A mathematical theory of microscale hydrodynamic cloaking and shielding by electro-osmosis

During the past 16 years, there has been rapid progress in rendering objects invisible by cloaking them with metamaterials in physics and mathematics subjects. One influential methodology to design metamaterials is the transformation-based approach. Since the pioneering works [16, 32, 22], many studies about cloaking based on the transformation theory appear in succession in various scientific areas, including acoustic waves [4, 14, 20, 25], conductive heat flux [13, 27], electromagnetic waves[7, 15, 17], stresses in an elastic medium [24, 33, 23], dc electric currents [37], and quantum mechanical matter waves [38]. Effectively, these cloaks are shells composed of metamaterials with a tailor-made distribution of the effective material parameters, determined by the invariance of governing equations under a transformation of the spatial coordinates. However, the requirement for inhomogeneous and anisotropic parameters makes it difficult to fabricate devices designed by transformation optics. Consequently, the feasibility of fabricating cloaks using composite natural materials has attracted great interest. For this purpose, scattering-cancellation technology has been developed and successfully used in electromagnetism [1] and other fields [11]. Generally speaking, this method can realize a similar function to transformation optics, while it only needs bilayer or monolayer structures and homogeneous isotropic bulk materials. In addition to the bulk materials, the metasurface cloak has also received much attention with the advantages of low weight and thinness. Instead of employing a shell, the metasurface cloak uses an ultrathin frequency-selective surface which is designed so that the induced currents along the surface cancel the scattering from the object to a cloak [2, 9, 10, 28]. Such a strategy is particularly relevant at microwave frequencies at which frequency-selective surfaces are readily available and easy to fabricate [26, 34].

In fluid, hydrodynamic cloaking has been a subject of intense research all the time. There are reports in the physics literature. The first systematic analysis of hydrodynamic cloaking is due to Urzhumov and Smith [35], based on transformation theory. Later, they further extended the analysis to the case of a two-dimensional flow around a cylinder in a medium with mixed positive and negative permeability [36]. For the high Reynolds numbers, it was demonstrated how to confine the water waves in a certain area for cloaking regions [40, 39]. More recently, another hydrodynamic model has been used to control fluid motion, i.e., the creeping flow or Stokes flow (Reynolds number $Re\ll 1$ ) inside two parallel plates, and a series of experimental works have been reported [29, 30, 31, 8]. The gap between two plates is much smaller than the characteristic length of the other two spatial dimensions, so the model is also called the Hele-Shaw flow or Hele-Shaw cell [18]. By using these microfluidic structures, Park et al [29, 30] have demonstrated by simulation that such anisotropic fluid media can be mimicked within the cloak, thereby producing the desired hydrodynamic cloaking effect. We should mention that a transformed fluid medium is not yet physically realizable; proof of concepts is so far by numerical simulation. The implementation of such a transformation-based fluid-flow cloak relied on 10 layers of metamaterial microstructures, as well as a fluid background filled with microcylinders to avoid impedance mismatch. Hence, there has been a growing interest in realizing metamaterial-less hydrodynamic cloaks.

In particular, in [8] Boyko et al present a new theoretical approach and an experimental demonstration of hydrodynamic cloaking and shielding in a Hele-Shaw cell that does not rely on metamaterials. In analogy to optics, “hydrodynamic cloaking” refers to a state wherein the (flow) field external to some region around an object is unaffected by the presence of the object. “Hydrodynamic cloaking” refers to the elimination of hydrodynamic forces on the object. The mechanism is based on the fact that flow fields on small scales are completely governed by momentum sources at boundaries and can be linearly superposed. An effective way to create such momentum sources is electro-osmotic flow (EOF)—an electrokinetic phenomenon that generates flow due to the interaction of an applied electric field with the native or induced net charge at liquid-solid interfaces [19]. The fluid pressure in the Hele-Shaw cell also satisfies a Laplacian-type equation, which has the analytical solutions for circular cylinder objects in the polar coordinate. With analytical solutions, Boyko et al first achieve both cloaking and shielding conditions for circular cylinder objects. Further, they establish approximate cloaking and shielding for the more complex shape of a slight deform from a perfectly cylindrical shape using shape perturbation theory.

In this paper, we greatly extend the results of Boyko et al [8] and establish a more general mathematical framework. For details, the contributions of this work are fourfold:

• •

  Based on the physics literature [8], we give the rigorous mathematical definition of hydrodynamic shielding and cloaking.

• •

  The representation formula of solution of the coupled system is obtained firstly, which gives a quantitative description of the hydrodynamic model.

• •

  By using the uniform approach–layer potential theory, we establish sharp conditions that can ensure the occurrence of the hydrodynamic cloaking and shielding for annulus (radial case) and confocal ellipses (non-radial case). Especially, for the confocal ellipses case which is not considered in [8], we introduce additional elliptic coordinates technique to overcome the difficulty caused by non-radial geometry.

• •

  For more general geometry, we further propose an optimization method to design the hydrodynamic shielding and cloaking, and prove the well-posedness of optimization problem. More important, the condition that can ensure the occurrence of approximate cloaks and shields for arbitrary-geometry are also established. A large amount of numerical experiments, which contains smooth objects, non-smooth objects and multiple objects, indicates that the optimized zeta potential can achieve the hydrodynamic shielding and cloaking effectively.

The paper is organized in the following way. We begin with the mathematical setting of the problem and statement of the main results in Section 2. In Section 3, we first present some preliminary knowledge on boundary layer potentials and then establish the representation formula of solution of the governing equations. Section 4 is devoted to the study of the perfect and approximate cloaking and shielding conditions by the analytical method and the optimization method, respectively. In Section 5, we conduct numerical experiments to corroborate our theoretical findings. The paper is concluded in Section 6 with some relevant discussions.

To begin, we consider the creeping flow of a viscous fluid of density $\rho$, viscosity $\mu$, and dielectric permittivity $\varepsilon$ within a narrow gap $\tilde{h}$ between two parallel plates of length $\tilde{L}$ and width $\tilde{W}$ forming a Hele-Shaw configuration in $\mathbb{R}^{3}$, as shown in Figure 2.1(a). We employ a Cartesian coordinate system $\tilde{x}=(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3})\in\mathbb{R}^{3}$, where the $\tilde{x}_{1}$ and $\tilde{x}_{2}$ axes lie at the lower plane and the $\tilde{x}_{3}$ axis is perpendicular thereto. Let $\tilde{\zeta}^{L}=\tilde{\zeta}^{L}(\tilde{x})$ and $\tilde{\zeta}^{U}=\tilde{\zeta}^{U}(\tilde{x})$ be an arbitrary zeta-potential distribution in the lower and upper plates. Let $\tilde{\bm{v}}=(\tilde{\bm{u}}(\tilde{x},\tilde{t}),\tilde{w}(\tilde{x},\tilde{t}))$ and $\tilde{p}=\tilde{p}(\tilde{x})$ denote the velocity field and the pressure, respectively, of the fluid. Here $\tilde{\bm{u}}(\tilde{x},\tilde{t})$ is the in-plane velocity field. When an electrostatic in-plane electric field $\tilde{\bm{E}}$ is applied parallel to the plates, the fluid motion is then governed by the continuity and momentum equations

with the Helmholtz–Smoluchowski slip boundary conditions

These equations contain three independent dimensional parameters, the density $\rho$, the viscosity $\mu$ and the dielectric permittivity $\varepsilon$, which are both constant and assumed to be known. For convenience in what follows, we mark the dimensional variables that appear in these equations with a tilde, for instance, $\tilde{\bm{u}}$, $\tilde{p}$, and so on. In addition, we also denote the dimensional gradient and Laplace operator as $\tilde{\nabla}$ and $\tilde{\Delta}$.

By making use of the Helmholtz–Smoluchowski slip condition, we have implicitly assumed that surface conduction is negligible and thus consider asymptotically small Dukhin numbers. Under this assumption, the fluid in the bulk is electrically neutral and the electric field $\tilde{\bm{E}}$ is solenoidal, i.e., $\tilde{\nabla}\cdot\tilde{\bm{E}}=0$. The electric field can be expressed through an electrostatic potential $\tilde{\varphi}$, $\tilde{\bm{E}}=-\tilde{\nabla}\tilde{\varphi}$, that is governed by the Laplace equation $\Delta\tilde{\varphi}=0$, which satisfies the insulation boundary conditions on the walls, $\frac{\partial\tilde{\varphi}}{\partial\tilde{x}_{3}}|_{\tilde{x}_{3}=0,\tilde{h}}=0$.

In microscale flows, fluidic inertia is commonly negligible compared to viscous stresses. Further, the gap is narrow. Therefore, we restrict our analysis to shallow geometries ($\tilde{h}\ll\tilde{L},\tilde{W}$) and neglect fluidic inertia, represented by a small reduced Reynolds number. Applying the lubrication approximation, we average over the depth of the cell and reduce the analysis to a two-dimensional problem in $\mathbb{R}^{2}$. The governing equations for the depth-averaged velocity $\bm{\tilde{}}{\bm{u}}_{aver}$, the pressure $\tilde{p}$, and the electrostatic potential $\tilde{\varphi}$ in $\tilde{x}_{1}$–$\tilde{x}_{2}$ plane are (see [8] )

where $\bm{\tilde{}}{\bm{u}}_{aver}$ is the average value of $\tilde{\bm{u}}$ over $\tilde{x}_{3}$ by integration, $\tilde{\bm{u}}_{slip}=\frac{\tilde{\varepsilon}\langle\tilde{\zeta}\rangle}{\tilde{\mu}}\tilde{\nabla}\tilde{\varphi}$ is the depth-averaged Helmholtz-Smoluchowski slip velocity, ${\zeta}_{mean}$ is the arithmetic mean value of the zeta potential on the lower and upper plates, i.e., ${\zeta}_{mean}=(\tilde{\zeta}^{L}+\tilde{\zeta}^{U})/2$. In the later analysis, we assume the zeta potential on the lower and upper plates is same. Scaling by the characteristic dimensions, we introduce the following dimensionless equations:

where $\bm{u}_{aver}$, $p$, $\varphi$ and $\zeta_{mean}$ are non-dimensional normalized variables.

In addition to these equations and the geometry of the flow domain, we also require the boundary conditions that apply at the boundaries of the domain to completely characterize the flow. These boundary conditions include the form and magnitudes of the velocity on the left inlet and right outlet to the domain. In this paper, we mainly consider a pillar-shaped object with arbitrary cross-sectional shape confined between the walls of a Hele-Shaw cell and subjected to a pressure-driven flow with an externally imposed mean velocity $\tilde{u}_{ext}$ and electric field $\tilde{E}$ along the $\tilde{x}_{1}$–axis, as shown in Figure 2.1. The reduced two-dimensional problem is shown in Figure 2.1(b). To cloak and shield the object hydrodynamically, we are concerned with the scattering by the object surrounded by a region, i.e., with exterior boundary value problems for the Laplace equation. We solve the problem assuming an unbounded domain, enabled by the fact that the boundaries of the chamber are located far from the cloaking and shielding region.

To mathematically state the problem, let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$ and let $D$ (object) be a domain whose closure is contained in $\Omega$. Throughout this paper, we assume that $\Omega$ and $D$ are of class $C^{1,\alpha}$ for some $0<\alpha<1$. Let $H(x)$ and $P(x)$ be the harmonic function in $\mathbb{R}^{2}$, denoting the background electrostatic potential and pressure field. For a given constant parameter $\zeta_{0}\in\mathbb{R}$, the zeta potential distribution in $\mathbb{R}^{2}\setminus\overline{D}$ is given by

We may consider the configuration as an insulation and no-penetration core coated by the shell (control region) $\Omega\setminus\overline{D}$ with zeta potential $\zeta_{0}$. From the equations (2.2) and the assumption of unbounded domain, the governing equations for non-uniform electro-osmotic flow via a Hele-Shaw configuration is modeled as follows:

where $\frac{\partial}{\partial\nu}$ denote the outward normal derivative and we use the notation $\frac{\partial p}{\partial\nu}\big{|}_{\pm}$ indicating

where $\nu$ is the outward unit normal vector to $\partial\Omega$.

We are now in a position to introduce the definition of cloaking and shielding which play a central role in this paper.

Let the depth-averaged hydrodynamic force acting on the object be given by

where $\bm{\sigma}=-pI$ is the normalized stress tensor.

Outside the cloaking region, the pressure is related to the velocity field through $\langle\bm{u}\rangle=-\nabla p/12$ subjected to the boundary condition $p(x)=P(x)$ as $|x|\rightarrow\infty$, and therefore, according to the Definition 2.1, the condition (2.4) can be expressed in terms of the pressure as

Similarly, from (2.5) it follows that the depth-averaged hydrodynamic force $\bm{F}$ vanishes provided

According to Definition 2.2, hydrodynamic shielding occurs. In this paper, we assume $D$ and $\Omega$ are known, and want to find appropriate zeta potential $\zeta_{0}$ to achieve the hydrodynamic shielding and cloaking effectively.

Our main results in this paper are given in the following theorems. The proofs are given in Section 4.1 4.2 and 4.3, respectively.

For the arbitrary-geometry for $D$ and $\Omega$, by using the optimization approach, we shall establish a sufficient condition for the occurrence of approximate hydrodynamic cloaking (or shielding) in the following theorem.

In this section, we first collect some preliminary knowledge on boundary layer potentials and then establish the representation formula of solution of the governing equations. Let $\Gamma_{i}:=\partial D$ and $\Gamma_{e}:=\partial\Omega$. For $\Gamma=\Gamma_{i}$ or $\Gamma_{e}$, let us now introduce the single-layer potential by

where $\vartheta\in L^{2}(\Gamma)$ is the density function, and the Green function $G(x,y)$ to the Laplace in $\mathbb{R}^{2}$ is given by

Then the following jump relation holds :

where $\mathcal{K}^{*}_{\Gamma}$ denote Neumann-Poincar$\acute{e}$ (NP) operator defined by

To establish the representation formula of the solution, we make use of the following lemma.

By the layer potential theory, we can obtain the following theorem using Lemma 3.1.

The following corollary is a direct consequence of Theorem 3.2, and thus its proof is skipped.

According to Theorem 3.2 and Corollary 3.3, we obtain the representation formula and quantitative estimation of solution of the electro-osmosis model. It will be used to deduce the hydrodynamic cloaking and shielding conditions in the following sections.

This section is devoted to the proofs of Theorems 2.1 and 2.2, which determine the conditions for hydrodynamic cloaking and shielding. We first consider the microscale hydrodynamic cloaking via electro-osmosis when the domains $D$ and $\Omega$ are concentric disks and confocal ellipses in Subsections 4.1 and 4.2, respectively. We calculate the explicit form of the solution to (2.3) for this two special cases. In Subsection 4.3, we then consider hydrodynamic cloaking and shielding for general shapes via an optimization method.