about 90 mm to about 100 mm, from about 1 mm to about 80 mm, from about 1 mm to about 60 mm, from about 1 mm to about 50 mm, from about 1 mm to about 30 mm, or from about 1 mm to about 10 mm, but may not be limited thereto. The length of the microstructure of each layer may gradually decrease from about 200 mm to about 100 mm, from about 190 mm to about 100 mm, from about 180 mm to about 100 mm, from about 170 mm to about 100 mm, from about 160 mm to about 100 mm, from about 150 mm to about 100 mm, from about 140 mm to about 100 mm, from about 120 mm to about 100 mm, from about 200 mm to about 110 mm, from about 200 mm to about 120 mm, from about 200 mm to about 130 mm, from about 200 mm to about 140 mm, or from about 200 mm to about 150 mm, but may not be limited thereto.

In an embodiment of the present disclosure, the metamaterial may further include at least one isotropic unit cell in addition to the at least one unit cell including the microstructure, but may not be limited thereto. The isotropic unit cell is applied in order to resolve the impedance mismatching problem originated from errors in the effective medium approximation. For example, the isotropic unit cell is arrayed outside the array of the at least one unit cell including the microstructure.

In an embodiment of the present disclosure, the fluid includes a liquid, gas or plasma. For example, the liquid may include various ones such as water, a polar or nonpolar solvent, an organic or inorganic solution, or oils , but may not be limited thereto. For example, the gas may include various ones such as air, nitrogen, oxygen, helium, a natural gas, or the like, but may not be limited thereto. For example, the plasma may include various ones such as arc plasma, glow discharge, corona, gliding arc plasma, dusty plasma, or the like but may not be limited thereto.

A second aspect of the present disclosure relates to a hydrodynamic cloaking metamaterial including at least one unit cell configured in a second region surrounding a first region including a target object or target space present in a fluid flow, wherein each of the at least one unit cell included in the metamaterial has a pre-determined effective viscosity so as to form a drag-free space in the first region for hydrodynamically cloaking the first region including the target object or target space.

In an embodiment of the present disclosure, the hydrodynamic cloaking metamaterial may be designed by the method in accordance with the first aspect of the present disclosure. Accordingly, all the above descriptions of the first aspect of the present disclosure can be applied to the second embodiment of the present disclosure, even though they are omitted hereinafter.

In an embodiment of the present disclosure, the hydrodynamic cloaking metamaterial may form a hydrodynamically cloaking shell in the second region when the target object or target space is present in a fluid flow, but may not be limited thereto.

In an embodiment of the present disclosure, since the hydrodynamically cloaking shell is formed, the target object or target space may not be affected by any drag force exerted by the fluid flow, but may not be limited thereto.

In an embodiment of the present disclosure, when the target object or target space is present in the fluid flow, the pre-determined effective viscosity of the metamaterial may be set to apply zero drag force to a space around the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, the pre-determined effective viscosity of the metamaterial may be calculated by using a coordinate-transformed viscosity tensor obtained via transformation of a fluidic coordinate system to distort a space surrounding the target object or target space so as to impart zero drag force in the first region for forming a hydrodynamically cloaking shell in the second region, but may not be limited thereto.

In an embodiment of the present disclosure, a value of the fluid viscosity tensor of the metamaterial may be assigned to change in the second region in a radial direction from the target object or target space such that a velocity contour and and/or a pressure contour of the fluid outside the second region becomes uniform to form the drag-free space in the first region, but may not be limited thereto.

In an embodiment of the present disclosure, a value of a viscosity tensor in each of the at least one unit cell included in the metamaterial may gradually decrease in a radial direction from the target object or target space in the second region, but may not be limited thereto. In this case, the value of the effective viscosity of each layer may be assigned constant in an azimuthal direction from the target object or target space.

In an embodiment of the present disclosure, a flow rate in a region of the hydrodynamic cloaking shell may be set to abruptly increase such that a uniform flow rate outside the second region maintains for hydrodynamically cloaking the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, the at least one unit cell may be arrayed around the target object or target space in the second region, but may not be limited thereto.

In an embodiment of the present disclosure, if the target object or target space has a circular shape in the second region, the at least one unit cell may be configured in the second region isotropically around the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, simulation including setting the at least one unit cell in a multilayer and setting a value of the effective viscosity of each layer to decrease in a radial direction from the target object or target space may be performed such that a velocity field and and/or a pressure field of the fluid outside the second region becomes uniform, but may not be limited thereto. In this case, the value of the effective viscosity of each layer may be set constant in an azimuthal direction from the target object or target space.

In an embodiment of the present disclosure, the microstructure may have a cylindrical, spherical or elliptical shape, or various polyhedral shapes, but may not be limited thereto. For example, the polyhedral shapes may include a rectangular parallelepiped, a tetrahedron, a cube, an octahedron, a dodecahedron, an icosahedron, a triangular pillar, a square pillar, a pentagonal pillar, a tetragonal pyramid, or a pentagonal pyramid, but may not be limited thereto. For example, the cylindrical shapes may include a circular cylinder or elliptic cylinder , but may not be limited thereto.

In an embodiment of the present disclosure, a thickness of the microstructure included in each of the at least one unit cell may be from about 1 mm to about 100 mm, and the length of the microstructure of each layer in a radial direction from the target object or target space may gradually decrease from 200 mm to about 100 mm, but may not be limited thereto. For example, the thickness of the microstructure may be from about 1 mm to about 100 mm, from about 10 mm to about 100 mm, from about 20 mm to about 100 mm, from about 30 mm to about 100 mm, from about 40 mm to about 100 mm, from about 50 mm to about 100 mm, from about 60 mm to about 100 mm, from about 70 mm to about 100 mm, from about 80 mm to about 100 mm, from about 90 mm to about 100 mm, from about 1 mm to about 80 mm, from about 1 mm to about 60 mm, from about 1 mm to about 50 mm, from about 1 mm to about 30 mm, or from about 1 mm to about 10 mm, but may not be limited thereto. The length of the microstructure of each layer may gradually decrease from about 200 mm to about 100 mm, from about 190 mm to about 100 mm, from about 180 mm to about 100 mm, from about 170 mm to about 100 mm, from about 160 mm to about 100 mm, from about 150 mm to about 100 mm, from about 140 mm to about 100 mm, from about 120 mm to about 100 mm, from about 200 mm to about 110 mm, from about 200 mm to about 120 mm, from about 200 mm to about 130 mm, from about 200 mm to about 140 mm, or from about 200 mm to about 150 mm, but may not be limited thereto.

In an embodiment of the present disclosure, the isotropic unit cell may have the same or smaller size than the unit cell including the microstructure. For example, the unit cell including the microstructure and the isotropic unit cell have a size ratio of about 1: from about 0.1 to about 1.

In an embodiment of the present disclosure, the at least one unit cell in the metamaterial may be set in a multilayer and a value of the viscosity tensor of each layer may be set to gradually decrease in a radial direction from the target object or target space, but may not be limited thereto.

In an embodiment of the present disclosure, the metamaterial may be prepared using various materials such as polymer, ceramic, semiconductor, metallic material, glass, or the like. For example, the metamaterial may be prepared by designing a detailed structure of the metamaterial to have a desired effective viscosity, selecting an appropriate material from the above-described materials and performing various methods such as lithography, 3D-additive manufacturing, injection molding, and the like.

In an embodiment of the present disclosure, the fluid includes a liquid, gas or plasma. The hydrodynamic cloak according to the embodiments of the present disclosure will offer a systematic strategy that can induce flow behavior which is not found in nature. Since the cloaking does not rely upon the absolute value of viscosity, it can be applied to the mechanics of fluids including gases, liquids, and plasmas For example, the liquid may include various ones such as water, a polar or nonpolar solvent, an organic or inorganic solution, or oils, but may not be limited thereto. For example, the gas may include various ones such as air, nitrogen, oxygen, helium, a natural gas, or the like, but may not be limited thereto. The hydrodynamic cloak according to the embodiments of the present disclosure will offer a systematic strategy that can induce flow behavior which is not found in nature. Since the cloaking does not rely upon the absolute value of viscosity, it can be applied to the mechanics of fluids including gases, liquids, and plasmas. For example, the plasma may include various ones such as arc plasma, glow discharge, corona, gliding arc plasma, dusty plasma, or the like but may not be limited thereto.

A wide range of fluid flow problems in mechanical, civil, chemical, and biomedical engineering applications can be solved by adopting the hydrodynamic cloak. For example, hydrological and meteorological disasters can be minimized and drag-free vehicles such as aircrafts, automobiles, and submarines can be designed by applying the cloaking materials.

Hereinafter, the present disclosure will be explained in more detail with reference to Examples. However, the following Examples are illustrative only for better understanding of the present disclosure but do not limit the present disclosure.

[EXAMPLES]

Numerical simulation was performed to predict pressure and velocity fields of fluid flow by using COMSOL Multiphysics, a commercial finite-element-based software. Four cases were simulated: the bare space, with obstacle, with obstacle and cloak, and with cloak only.

A fluidic domain with a dimension of 1 cm x 1 cm x 50 μm was generated. A circular ring indicating the cloaking shell was considered in the domain. The ring had an inner radius of 2 mm and an outer radius of 4 mm.

Non-slip boundary condition (

) was applied to the surface of solid wall in the domain. A pressure difference (Δp=1 kPa) was assumed along y-axis between the inlet and the outlet. Water was employed as the working fluid with the viscosity of 1 mPa·s at room temperature. To compute flow fields in the domain, the Navier-Stokes equation was considered as the form of differential equation for numerical simulation.

The microfluidic devices were designed by using a commercial CAD program, AutoCAD 2017 (Autodesk, United States). The drawings are shown in FIG. 3A. FIG. 3 shows demonstration of experiments for hydrodynamic cloaks. The four microchannels were placed in the two circles representing 4-inch silicon (Si) wafers. Each microfluidic device had inlet and outlet with a diameter of 2 mm. The length and the width of the channels were 5 cm and 1 cm, respectively. The length of the pillar-arrayed section was set to be 3 cm to eliminate unexpected inlet and outlet effects and to secure stable laminar flow.

The Si master was fabricated by using photolithography. After the Si wafer was covered with SU-8, an epoxy-based negative photoresist, UV (ultraviolet) light was irradiated over the patterned chromium (Cr) photomask. The Cr photomask possesses the microchannel patterns as shown in FIG. 3B. Soft lithography was used to fabricate the microfluidic metamaterials. The metamaterials were manufactured by replicating the patterned Si masters with polydimethylsiloxane (PDMS). The corresponding procedure is as follows: Sylgard 184 A and B (Dow Corning, United States), a prepolymer and a curing agent respectively, were mixed thoroughly at a mass ratio of 10:1. After the Si master was attached to a Petri dish container (FIG. 3C), the Sylgard mixture was poured onto the Si master-attached container. The container was placed in a vacuum oven for 2 hours to remove air bubbles from the Sylgard mixture generated during the mixing process. The curing process of the Sylgard mixture was conducted by placing the container in a heating oven at 60℃ for 4 hours.

The cured PDMS was separated from the Si master and rinsed with ethanol (Daejung Chem., Republic of Korea) to remove dusts and unreacted residues. Inlet and outlet of the microchannels were created by punching 1.5 mm diameter holes for the reservoir of the cured PDMS. The PDMS was bonded to a slide glass after corona treatment for 5 minutes to oxidize the surface, and heat-treatment for 1 hour at 110℃ for strong adhesion. The corona treatment was conducted by using a high frequency generator (model BD-10AV, Electro-Technic products Inc., United States). Finally, the microdevices for the microfluidic experiment were constructed as shown in Fig. 3d.

Hydrodynamic cloaking was experimentally implemented using the microfluidic devices. The inlet and outlet of the microchannels were punched with a Harris Uni-Core 1.5-mm-diameter puncher (Ted Pella, Inc., United States) and connected to flexible plastic tubes (Tygon®, Saint-Gobain Performance Plastics, France) to infuse a fluid.

The fluid used in the experiment was water, a typical Newtonian fluid with a 1 mPa·s viscosity at room temperature. Fluorescent microparticles with a radius of 3.2 μm (Red fluorescent, Fluoro-Max^TM, CAT.NO.RO300, LOT NO.42259, Thermo Scientific, United States) were dispersed in the water to visualize the streamlines of fluid flow. The volumetric concentration of the microparticle/water solution was 0.3 μL/mL. TWEEN^® 20 (Polyethylene glycol sorbitan monolaurate, Sigma Aldrich, United States), a nonionic surfactant, was added to the microparticle/water solution with a volume concentration of 0.1 μL/mL to help dispersion and avoid aggregation of the microparticles. The fluid with microparticles was injected into the reservoirs of the microchannel through a 3 mL syringe (3 mL sterile, non-toxic, and non-pyrogenic Kovax-Syringe, Korea Vaccine Co., Ltd, Republic of Korea).

The total experimental set-up is shown in FIG. 3E. Particle streamlines were observed by using an inverted fluorescence microscope (IX53, Olympus Corporation, Japan). UPlanFL N 4x/0.13 PhP (Olympus Corporation, Japan) was used as a microscope objective lens. A color CCD (charge-coupled device) camera (AcquCAM 23G, JNOpTIC Corporation, Republic of Korea) was installed to the microscope with a low-magnification C-mount adapter (U-TV0.5XC, Olympus Corporation, Japan). Pressure-driven fluid flow was generated by using a N₂ gas pressure pump with a pressure of 5 kPa. A digital pressure gauge (DPG8001-60, OMEGA Engineering, United States) was used to control pressure values with a precision regulator (100LR, ControlAir Inc., United States). After filling the syringe with the microparticle/water solution, the imposed pressure drove a Poiseuille flow in the microchannel. The streamlines of microparticles were observed by using a CCD camera. The employed light conditions in the experiment were a gain of 40.3 dB and an exposure time of 0.125 s.

Images of the particle streamlines were captured by using a commercial CCD camera-related program, JNOPTIC Capture 2.4 (JNOpTIC Corporation, Republic of Korea). The captured images were processed to clearly show the streamlines by using Adobe Photoshop CS6 (Adobe Systems, United States). Color of the streamlines was changed from red to green, brightness and contrast of the images were optimized, and background noise was removed.

A basic concept of the hydrodynamic cloak is illustrated by the expected pressure contours in the fluid (Fig. 1). FIG. 1 is a schematic illustration of a hydrodynamic cloak with a drag-free space. Compared with the bare case shown in Fig. 1a, the obstacle placed in the fluid is subject to a frictional drag (Fig. 1b). As a result, the pressure field developed around it is distorted, and the surrounding fluid can be splashed. Herein, the strategy of the inventors of the present disclosure is to create a drag-free space and to locate the obstacle within the hydrodynamic cloak. The fluidic space is compressed from the cylindrical region (0<r<b) into the annular region (a<r'<b) and then the empty space (0<r'<a) is generated in the coordinate system (FIG. 1C and FIG. 1D).

(1)

(2)

(3)

Herein, ∇´,

, and

is defined as

=

, where

The transformed viscosity tensor,

, for the ideal cloak is given by

, where a is an inner radius of the cloaking shell, and r' is a radial axis in the coordinate transformed physical space (see FIG. 4). FIG. 4 shows modeling of ideal hydrodynamic cloak. On the other hand, the ideal cloak is not experimentally feasible due to extremely large variation in material properties along both principal axes. In this respect, the viscosity tensor is manipulated to obtain the "reduced" cloak with mitigated material parameters. It possesses the similar dispersion in the space as the ideal case although the impedance mismatch at the outer boundary of the cloak is resolved by multiplying the viscosity tensor by a factor of 1.58. The reduced form of the transformed viscosity tensor (

) is expressed as follows:

(4)

The radial profiles of

are shown in Fig. 2b. Herein,

is a function of the radial distance, but

is a constant lower than the original viscosity.

Once theoretical modeling for the viscosity tensor is achieved, the following numerical simulation is conducted to confirm the hydrodynamic cloak. FIG. 2C presents the pressure, pressure gradient, and velocity fields of the bare space case without cloaking, respectively. Since there is no perturbation in the domain, uniform pressure and velocity contours are obtained. When a cylindrical obstacle with a radius of 2 mm and a height of 50 μm is placed in the fluid, the obstacle cannot avoid the drag force exerted by the fluid flow and the streamlines are disturbed severely (see Fig. 2d).

The simulation results acquired for the case of cloaking the obstacle are shown in Fig. 2e. The applied pressure and pressure gradient fields are not perturbed by the obstacle encircled with the hydrodynamic cloak. Consequently, the pressure distribution outside the cloak remains unchanged compared with the bare space case. Furthermore, the pressure around the obstacle surrounded by the cloak is constant resulting in zero drag force. The hydrodynamic stress induced by the inclusion of the obstacle does not influence the central region. This implies that since an external observer is unaware of the presence of the obstacle in the flow due to the unchanged velocity fields and streamlines outside the cloak, the object amid the flow can be hidden hydrodynamically (see Fig. 2e). Interestingly, a flow rate in the cloaking shell region (a<r'<b) is drastically increased to maintain the uniform flow rate outside the cloak. The exclusion of the fluid momentum in the center region leads to such increase in the velocity in the shell region for the flow compensation, which is a physical driving force for the fluidic momentum to bypass the obstacle.

In the next case, the obstacle is removed from the flow domain to demonstrate the cloaking more clearly (Fig. 2f). The flow behavior for the case of the cloak only is the same as that for the case of with cloak and obstacle. This indicates that the hydrodynamic cloak is independent of the shape of the obstacles and the presence of the obstacle is not required for cloaking operation. As stated above, the cloaking is solely determined by the characteristics of the form-invariance of governing equations. As a result, a drag-free space is generated in the center region with zero pressure gradient. The velocity inside the cloak is negligible along the flow direction. Indeed, the fluidic inertia cannot penetrate the cloak and reach the center region even if there is no solid barrier in the flow domain.

The drag force (

) and the drag coefficient (

) of the obstacle are calculated numerically, where

is the normal stress in y direction acting on the obstacle surface,

is the mass density of fluid,

is the velocity of the obstacle with respect to fluid flow, s is the surface area of the obstacle, and A is the reference area of the obstacle. The calculated drag force imposed on the obstacle is initially 120 μN but decreases by 7 times (i.e., 18 μN) after applying the cloak. Besides, the dimensionless quantity, the drag coefficient is reduced by 10 times (i.e., from 4569.7 to 440.8). This finding shows a possibility of the hydrodynamic cloak not only theoretically but also numerically. While the calculated drag values do not become zero due to the mathematical iteration, the dramatic reduction in the fluidic drag is very significant. Furthermore, if the obstacle is smaller in size than the cloaked central region, the resulting drag force is negligible.

To implement the hydrodynamic cloak experimentally, a multilayered cloak is designed based on the homogenized layer method and analyzed numerically. The cloaking shell region (a <r'<b) is divided into ten annular layers, and the averaged viscosity components of the scaled

are assigned to each layer as plotted in Fig. 2b. Therefore, the viscosity distribution in the shell region becomes discretized in a stepwise manner. The multilayered cloak is modeled numerically to validate the design, and the simulation results are shown in FIG. 5. FIG. 5 shows modeling of multilayered hydrodynamic cloak. While a slightly distinguishable non-uniformity was observed at the interface of the layers, the drag-free space was created successfully.

The cloaking shell layers have different anisotropic viscosity tensors, which are given by controlling the structure of each unit cell. Thus far, a few unit cells have been devised based on effective medium theory for fabricating metamaterial cloaks. However, a unit cell for hydrodynamic properties has never been reported. The inventors of the present disclosure adopted homogenization theory in the present disclosure to design and fabricate a unit cell with effective viscosity, which is acquired by manipulating superficial velocity in the unit cell. The effective viscosity of a unit cell can be defined as

, where

and

are the velocity fields with and without a micropillar, respectively,

is the superficially averaged quantity, and

is the intrinsic viscosity of the fluid. The dimension of the unit cell is 200 x 200 x 50 μm³ (see FIG. 6A). FIG. 6 shows design and fabrication of hydrodynamic cloak. Micropillars are embedded in each unit cell to control the superficial velocity in the cell. The simulated velocity field and the corresponding effective viscosity for each unit cell are shown in FIG. 7. FIG. 7 shows modeling of effective viscosity tensor using unit cells. The size of the unit cell should be smaller than the characteristic scale of momentum diffusivity of fluid. In the present disclosure, water with a kinematic viscosity of 0.9 mm²/s is selected as the fluid for the homogenization. All the viscosity values were adjusted 3.47 times higher to employ experimentally feasible azimuthal viscosity values. Hence, the length of the micropillar (ℓ) is varied from 188 μm to 159 μm, while the thickness and height are fixed at 50 μm. As a result, the azimuthal viscosity is 1.3 mPa·s, and the radial component varies from 5.3 to 79.27 mPa·s (FIG. 6B and Table. 1).

Layer No.	ℓ (㎛)	μ_eff,˚r (mPa·s)	μ_eff,˚θ (mPa·s)
Center region	-	NONE	NONE
1	188	79.269	1.292
2	188	79.269	1.292
3	183	33.623	1.302
4	178	18.760	1.311
5	174	13.126	1.318
6	171	10.374	1.322
7	168	8.551	1.328
8	164	6.803	1.332
9	161	6.094	1.338
10	159	5.300	1.338
Background	-	2.758	2.758

Table 1 shows mapping of effective viscosities onto each layer for modeling hydrodynamic cloak.These viscosity values are assigned onto the layers for the hydrodynamic cloak (Fig. 6c). The layers of the cloak from inside to outside regions consist of 65, 72, 79, 85, 91, 97, 104, 110, 116, and 120 unit cells, respectively. To resolve the impedance mismatching problem originated from errors in the effective medium approximation, the effective viscosity of the background is given as 2.76 mPa·s by applying an isotropic unit cell composed of a cylinder with a 150 μm diameter.

The designed metamaterial cloak is realized and validated using a microfluidic device. Silicon (Si) masters for the cases of bare space, with obstacle, with obstacle and cloak, and with cloak only are patterned by using photolithography. The microfluidic devices are manufactured by replicating the silicon masters with polydimethylsiloxane (PDMS) and bonding it on slide glasses.

The designed microfluidic metamaterial is simulated numerically to confirm the hydrodynamic cloaking. The predicted pressure fields, velocity fields, and flow streamlines are presented in FIG. 8A through FIG. 8C. FIG. 8 shows prediction and experimental validation of hydrodynamic cloak. The simulation results coincide with experimental observation of the cloaks designed using continuous and discretized viscosity distributions (see FIG. 9 through FIG. 11). FIG. 8A and FIG. 8B show the microfluidic metamaterials, i.e., the hydrodynamic cloaking cases, constructed with help of the effective viscosity yield pressure and velocity distributions identical to the theoretically implemented cases shown in FIG. 3C through FIG. 3F, upper and lower. The pressure and velocity contours outside the cloak are not disturbed by the existence of the cloaked obstacle. This implies that a drag-free space is created in the central region.

The streamlines in the microfluidic devices are visualized to verify the hydrodynamic cloak experimentally by using fluorescent microparticles. FIG. 8D shows that the streamlines obtained experimentally are the same as those predicted numerically (see Fig. 8c). These results confirm that the hydrodynamic behavior in a microfluidic device can be manipulated using the effective viscosity defined by homogenizing flow in a unit cell. In comparison with the bare space case, the streamlines observed for the case of the obstacle only are disturbed by the fluidic resistance. This drag phenomenon can be prevented by encircling the obstacle with the hydrodynamic cloak since the pressure field outside the cloak is maintained. Subsequently, the obstacle is free from the drag force. The drag-free space is visualized experimentally by removing the obstacle. This, indeed, indicates that the hydrodynamic cloak is materialized successfully in a microfluidic device albeit a flow rate of 0.5 mm/s is measured in the central region.

The elements of coordinate transformations should be defined before the development of transformation hydrodynamics. Basically, a coordinate transformation method demonstrates the relationship between two coordinate systems, old and new coordinate systems with two bases, Cartesian and orthogonal bases. The position vector

is expressed as unprimed (

) in the old coordinate system and primed (

) in the new coordinate system. Each position vector is composed of contravariant components (

or

) and unit covariant vectors (

or

) as follows:

(5)

(6)

Then, connection between the two coordinate systems is defined as follows:

(7)

In this relationship, the Jacobian matrix (

), as a backward transformation matrix, which transforms the coordinate basis from new to old, is as follows:

(8)

The inventors of the present disclosure thought that transformation optics can be translated into hydrodynamics governed by Navier-Stokes equations. A form invariance of Navier-Stokes equations is proved below.

Navier-Stokes equations consist of the momentum balance equation (Equation 9) and the mass balance equation (Equation 10) as follows:

(9)

(10)

Herein,

is the density of fluid,

is the velocity field,

is the viscous stress tensor, p is the hydrostatic pressure,

is the unit identity tensor, and

is the term about body acceleration (generally, gravity). By assuming an incompressible fluid and neglecting gravity, Navier-Stokes equations can be simplified to as follows:

(11)

(12)

In Equation 11, the viscous stress tensor (

) can be expressed as a Stokes' stress constitutive equation shown below.

(13)

Herein,

is the viscosity of fluid.

For transformation optics of Maxwell's equations, coordinate transformations of the curl and divergence of vector fields should be defined. Unfortunately, this mathematical approach cannot be directly applied to transformation hydrodynamics based on the Navier-Stokes equations since the momentum equation (Equation 11) contains the divergence term of the second order stress tensor (

), not the curl or divergence of first-order tensors. Accordingly, the coordinate transformation of the divergence of the stress tensor should be derived. By the divergence theorem, 0

(14)

Then, the left-hand-side of Equation 14 is expressed with the stress tensor in the new coordinate system (

) as follows:

(15)

And, the right-hand-side of Equation 14 can be transformed as follows:

(16)

Consequently, from the equations above, the following relationship is obtained:

(17)

Herein, the coordinate transformation of the del operator (▽) is defined as

. Therefore, the divergence of the stress tensor in the physical space (the new coordinate system) is related with the equation in the virtual space (the old coordinate system) as follows:

(18)

From Equation 11 and Equation 18, the following relationship is derived:

(19)

Herein,

and

. Meanwhile, the viscous stress tensor in the physical space (

in Equation 19) can be expressed with the strain tensor as follows:

(20)

Herein,

by a transformation rule. Hence, from Equation 19 and Equation 20, the following relationship is accomplished:

(21)

If the inverse matrix of

is multiplied on both sides of Equation 21, the following equation can be obtained:

(22)

Consequently, the momentum equation in the physical space is derived with the transformed parameters as follows:

(23)

Herein,

and

are the fluid viscosity and density as a form of the 2nd-order tensor defined as follows:

(24)

(25)

Meanwhile, a mathematical form of the mass balance equation in the physical space (Equation 26) is easily satisfied because this equation does not include a high order tensor term but only the velocity vector term.

(26)

For the cloaking phenomena, the Jacobian matrix

, which connects the virtual space (

for the Cartesian coordinate system and

for the orthogonal coordinate system) and the physical space (

for the Cartesian coordinate system and

for the orthogonal coordinate system), should be defined as follows:

(27)

Herein,

is the Jacobian matrix between two orthogonal coordinate systems (

) and

is the Jacobian matrix between the orthogonal coordinate system and the Cartesian coordinate system in virtual space. The total Jacobian matrix (

) transforms the coordinate systems in the order of

.

The next step is to obtain the total Jacobian matrix (

) for cloaking a cylindrical object. At first, the linear geometric transformation, i.e., a radial stretch, was assumed under the two-dimensional cylindrical case. When the inner radius was a and the outer radius was b, the relation between the two spaces that compresses a fluidic space from the cylindrical region (0<r <b) into the annular region (a<r'<b) is defined as follows:

(28)

(29)

(30)

Then, the Jacobian matrix for the radial stretch (

) is expressed in a matrix form as shown below:

(31)

Also, the relationship between the cylindrical coordinate system and the Cartesian coordinate system is defined as follows:

(32)

(33)

The corresponding Jacobian matrix for the backward transformation (

) is as follows:

(34)

Finally, the total Jacobian matrix for the cylindrical cloak (

) and its determinant are obtained as follows:

(35)

(36)

All the pieces required for deriving the transformed viscosity tensor are collected. For feasible experimental implementation, the steady-state creeping flow case of Navier-Stokes equations was chosen not to take into account the mapping of the fluidic density tensor (

).

(37)

Following Equations 24, 35, and 36, the transformed viscosity tensor (

) for the cylinderical cloak is derived as follows:

(38)

The viscosity tensor in the ideal case (

, Equation 38) is very difficult to be realized experimentally since the tensor components at specific positions (e.g., at

or

) have infinite values and vary extremely in each axial direction. In other words, a matrix singularity strictly hinders design and fabrication of a metamaterial cloak based on the ideal parameters. Instead of, a reduced set of material parameters has been considered to mitigate this condition in the previous researches. The reduced transformed viscosity tensor (

) is calculated by muliplying

by

as follows:

(39)

The z-directional components were not under consideration since the inventors of the present disclosure solved 2-D fluid flow.

In the present disclosure, the inventors propose a novel strategy for the design and fabrication of a hydrodynamic metamaterial cloak which can create a drag-free space in it. Virtual fluidic space was controlled using transformation hydrodynamics based on the form invariance of the Navier-Stokes equations. The hydrodynamic cloaking and the resulting drag-free space were numerically simulated and experimentally realized with a microfluidic device.

Further, in the present disclosure, the inventors introduce a new class of metamaterial, the hydrodynamic cloak. Theoretical and numerical analyses of flow behavior were carried out for design of the cloak. The designed metamaterial was realized in a microfluidic device, and the fluidic cloaking was validated experimentally. The hydrodynamic cloak will offer a systematic strategy that can induce flow behavior which is not found in nature. Since the cloaking developed in the present disclosure does not rely upon the absolute value of viscosity, it can be applied to the mechanics of fluids including gases, liquids, and plasmas. A wide range of fluid flow problems in mechanical, civil, chemical, and biomedical engineering applications can be solved by adopting the hydrodynamic cloak. For instance, hydrological and meteorological disasters can be minimized and drag-free vehicles such as aircrafts, automobiles, and submarines can be designed by applying the cloaking materials.

In summary, fluidic drag is an inevitable phenomenon in aerodynamics and hydrodynamics when a solid object moves relatively with respect to a surrounding fluid. In this sense, exclusion of such a drag force is a great challenge to achieve for scientific and engineering applications. The inventors of the present disclosure devised a hydrodynamic cloak that conceals an object in a flowing fluid via transformation of fluidic coordinate system. Transformation hydrodynamics was introduced in a bid to create a desired virtual space and to control fluidic energy in the space. The present disclosure proposes a strategic design for the cloaking metamaterial with a drag-free space, and the resulting structure can be implemented with the use of a microfluidic device. The metamaterial fabricated by theoretical mapping of effective viscosity tensors enables the cloaking of fluid dynamics.

The above description of the present disclosure is provided for the purpose of illustration, and it would be understood by a person with ordinary skill in the art that various changes and modifications may be made without changing technical conception and essential features of the present disclosure. Thus, it is clear that the above-described embodiments are illustrative in all aspects and do not limit the present disclosure. For example, each component described to be of a single type can be implemented in a distributed manner. Likewise, components described to be distributed can be implemented in a combined manner.

The scope of the present disclosure is defined by the following claims rather than by the detailed description of the embodiment. It shall be understood that all modifications and embodiments conceived from the meaning and scope of the claims and their equivalents are included in the scope of the present disclosure.

Claims

A method for designing a hydrodynamic cloaking metamaterial, the method comprising:

providing a metamaterial including at least one unit cell in a second region surrounding a first region including a target object or target space present in a fluid flow,

wherein each of the at least one unit cell included in the metamaterial has a pre-determined effective viscosity so as to form a drag-free space in the first region for hydrodynamically cloaking the first region including the target object or target space.
The method of Claim 1,

wherein the pre-determined effective viscosity of each of the at least one unit cell included in the metamaterial is calculated by using a coordinate-transformed viscosity tensor obtained via transformation of a fluidic coordinate system to distort a space surrounding the target object or target space so as to impart zero drag force in the first region for forming a hydrodynamically cloaking shell in the second region.
The method of Claim 2,

wherein the pre-determined effective viscosity is set by numerical simulation for a fluid viscosity tensor represented by the following equation 1:

[Equation 1]

wherein a is an inner radius of the cloaking shell, and r' is a radial axis in the coordinate transformed physical space.
The method of Claim 3,

wherein a value of the fluid viscosity tensor is set to change in the second region in a radial direction from the target object or target space such that a velocity contour and and/or a pressure contour of the fluid outside the second region becomes uniform to form the drag-free space in the first region.
The method of Claim 2,

wherein a flow rate in a region of the hydrodynamic cloaking shell formed in the second region is set to abruptly increase such that a -uniform flow rate outside the second region maintains for hydrodynamically cloaking the target object or target space.
The method of Claim 1,

wherein the at least one unit cell is configured in a spaced apart array to each other.
The method of Claim 1,

wherein the at least one unit cell is arrayed in a monolayer or multilayer in the second region -around the target object or target space.
The method of Claim 7, further comprising:

simulation which includes setting the at least one unit cell in a multilayer and assigning a different value of the effective viscosity of each layer to decrease in a radial direction from the target object or target space such that a velocity field and and/or a pressure field of the fluid outside the second region becomes uniform.
The method of Claim 8,

wherein the value of the effective viscosity of each layer is assigned constant in an azimuthal direction from the target object or target space.
The method of Claim 1,

wherein each of the at least one unit cell includes a microstructure therein.
The method of Claim 10,

wherein the effective viscosity of the each unit cell is represented by the following equation 2:

[Equation 2]

wherein
and
are the viscosity fields with and without the microstructure, respectively,
is a superficially averaged quantity, and
is an intrinsic viscosity of the fluid.
The method of Claim 10,

wherein the at least one unit cell is arrayed in a multilayer in the second region -around the target object or target space, the microstructure included in each of the at least one unit cell has the same thickness and height to another, and a length of the microstructure gradually decreases in a radial direction from the target object or target space.
The method of Claim 1,

wherein the fluid includes a liquid, gas or plasma.
The method of Claim 1,

wherein the metamaterial further includes at least one isotropic unit cell in addition to the at least one unit cell included having the effective viscosity.
A hydrodynamic cloaking metamaterial, comprising:

at least one unit cell configured in a second region surrounding a first region including a target object or target space present in a fluid flow,

wherein each of the at least one unit cell included in the metamaterial has a pre-determined effective viscosity so as to form a drag-free space in the first region for hydrodynamically cloaking the first region including the target object or target space.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein the hydrodynamic cloaking metamaterial is designed by a method in accordance with any one of Claims 1 to 14.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein the hydrodynamic cloaking metamaterial forms a hydrodynamically cloaking shell in the second region when the target object or target space is present in a fluid flow.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein the pre-determined effective viscosity is calculated by using a coordinate-transformed viscosity tensor obtained via transformation of a fluidic coordinate system to distort a space surrounding the target object or target space so as to impart zero drag force in in the first region for forming a hydrodynamically cloaking shell in the second region.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein the at least one unit cell is configured in a spaced apart array to each other in a monolayer or multilayer.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein the at least one unit cell is configured in the second region isotropically around the target object or target space.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein each of the at least one unit cell includes a microstructure therein.
The hydrodynamic cloaking metamaterial of Claim 21,

wherein the microstructure has a polyhedral, cylindrical, spherical or elliptical shape.
The hydrodynamic cloaking metamaterial of Claim 21,

wherein the at least one unit cell is arrayed in a multilayer in the second region around the target object or target space, the microstructure included in each of the at least one unit cell has the same thickness and height to another, and a length of the microstructure gradually decreases in a radial direction from the target object or target space.
The hydrodynamic cloaking metamaterial of Claim 15,

wherein the metamaterial further includes at least one isotropic unit cell in addition to the at least one unit cell included having the pre-determined effective viscosity.