Manipulating local heat flux with different patterns

Since the thermal conduction equation has form invariance under coordinate transformation, one can design thermal metamaterials with novel functions by tailoring materials' thermal conductivities. In this work, we establish a different transformation theory, and propose a layered device with anisotropic thermal conductivities. The device is able to convert heat flux from parallel patterns into non-parallel patterns and vice versa. In the mean time, the heat flux pattern outside the device keeps undisturbed as if this device is absent. We perform finite-element simulations to confirm the converting behavior. This work paves a different way to manipulate the flow of heat at will.

2 N.Q. Zhu, X.Y. Shen, and J.P. Huang: Manipulating local heat flux with different patterns acoustics [17]) where wave equations dominate. Obviously, an illusion of thermal conduction not only offers a different way for controlling heat conduction but also has extensive applications in misleading the detectors of temperature distribution signatures. Just as our recent work shows, a thermal illusion device based on the thermal conduction equation can be created by adopting a complementary layer [18]. However, the thermal conductivity of the complementary layer should be negative. For complying with the second law of thermodynamics, one must apply external work on the system [9,12]. Inspired by the former work [18], in order to build thermal illusion devices without using negative-conductivity materials, here we attempt to propose a new kind of thermal illusion device. To this end, we shall change the flow direction of local heat flux inside the device whereas the pattern of heat flux outside the device keeps unchanged (as if the device does not exist). That is, a phenomenon of thermal illusion is created through the thermal illusion device. In order to check whether the device works or not, we shall resort to two-dimensional finite-element simulations.

Coordinate Transformation Approach
Considering a typical thermal conduction process, heat flux is proportional to a temperature gradient. The thermal conduction equation can then be written as where ρ and C are the density and heat capacity respectively and T represents temperature evolving with time t at each point X = (x, y) in the space. In equation (1), κ is thermal conductivity, and Q is a heat source. For a steady state process, the distribution of temperature T is independent of time t, and thus the first term in equation (1) vanishes. Throughout this work, we suppose there is no heat source, Q = 0. Therefore, equation (1) can be reduced to ∇ · (−κ∇T ) = 0.
Upon a change of variable X = (x, y) → X ′ = (x ′ , y ′ ) described by a Jacobian transformation matrix J, this equation, ∇ · (−κ∇T ) = 0, takes the following form, where J T is the transposed matrix of J, and det(J) is the determinant of J. Thus the new thermal conductivity in the transformed space X ′ can be expressed as

Transformation 1
The coordinate transformation is constructed in two-dimensional Cartesian coordinate systems and schematically presented in Similarly, we use X = (x, y) to represent an arbitrary point in the original space, and X ′ = (x ′ , y ′ ) to denote the corresponding point in the transformed space. According to the geometrical relation that r = x 2 + y 2 and r ′ = x ′ 2 + y ′ 2 , we can derive the mapping transformation in Regions I and II, Clearly, equation (4) is used to geometrically compress the triangle OCD (original space) to the sector OAB (transformed space). Accordingly the vertical lines in the original space is distorted to arcs in the transformed space. If we fill Regions I and II with appropriate anisotropic thermal conductivities developed by transformation norms [1], parallel heat flux propagating through these regions can be converged.
Equation (4) can then be easily derived as Therefore, we are able to obtain the anisotropic thermal conductivity tensor κ ′ 1 of the illusion device, The values of the components of κ ′ 1 have no relation with either a or b and are determined only by the coordinates of the points.
Note there is det(κ ′ 1 )=1, which is positive. It should also be noted that the conductivity tensor above satisfy a symmetrical To illustrate this conductivity tensor is positive and provide practicable parameters for experiment, we can diagonalize it by rotating the principle axis by an appropriate angle α. This angle can be calculated by using the relation: where κ xy , κ xx , κ yy are the elements of the tensor κ ′ 1 . As a result, we obtain the diagonalized tensor as Obviously, owing to 2x 2 + y 2 > y(4x 2 + y 2 ) 1 2 , the tensor is positive as expected.

Transformation 2
Transformation 1 mentioned above deals with the steady-state thermal conduction and converges the parallel heat flux on the vertex of the triangle, namely, from parallel patterns to nonparallel patterns. For the sake of completeness, here we want to attempt an inverse behavior, i.e., from non-parallel patterns to parallel patterns. To proceed, we assume a temperature field associated with an extremely small ring located at the centre where heat flow diverges from it, and we aim to design a device converting the dispersed heat flux (non-parallel patterns) into parallel patterns.
A proper approach to achieve this goal is to make the following transformation that is illustrated in Figure 2. Suppose the radius of the small circle is a, the side length of the square is 2b and the radius of the big circle is √ 2b. We extend the sector OAB (original space) to the triangle OCD (transformed space). The geometrical relation of the transformation means that each point in an arc located in the original space will be mapped to a vertical segment in the transformed space. As a result, when the divergent heat flux reaches the device, they will change to a parallel pattern due to the distorted coordinates.
Also, we use X = (x, y) and X ′ = (x ′ , y ′ ) to respectively represent an arbitrary point in the original space and the corresponding point in the transformed space. We obtain the mapping transformation in Regions I and III, The thermal conductivity κ ′ 2 for Regions I and III can be derived from equation (3), With the help of the rotation matrix, we can derive the thermal conductivity κ ′′ 2 for Regions II and IV,

Results
We perform finite element simulations based on commercial software COMSOL Multiphysics (http : //www.comsol.com) to check whether the device works indeed.

Transformation 2
In Figure 5a Regions I and III are filled up with a material whose thermal conductivity tensor is κ ′ 2 derived from equation (8).

Conclusion
Since all the positive conductivity tensors we derived above are symmetrical matrices, which can be diagonalized by rotating the principle axis by an appropriate angle. The new simplified conductivity matrices can be easily used to implement the device with effective medium theories [19]. Thus, it should be more convenient to build such a thermal illusion device.      for Transformation 1, i.e., r 2 /x 2 − 3y 2 /r 2 .   (3) for Transformation 2, namely, r 2 /x 2 + y 2 /r 2 (for Regions I and III) and y 2 /r 2 (for Regions II and IV). Note that (c) can be obtained by rotating (a) for 90 degrees.