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We present a finite element analysis of a diffusion problem involving a coated cylinder enabling the rotation of heat fluxes. The coating consists of a heterogeneous anisotropic conductivity deduced from a geometric transformation in the time dependent heat equation. In contrast to thermal cloak and concentrator, specific heat and density are not affected by the transformation in the rotator. Therein, thermal flux diffuses from region of lower temperature to higher temperature, leading to an apparent negative conductivity analogous to what was observed in transformed thermostatics. When a conducting object lies inside the rotator, it appears as if rotated by certain angle to an external observer, what can be seen as a thermal illusion. A structured rotator is finally proposed inspired by earlier designs of thermostatic and microwave rotators.
© 2013 Optical Society of America
1. Introduction
Six years ago, it was suggested by Pendry, Schurig and Smith that an object surrounded by a coating consisting of an heterogeneous anisotropic material becomes invisible for electromagnetic waves [1]. The theoretical idea based on geometric transformations in the Maxwell’s equations [2] has been since then confirmed by some experiments. Leonhardt independently analyzed conformal invisibility devices using the Schrödinger equation, which suppresses the anisotropy of the cloak at the cost of constraining the cloaking to relatively small wavelengths compared to the object to hide [3]. A plasmonic route to invisibility was proposed by Alu and Engheta [4] but it relies on a specific knowledge of the shape and material properties of the object being concealed. Last, Milton and Nicorovici proposed to cloak a countable set of line sources using anomalous resonance when it lies in the close neighborhood of a cylindrical coating filled with a negative refractive index material [5, 6].
In the present article, we build upon our recent proposal of cloaking in thermodynamics [7], which relies on a covariant formulation of the heat equation [8] and study a rotator for heat fluxes, which is a counterpart of the transformed medium Chen and Chan introduced in optics [9, 10]. It is also worthwhile noticing mathematicians working in the field of inverse problems were already aware of the counter-intuitive physics of the anisotropic conducting equation, albeit in a different context: cloaking in electric impedance tomography [11]. Interestingly, the isomorphism between the anisotropic conductivity and thermostatic equations make it possible to control the pathway of heat flux in a stationary setting, as observed by Fan et al. [12] (see also [6] for analogous cloaking in electrostatics) and experimentally validated by Narayano and Sato [13]. However, unlike for [12, 13], the object and rotator which we study here lie well within the intense near field of the heat source, and time plays an essential role, as in experiments on thermodynamics cloaking [14] from Wegener’s group.
2. Transformed heat equation for an area preserving transform
We consider the two-dimensional diffusion equation in a bounded cylindrical domain Ω with a source p
where u represents the distribution of temperature evolving with time t > 0, at each point x = (x, y) in Ω. Moreover, κ is the thermal conductivity (W.m−1.K−1 i.e. watt per meter kelvin in SI units), ρ is the density (kg.m−3 i.e. kilogram per cubic meter in SI units) and c the specific heat (or thermal) capacity (J.K−1.kg−1 i.e. joule per kilogram kelvin in SI units). It is customary to let κ go in front of the spatial derivatives when the medium is homogeneous. However, here we consider a heterogeneous medium, hence the spatial derivatives of κ are derivatives taken in distributional sense, thereby ensuring continuity of the heat flux κ∇u is encompassed in Eq. (1)).In this article, we further consider a source with a time step (Heaviside) variation and a singular (Dirac) spatial variation, that is: p(x, t) = p0H(t)δ(x−x0), with H the Heaviside function and Delta the Dirac distribution. This means that the source term is constant throughout time t > 0, while it is spatially localized on the line x = x0.
In the sequel, we adopt a covariant approach of the heat equation. For this, let us consider a map from a co-ordinate system {x′, y′} to the co-ordinate system {x, y} given by the transformation characterized by x(x′, y′) and y(x′, y′). Note that it is the transformed domain and co-ordinate system that are mapped onto the initial domain with Cartesian coordinates, and not the opposite. This change of co-ordinates is characterized by the transformation of the differentials through the Jacobian:
The only thing to do in the transformed coordinates is to replace the materials (often homogeneous and isotropic) by equivalent ones that are inhomogeneous (their thermal characteristics are no longer piecewise constant but merely depend on x′, y′ co-ordinates) and anisotropic ones (tensorial nature) whose properties are given by a new conductivity in Eq. (1) [7]
which is matrix-valued (anisotropic and hetereogeneous). Here, T = JTJ/det(J) is a representation of the metric tensor, which is associated with a distorted mesh, see Fig. 1(d) for the case of a rotator. Moreover, the product of density by heat capacity in the left hand side of Eq. (1) is now multiplied by the determinant of the Jacobian matrix J of the transformation.
Fig. 1 Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/(cρ) = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 5.10−5m and R2 = 3.10−4m, and rotation angle θ0 = 3π/4. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), t = 0.01s (b) and t = 0.1s (c); (d) The mesh formed by streamlines and isothermal values in the long time regime t ≥ 0.1s illustrates the deformation of the transformed thermal space.
The associated time-dependent transformed heat equation takes the form [7]:
Here, we would like to consider a geometric transform which is area preserving i.e. such that det(J) = 1. This greatly simplifies the control of heat fluxes in transient regimes, compared to what was done in [7]. For this, let us now consider the following transform [9] which rotates the inner disc r = R1 through an angle θ0, and gradually diminishes the rotation angle through the coronna R1 < r < R2 up to a vanishing rotation angle on the outer boundary of the disc r = R2. The effect on the metric is shown for 2 angles θ0 in Fig. 1(d) and Fig. 2(d).Fig. 2 Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/(cρ) = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 1.10−4m and R2 = 3.10−4m, and rotation angle θ0 = π/2. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), t = 0.01s (b) and t = 0.1s (c); (d) The mesh formed by streamlines and isothermal values in the long time regime t ≥ 0.1s illustrates the deformation of the transformed thermal space.
It remains to compute the transformation matrix T associated with this transformation and to express it in the Cartesian co-ordinates (x′, y′). As a result, we shall obtain the anisotropic conductivity characterizing the thermic metamaterial in the cylindrical rotator defined by the radii R1 (interior radius) and R2 (exterior radius). The Jacobian matrix Jxx′ is therefore merely the product of three elementary Jacobians : Jxx′ = JxθJθθ′Jθ′x′, where Jxθ = ∂(x, y)/∂(r, θ) = R(θ)diag(1, r) is the Jacobian associated with the change to cylindrical co-ordinates (r, θ), which involves the rotation (unimodular) matrix R(θ). In the same way, the Jacobian Jθ′x′ associated with the change to Cartesian co-ordinates (x′, y′) is such that Jθ′x′ = ∂(r′, θ′)/∂(x′, y′) = diag(1, 1/r′)R(−θ′), using the fact that R(θ′)−1 = R(−θ′). Finally, Jθθ′ = ∂(r, θ)/∂(r′, θ′) is the azimuthally stretched cylindrical co-ordinates as proposed in [9]. Importantly, det(Jθθ′) = 1 as the azimuthal stretch preserves the overall volume within the disc r = r′ ≤ R2 (unitary transform). As a first consequence, the transformation does not affect the product of heat capacity and density, which do not play a particular role in the design of thermic rotator.
Altogether, the material properties of the thermic rotator are described by the transformation matrix T through its inverse, and using again the fact that θ = θ′, we obtain that we give explicitly in Cartesian coordinates:
where and t = θ0rf′(r)/(f(R2) − f (R1)) = θ0r/(R2 − R1).3. Illustrative numerical examples
The anisotropic heterogeneous conductivity in the rotator should be now inserted in the transformed heat equation, which we want to solve numerically. The finite element method is the tool of choice, and is implemented in the commercial package COMSOL MULTIPHYSICS.
In Fig. 1, we show the result of considering a rotation angle 3π/4 in the transformation matrix. One can clearly see that the isotherms are tilted through an angle 3π/4 inside the inner disc r′ < R1 compared to the isotherms generated by the heat source outside the rotator: This leads to an apparent negative conductivity, whereby heat diffuses from regions of low to high temperature, a counterintuitive effect already encountered in thermostatic contexts [12, 13]. We note that this apparent negative conductivity effect is markedly enhanced at short times: Thermal exchanges are enhanced in the transient regime. Importantly, heat fluxes are smoothly rotated within the region R1 < r′ < R2, so that the rotator is itself invisible thermally for an external observer.
In Fig. 2 we consider a rotation angle θ0 = π/2 which results in a smaller anisotropy along the azimuthal angle, and a slightly less enhanced apparent negative conductivity and thermal exchanges. In Fig. 3, we consider a conducting object inside the inner disc, and we show that it appears as if rotated by an angle π/2 to an external observer when we take θ0 = π/2. Finally, we propose a structured design of a rotator for thermodynamics in Fig. 4.
Fig. 3 Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/(cρ) = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 2.10−4m and R2 = 3.10−4m, and rotation angle θ0 = π/2. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), (b) and t = 0.1s (c), (d); The rectangular object has a diffusivity one hundred times smaller than that of the surrounding medium and is rotated by an angle π/2 in (b) and (d) compared to (a) and (c); The temperature field in (a),(b) and (c),(d) is exactly the same oustide the rotator.
Fig. 4 Normalized temperature field for a source located on the left-hand side i.e. along the line x0 = (−7.10−4m, y) which diffuses heat in a medium with diffusivity κ/(cρ) = 1.10−5m2.s−1 containing a structured ring of center (0, 0) and radii R1 = 2.10−4m and R2 = 4.10−4m consisting of four rows with 50 insulating sectors (diffusivity one hundred times smaller than that of the surrounding medium) in each row; Rows are slightly rotated with respect to one another (through an angle θ = π/20), and separated by three thin highly conducting wires (diffusivity 100 times that of surrounding medium) directed along the azimuthal angle. The temperature is normalized throughout time on the left-hand side of the cell. Snapshots of temperature distribution at t = 0.005s (a) and t = 0.02s (b); White curves represent the trajectories of heat fluxes. The achieved rotation of heat flux is π/4.
4. Conclusion
We used the equivalence between a geometric transformation and a change of characteristics of a metamaterial [7] (anisotropic conductivity) to compute the thermal response of the rotator proposed in [9] in the context of optics. Importantly, the rotator design is based upon an area preserving transform, which greatly simplifies the transformed heat equation in transient regime. We proved that the rotator works even if the heat source is very close to the cloak (a fraction of thermal isovalues away), and the apparent negative thermal conductivity [12, 13] is markedly enhanced at short times. The rotator enhances exchanges by smoothly rotating heat fluxes. We observed some mirage effect whereby a conducting object located inside the rotator seems to be tilted in accordance with the involved geometric transformation. Finally, a structured rotator was proposed, following designs for thermostatics [13] and microwaves [10].
Acknowledgments
S.G. is thankful for an ERC funding through ANAMORPHISM project. The authors are also grateful for stimulating discussions with M. Zerrad.
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