We adapt tools of transformation optics, governed by a (elliptic) wave equation, to thermodynamics, governed by the (parabolic) heat equation. We apply this new concept to an invibility cloak in order to thermally protect a region (a dead core) and to a concentrator to focus heat flux in a small region. We finally propose a multilayered cloak consisting of 20 homogeneous concentric layers with a piecewise constant isotropic diffusivity working over a finite time interval (homogenization approach). We have studied analytically and numerically the extension of transformation optics to the domain of thermodynamics. We have proposed to design an invisibility cloak, which reduces the temperature inside an arbitrary region, and a concentrator which focuses heat flux in a given region.We stress that the efficiency of the thermal protection with the invisibility cloak depends upon the position of its center, because the maximum value of the constant field inside the invisibility region corresponds to the value of the temperature at this point before the geometric transform is applied. We have finally proposed a design of multi-layered cloak consisting of homogeneous isotropic layers of materials with realistic diffusivities. Applications may address the field of microelectronics, for which reason the diameters of metamaterials are 400 to 600 micrometers and the time scale considered is 20 to 50 milliseconds in our numerical examples. However the key parameter is the ratio of time to length, so that the scale can be directly modified for most general thermodynamics applications. Notice here that we assumed the materials not to be temperature dependent, but this can be directly included within our approach. Moreover, analysis of cloaking for other types of heat sources, such as instantaneous ones, or harmonic ones, is also a possible extension of this work.
The key goal with this research was to control the way heat diffuses in a manner similar to those that have already been achieved for waves, such as light waves or sound waves, by using the tools of transformation optics," says Guenneau. Though this technology uses the same fundamental theories as recent advances in optical cloaking, there is a key difference. Until now, he explains, cloaking research has revolved around manipulating trajectories of waves. These include electromagnetic (light), pressure (sound), elastodynamic (seismic), and hydrodynamic (ocean) waves. The biggest difference in their study of heat, he points out, is that the physical phenomenon involved is diffusion, not wave propagation. "Heat isn't a wave—it simply diffuses from hot to cold regions," Guenneau says. "The mathematics and physics at play are much different. For instance, a wave can travel long distances with little attenuation, whereas temperature usually diffuses over smaller distances." To create their thermal invisibility cloak, Guenneau and colleagues applied the mathematics of transformation optics to equations for thermal diffusion and discovered that their idea could work. In their two-dimensional approach, heat flows from a hot to a cool object with the magnitude of the heat flux through any region in space represented by the distance between isotherms (concentric rings of diffusivity). They then altered the geometry of the isotherms to make them go around rather than through a circular region to the right of the heat source—so that any object placed in this region can be shielded from the flow of heat (see image). "We can design a cloak so that heat diffuses around an invisibility region, which is then protected from heat. Or we can force heat to concentrate in a small volume, which will then heat up very rapidly," Guenneau says. The ability to shield an area from heat or to concentrate it are highly desirable traits for a wide range of applications. Shielding nanoelectronic and microelectronic devices from overheating, for example, is one of the biggest challenges facing the electronics and semiconductor industries, and an area in which thermal cloaking could have a huge impact. On a larger scale and far into the future, large computers and spacecraft could also benefit greatly. And in terms of concentrating heat, this is a characteristic that the solar industry should find intriguing. Guenneau and colleagues are now working to develop prototypes of their thermal cloaks for microelectronics, which they expect to have ready within the next few months.
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