Keywords

1 Opening Remarks

Electromagnetic waves are dominated by the Maxwell equations, which have the hyperbolic feature. Due to the generality of hyperbolic equations, electromagnetic phenomena can be extended to other physical fields like acoustics without much difficulty. However, it is crucially different from the Fourier conduction because the Fourier equation is parabolic [1,2,3]. By modulating the Fourier equation with thermal relaxation [4,5,6,7,8,9], the parabolic equation can become hyperbolic, which can support the propagation of thermal waves and avoid the infinite speed of thermal propagation. Thermal waves refer to wave-like temperature profiles. The refractive behaviors can thus be studied [10,11,12,13,14,15].

Besides thermal relaxation, by combining the Fourier equation with convection, the dominant equation can also have the hyperbolic feature [16], which leads to novel thermal phenomena such as nonreciprocity [17, 18], anti-parity-time symmetry [19,20,21], negative transport [22], cloaks [23,24,25], and crystals [26]. However, the refractive behaviors have been rarely touched, let alone the negative refraction of thermal waves. Although it is difficult to discuss the refractive behaviors of diffusion waves [1,2,3], we introduce advection to provide the hyperbolic property for heat transfer [16]. Therefore, it is possible to discuss the refractive behaviors of thermal waves based on conduction and convection. Despite the hyperbolic property, the entire investigation is not related to thermal relaxation [4,5,6,7,8,9]. In addition to the hyperbolic property, advection is also ubiquitous, so taking it into account is necessary and meaningful. Moreover, advection also clarifies the concept of thermal wave vectors, so different refractive behaviors can be visualized. We do not consider radiation (another basic mode of heat transfer) because it essentially belongs to electromagnetic waves.

We then consider conduction and advection with a time-harmonic temperature source to generate thermal waves. Since the heat transfer efficiency of advection is much higher than that of conduction, we can generally conclude that conduction leads to the decay of thermal waves, and advection contributes to the propagation of thermal waves [22]. Therefore, controlling thermal waves is essentially controlling advection velocities. We then focus our discussions on advection velocities which are preset (ideal model), determined by the Darcy equation (practical model), and controlled by layered structures (experimental suggestion).

2 Theoretical Foundation

We firstly consider an ideal model base on conduction and convection in a porous medium (composed of fluid and solid) whose dominant equation is

$$\begin{aligned} \rho _m C_m \frac{\partial T}{\partial t}+\boldsymbol{\nabla }\cdot \left( -\kappa _m \boldsymbol{\nabla }T\right) +\rho _f C_f \boldsymbol{v}\cdot \boldsymbol{\nabla }T=0, \end{aligned}$$
(16.1)

where T and t are temperature and time, respectively. \(\rho _m C_m\) is the product of the density and heat capacity of the porous medium, and \(\kappa _m\) is the thermal conductivity of the porous medium, which can be both calculated by the weighted average of the fluid and solid [28]. \(\rho _f C_f\) is the product of the density and heat capacity of the fluid, and \(\boldsymbol{v}\) is the convective velocity of the fluid. When we discuss the pure fluid shown in Fig. 16.1a, the solid does not exist, so we can derive \(\rho _m C_m=\rho _f C_f\) and \(\kappa _m=\kappa _f\) with \(\kappa _f\) being the thermal conductivity of the fluid. The convective term \(\rho _f C_f \boldsymbol{v}\cdot \boldsymbol{\nabla }T\) introduces the hyperbolic property to heat transfer. A heat source with time-harmonic temperature (THT) is located on the upper-left boundary. The time-harmonic temperature can be expressed as \(T=A\cos \left( \omega t\right) +B\), where A, \(\omega \), and B are the temperature amplitude, circular frequency, and reference temperature of the thermal wave, respectively. The other boundaries are set with open boundary condition (OBC), indicating no reflection of heat energy (i.e., with semi-infinite length). The ideal model is composed of pure fluid with incident velocity \(\boldsymbol{v}_i\) and refractive velocity \(\boldsymbol{v}_r\), whose angles to the normal are \(\theta _i\) and \(\theta _r\), respectively. The heat source with time-harmonic temperature generates thermal waves with the assistance of convection. Therefore, the direction of thermal waves follows that of convective velocities, which can be concluded as

$$\begin{aligned} \tan \theta _i&=\frac{v_{ix}}{-v_{iy}},\end{aligned}$$
(16.2a)
$$\begin{aligned} \tan \theta _r&=\frac{v_{rx}}{-v_{ry}}, \end{aligned}$$
(16.2b)

where minus signs ensure \(-v_{iy}>0\) and \(-v_{ry}>0\). The subscript x (or y) denotes the x-component (or y-component) of convective velocities. The continuity of convective velocities along the y axis gives

$$\begin{aligned} v_{iy}=v_{ry}. \end{aligned}$$
(16.3)

Equation (16.2) can then be simplified as

$$\begin{aligned} \frac{\tan \theta _i}{\tan \theta _r}=\frac{v_{ix}}{v_{rx}}, \end{aligned}$$
(16.4)

which is similar to the tangent law for conductive refractions [29,30,31], but also has a different physical connotation. Compared with conductive refractions where thermal conductivities are critical, thermal wave refractions are mainly determined by convective velocities.

Fig. 16.1
figure 1

Adapted from Ref. [27]

Schematic diagrams of a ideal model, b practical model, and c layered structure. THT, time-harmonic temperature; OBC, open boundary condition.

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We then further discuss convective velocities with a practical model, as presented in Fig. 16.1b. Compared with the ideal model where convective velocities are preset, we explain the origination of convective velocities in the practical model. Although many fluid models [32,33,34,35,36,37] are applicable, we use the Darcy equation in porous media [32,33,34,35] for brevity, i.e., \(\boldsymbol{v}=-\left( \boldsymbol{\sigma }/\mu \right) \cdot \boldsymbol{\nabla }P\), where \(\boldsymbol{v}\) is convective velocity, \(\sigma \) is permeability, \(\mu \) is dynamic viscosity, and P is pressure. The upper and lower boundaries are additionally set with high pressure \(P_h\) and low pressure \(P_l\). The anisotropic permeabilities of the incident region \(\boldsymbol{\sigma }_i\) and refractive region \(\boldsymbol{\sigma }_r\) are, respectively, expressed as

$$\begin{aligned} \boldsymbol{\sigma }_i&=\left( \begin{matrix}\sigma _{ixx}&{}\sigma _{ixy}\\ \sigma _{iyx}&{}\sigma _{iyy}\end{matrix}\right) ,\end{aligned}$$
(16.5a)
$$\begin{aligned} \boldsymbol{\sigma }_r&=\left( \begin{matrix}\sigma _{rxx}&{}\sigma _{rxy}\\ \sigma _{ryx}&{}\sigma _{ryy}\end{matrix}\right) . \end{aligned}$$
(16.5b)

We can then express the incident velocity \(\boldsymbol{v}_i\) and refractive velocity \(\boldsymbol{v}_r\) with the Darcy equation as

$$\begin{aligned} \nonumber \boldsymbol{v}_i&=-\frac{\boldsymbol{\sigma }_i}{\mu }\cdot \boldsymbol{\nabla }P_i=\frac{-1}{\mu }\left( \begin{matrix}\sigma _{ixx}&{}\sigma _{ixy}\\ \sigma _{iyx}&{}\sigma _{iyy}\end{matrix}\right) \left( \begin{matrix}\nabla P_{ix}\\ \nabla P_{iy}\end{matrix}\right) \\&=\frac{-1}{\mu }\left( \begin{matrix}\sigma _{ixx}\nabla P_{ix}+\sigma _{ixy}\nabla P_{iy}\\ \sigma _{iyx}\nabla P_{ix}+\sigma _{iyy}\nabla P_{iy}\end{matrix}\right) =\left( \begin{matrix}v_{ix}\\ v_{iy}\end{matrix}\right) , \end{aligned}$$
(16.6a)
$$\begin{aligned} \nonumber \boldsymbol{v}_r&=-\frac{\boldsymbol{\sigma }_r}{\mu }\cdot \boldsymbol{\nabla }P_r=\frac{-1}{\mu }\left( \begin{matrix}\sigma _{rxx}&{}\sigma _{rxy}\\ \sigma _{ryx}&{}\sigma _{ryy}\end{matrix}\right) \left( \begin{matrix}\nabla P_{rx}\\ \nabla P_{ry}\end{matrix}\right) \\&=\frac{-1}{\mu }\left( \begin{matrix}\sigma _{rxx}\nabla P_{rx}+\sigma _{rxy}\nabla P_{ry}\\ \sigma _{ryx}\nabla P_{rx}+\sigma _{ryy}\nabla P_{ry}\end{matrix}\right) =\left( \begin{matrix}v_{rx}\\ v_{ry}\end{matrix}\right) . \end{aligned}$$
(16.6b)

With Eq. (16.6), we can obtain

$$\begin{aligned} \tan \theta _i&=\frac{v_{ix}}{-v_{iy}}=-\frac{\sigma _{ixx}\nabla P_{ix}+\sigma _{ixy}\nabla P_{iy}}{\sigma _{iyx}\nabla P_{ix}+\sigma _{iyy}\nabla P_{iy}},\end{aligned}$$
(16.7a)
$$\begin{aligned} \tan \theta _r&=\frac{v_{rx}}{-v_{ry}}=-\frac{\sigma _{rxx}\nabla P_{rx}+\sigma _{rxy}\nabla P_{ry}}{\sigma _{ryx}\nabla P_{rx}+\sigma _{ryy}\nabla P_{ry}}, \end{aligned}$$
(16.7b)

which further yields

$$\begin{aligned} \frac{\nabla P_{iy}}{\nabla P_{ix}}&=-\frac{\sigma _{ixx}+\sigma _{iyx}\tan \theta _i}{\sigma _{ixy}+\sigma _{iyy}\tan \theta _i},\end{aligned}$$
(16.8a)
$$\begin{aligned} \frac{\nabla P_{ry}}{\nabla P_{rx}}&=-\frac{\sigma _{rxx}+\sigma _{ryx}\tan \theta _r}{\sigma _{rxy}+\sigma _{ryy}\tan \theta _r}. \end{aligned}$$
(16.8b)

The boundary conditions on the interface of incident and refractive regions are

$$\begin{aligned} v_{iy}&=v_{ry},\end{aligned}$$
(16.9a)
$$\begin{aligned} \nabla P_{ix}&=\nabla P_{rx}, \end{aligned}$$
(16.9b)

which indicate the continuities of convective velocities along the y axis (Eq. (16.9a)) and pressure gradients along the x axis (Eq. (16.9b)). With Eq. (16.9), we can derive

$$\begin{aligned} \sigma _{iyx}+\sigma _{iyy}\frac{\nabla P_{iy}}{\nabla P_{ix}}=\sigma _{ryx}+\sigma _{ryy}\frac{\nabla P_{ry}}{\nabla P_{rx}}. \end{aligned}$$
(16.10)

The substitution of Eq. (16.8) into Eq. (16.10) yields

$$\begin{aligned} \frac{\sigma _{ixx}\sigma _{iyy}-\sigma _{ixy}\sigma _{iyx}}{\sigma _{ixy}+\sigma _{iyy}\tan \theta _i}=\frac{\sigma _{rxx}\sigma _{ryy}-\sigma _{rxy}\sigma _{ryx}}{\sigma _{rxy}+\sigma _{ryy}\tan \theta _r}. \end{aligned}$$
(16.11)

Equation (16.11) is an extension of Eq. (16.4), revealing the refractive behaviors of thermal waves with a practical model determined by permeabilities. In other words, thermal wave refractions can be controlled by designing specific permeabilities.

We then suppose that the anisotropic permeabilities of porous media I and II have the same eigenvalues, i.e., \(\sigma _s\) and \(\sigma _p\). Therefore, \(\boldsymbol{\sigma }_i\) and \(\boldsymbol{\sigma }_r\) can be obtained by anticlockwise rotating the eigenvalues with angles of \(\gamma _1\) and \(\gamma _2\), respectively. The permeabilities can then be expressed as [38,39,40,41,42]

$$\begin{aligned} \boldsymbol{\sigma }_i=\left( \begin{matrix}\sigma _s\cos ^2\gamma _1+\sigma _p\sin ^2\gamma _1&{}\left( \sigma _s-\sigma _p\right) \cos \gamma _1\sin \gamma _1\\ \left( \sigma _s-\sigma _p\right) \cos \gamma _1\sin \gamma _1&{}\sigma _s\sin ^2\gamma _1+\sigma _p\cos ^2\gamma _1\end{matrix}\right) ,\end{aligned}$$
(16.12a)
$$\begin{aligned} \boldsymbol{\sigma }_r=\left( \begin{matrix}\sigma _s\cos ^2\gamma _2+\sigma _p\sin ^2\gamma _2&{}\left( \sigma _s-\sigma _p\right) \cos \gamma _2\sin \gamma _2\\ \left( \sigma _s-\sigma _p\right) \cos \gamma _2\sin \gamma _2&{}\sigma _s\sin ^2\gamma _2+\sigma _p\cos ^2\gamma _2\end{matrix}\right) . \end{aligned}$$
(16.12b)

The substitution of Eq. (16.12) into Eq. (16.11) yields

$$\begin{aligned} \frac{\sigma _s \sigma _p}{\sigma _{ixy}+\sigma _{iyy}\tan \theta _i}=\frac{\sigma _s \sigma _p}{\sigma _{rxy}+\sigma _{ryy}\tan \theta _r}, \end{aligned}$$
(16.13)

which can be further reduced to

$$\begin{aligned} \nonumber&\dfrac{\sigma _s}{\left( \sigma _s/\sigma _p-1\right) \cos \gamma _1\sin \gamma _1+\left( \sigma _s/\sigma _p\sin ^2\gamma _1+\cos ^2\gamma _1\right) \tan \theta _i}=\\&\dfrac{\sigma _s}{\left( \sigma _s/\sigma _p-1\right) \cos \gamma _2\sin \gamma _2+\left( \sigma _s/\sigma _p\sin ^2\gamma _2+\cos ^2\gamma _2\right) \tan \theta _r}. \end{aligned}$$
(16.14)

With the further assumption that the eigenvalues are highly anisotropic,

$$\begin{aligned} \sigma _p\gg \sigma _s\approx 0, \end{aligned}$$
(16.15)

we can obtain \(\sigma _s/\sigma _p\approx 0\) and reduce Eq. (16.14) to

$$\begin{aligned} \theta _i&\approx \gamma _1,\end{aligned}$$
(16.16a)
$$\begin{aligned} \theta _r&\approx \gamma _2. \end{aligned}$$
(16.16b)

Equation (16.16) is a further extension of Eq. (16.11), which indicates that the direction of thermal waves approximately follows the orientation of anisotropic permeabilities. Therefore, by designing \(\gamma _1\) and \(\gamma _2\), we can effectively control the angles of incidence and refraction.

Fig. 16.2
figure 2

Adapted from Ref. [27]

Simulations of a and b positive refraction, c and d vertical refraction, and e and f negative refraction. Colors denote temperatures, and solid arrows denote convective velocities. The simulation size is \(10\times 10\) cm\(^2\). The time-harmonic temperature is set at \(T=40\cos \left( \pi t/10\right) +323\) K with length 2 cm, whose left endpoint has a 1 cm distance from the left-upper corner. The initial temperature is set at 323 K. The advection velocities of the incident regions are \(\left( v_0\tan \theta _0,\,-v_0\right) ^\tau \) where \(\tau \) denotes transpose. The advection velocities of the refractive regions are \(\left( v_0\tan \theta _p,\,-v_0\right) ^\tau \) for (a) and (b), \(\left( v_0\tan \theta _v,\,-v_0\right) ^\tau \) for (c) and (d), and \(\left( v_0\tan \theta _n,\,-v_0\right) ^\tau \) for (e) and (f). Concrete parameters: \(v_0=1\) mm/s, \(\theta _0=4\pi /18\) rad, \(\theta _p=2\pi /18\) rad, \(\theta _v=0\) rad, \(\theta _n=-2\pi /18\) rad, \(\kappa _f=0.01\) W m\(^{-1}\) K\(^{-1}\), \(C_f=1000\) J kg\(^{-1}\) K\(^{-1}\), and \(\rho _f=1000\) kg/m\(^3\), respectively.

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3 Finite-Element Simulation

We further perform finite-element simulations to confirm the theory. We use three templates: heat transfer in fluids, porous media, and the Darcy law in COMSOL Multiphysics. We choose a free triangular mesh with a maximum element size of \(2\times 10^{-4}\) m, a minimum element size of \(10^{-6}\) m, a maximum element growth rate of 1.1, a curvature factor of 0.2, and a resolution of narrow regions of 1. The transient simulations are conducted with times from 0 s to 200 s with a step of 0.1 s, and the relative tolerance is 0.0001. We also introduce a dimensionless characteristic parameter to understand the two modes of heat transfer, i.e., the Peclet number \(\textrm{Pe}=|v|L/D\), where v is advection velocity, L is characteristic length, and \(D=\kappa _m/\left( \rho _m C_m\right) \) is heat diffusion coefficient. In our investigation, the Peclet number \(\textrm{Pe}=10^4\) is large, so our system is dominated by advection and featured by the hyperbolic property.

The results are presented in Figs. 16.2, 16.3 and 16.4. We also consider three typical cases. The first one is positive refraction, indicating that the incident and refractive thermal waves are at both sides of the normal. The second is vertical refraction with the refractive thermal wave along the normal. The third one is negative refraction, indicating that the incident and refractive thermal waves are on the same side of the normal.

Fig. 16.3
figure 3

Adapted from Ref. [27]

Simulations based on the practical model. The upper and lower boundaries are additionally set with \(P_h=15000\) Pa and \(P_l=0\) Pa, respectively. The parameters of thermal conductivity, heat capacity, and density of porous media I and II are 0.01 W m\(^{-1}\) K\(^{-1}\), 1000 J kg\(^{-1}\) K\(^{-1}\), and 1000 kg/m\(^3\), respectively. The anisotropic permeabilities of porous media I and II have the same eigenvalues, i.e., \(\sigma _s=10^{-13}\) m\(^2\) and \(\sigma _p=10^{-11}\) m\(^2\). The anisotropic permeabilities in the incident regions are described by Eq. (16.12a) with \(\gamma _1=4\pi /18\) rad, and those in the refractive regions are described by Eq. (16.12b) with \(\gamma _2=2\pi /18\) rad for (a) and (b), \(\gamma _2=0\) rad for (c) and (d), and \(\gamma _2=-2\pi /18\) rad for (e) and (f). The dynamic viscosity of the fluid is 0.001 Pa\(\cdot \)s.

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Fig. 16.4
figure 4

Adapted from Ref. [27]

Simulations based on layered structures. Anisotropic permeabilities are obtained with two isotropic porous media A and B (with porosity 0.1), whose permeabilities are \(\sigma _a=5\times 10^{-14}\) m\(^2\) and \(\sigma _b=2\times 10^{-11}\) m\(^2\), respectively. The fluid is water whose thermal conductivity, heat capacity, density, and dynamic viscosity are 0.6 W m\(^{-1}\) K\(^{-1}\), 4200 J kg\(^{-1}\) K\(^{-1}\), 1000 kg/m\(^3\), and 0.001 Pa\(\cdot \)s, respectively. The thermal conductivity, heat capacity, and density of the solid is 0.1 W m\(^{-1}\) K\(^{-1}\), 800 J kg\(^{-1}\) K\(^{-1}\), and 3000 kg/m\(^3\), respectively. Porous media A and B are supposed to have only a permeability difference.

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The simulations based on the ideal model are shown in Fig. 16.2. We take the positive refraction as an example. The thermal wave takes about 50 s to reach the interface (Fig. 16.2a). Since the convective velocities of the incident and refractive regions are different, the thermal wave changes its propagation direction (Fig. 16.2b). Since \(v_{ix}\) and \(v_{rx}\) are both positive, the result is positive refraction. We then maintain the incident velocity and change the refractive velocity. Suppose the direction of the refractive velocity is vertical to the interface. In that case, that of the refractive thermal wave is also vertical to the interface, yielding vertical refraction (Fig. 16.2c, d). We finally set the refractive velocity to have a negative x-component, so the refractive and incident thermal waves are at the same side of the normal, yielding negative refraction (Fig. 16.2e, f). Intuitively speaking, where thermal convection flows, where thermal waves propagate. The temperature evolutions in more detail are animated in the supplemental media.

The simulations in Fig. 16.2 are based on pure fluid. We then perform simulations based on the Darcy equation in porous media where anisotropic permeabilities can guide advection velocities. The positive, vertical, and negative refractions results are shown in Fig. 16.3, which are the same as Fig. 16.2. Convective velocities (denoted by arrows) are no longer ideally distributed, which, however, does not affect phenomenon observations. The propagation of thermal waves at 50 s is similar in Fig. 16.3a, c, e because of the same permeabilities in the incident regions. Positive, vertical, and negative refractions occur in Fig. 16.3b, d, f, respectively. Therefore, the simulations are consistent with the theoretical predictions.

Despite the practical model, the parameters are still too ideal. Therefore, we further use a layered structure to obtain the desired anisotropic permeability (Fig. 16.1c). Porous medium A with permeability \(\sigma _a\) and width a and porous medium B with permeability \(\sigma _b\) and width b are arranged alternately. The effective permeabilities with series connection \(\sigma _s\) and parallel connection \(\sigma _p\) can be expressed as [38,39,40,41,42]

$$\begin{aligned} \sigma _s&=\frac{a+b}{a/\sigma _a+b/\sigma _b}, \end{aligned}$$
(16.17a)
$$\begin{aligned} \sigma _p&=\frac{a\sigma _a+b\sigma _b}{a+b}. \end{aligned}$$
(16.17b)

In this way, we can use two isotropic porous media to realize the anisotropic permeability described by Eq. (16.12).

The simulations of positive, vertical, and negative refractions are shown in Fig. 16.4a–c, d–f, g–i, respectively. Since we keep the eigenvalues of the anisotropic permeabilities the same, only two porous media with isotropic permeabilities are required. We can obtain the expected permeabilities by alternately arranging porous media A and B and anticlockwise rotating the structure with different angles. Compared with the advection velocities in Figs. 16.2 and 16.3, those in Fig. 16.4 are more complicated, but the general directions are still as expected. Therefore, the expected control of thermal waves can be obtained. Meanwhile, the simulations in Fig. 16.4 can be regarded as experimental suggestions because we choose practical parameters like water. For this reason, the wavelength and decay rate of thermal waves are very different from those in Figs. 16.2 and 16.3.

Although thermal wave refractions have a phenomenal analog to electromagnetic ones, the underlying mechanisms are very different. The former requires anisotropy, but the latter does not. Therefore, anisotropy guides the direction of convective velocities and further affects the propagation of thermal waves. In other words, convection helps generate thermal waves. From this perspective, we further understand the negative refraction of thermal waves. Although we have observed that the incident and refractive thermal waves are on the same side of the normal, it is phenomenological negative refraction. Generally, negative refractions feature the opposite directions of energy flows and wave vectors. In our system, energy flows follow convective velocities, and wave vectors follow thermal waves. Therefore, energy flows and wave vectors are along the same direction because convective velocities yield thermal waves (i.e., the casualty in thermotics [22]). We can then conclude that the negative refraction of thermal waves is phenomenally observed. However, the wave vector and energy flow are still in the same direction, not violating the casualty.

Fig. 16.5
figure 5

Adapted from Ref. [27]

Thermal wave concentrator. a Schematic diagram with simulation size \(10\times 5\) cm\(^2\), inner radius 2 cm, and outer radius 3 cm. Simulations at b 100 s and c 200 s. d Temperature distribution and e energy flow distribution along \(x=0\) cm. The upper and lower boundaries are set with \(P_h=3750\) Pa and \(P_l=0\) Pa, respectively. The time-harmonic temperature is set at \(T=40\cos \left( \pi t/10\right) +323\) K with length 6 cm in the center of the upper boundary. Other boundaries are set with open boundary condition. Four porous media (A, B, C, and D) are applied. Parameters: The permeabilities of A and B are, respectively, \(5\times 10^{-14}\) and \(2\times 10^{-11}\) m\(^2\), and those of C and D are the same, i.e. \(\times 10^{-12}\) m\(^2\). The thermal conductivity, heat capacity, density, and dynamic viscosity of the fluid in A-D are 0.01 W m\(^{-1}\) K\(^{-1}\), 1000 J kg\(^{-1}\) K\(^{-1}\), 1000 kg/m\(^3\), and 0.001 Pa\(\cdot \)s, respectively. The thermal conductivity and density of the solid in A-D are 0.01 W m\(^{-1}\) K\(^{-1}\) and 1000 kg/m\(^3\), respectively. The heat capacity of the solid in A-C is 1000 J kg\(^{-1}\) K\(^{-1}\), and that for D is 2500 J kg\(^{-1}\) K\(^{-1}\).

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4 Model Application

The refractive behaviors of thermal waves can also be applied in practice, such as in designing a thermal wave concentrator. Since an anisotropic permeability can guide the direction of thermal waves, we are allowed to design the orientation of the anisotropic permeability to point towards the center, as schematically shown in Fig. 16.5a. Hence, thermal waves propagate along with the anisotropic permeability orientation, thus being guided and concentrated (see the solid lines with arrows in Fig. 16.5a). The simulations at 100 and 200 s are presented in Fig. 16.5b, c, respectively. We also plot the temperature and heat flux distributions at 200 s on the central dashed line, which are shown in Fig. 16.5d, e, respectively. The wave number and heat flux in the center are larger than those in the background. Therefore, this scheme provides guidance to control thermal waves beyond scattering cancellation [23, 24] and coordinate transformation [25].

5 Conclusion

We reveal the refractive behaviors of thermal waves between different media and present the thermal wave counterpart of electromagnetic wave refractions, including the positive, vertical, and negative refractions. We also design convection-assisted metamaterials to control thermal wave refractions and provide experimental suggestions to observe the desired phenomena. We further propose a potential application of thermal wave concentrators, which can be used for energy collection. These results are helpful in understanding and controlling the refractive behaviors of thermal waves and may have potential applications in thermal wave imaging [46,47,48,49,50] and intelligent thermal management.

6 Exercise and Solution

Exercise

  1. 1.

    Prove Eq. (16.16) in detail.

Solution

  1. 1.

    With Eqs. (16.15) and (16.14) can be reduced to

    $$\begin{aligned} \frac{\sigma _s}{-\cos \gamma _1\sin \gamma _1+\cos ^2\gamma _1\tan \theta _i}=\frac{\sigma _s}{-\cos \gamma _2\sin \gamma _2+\cos ^2\gamma _2\tan \theta _r}. \end{aligned}$$
    (16.18)

    Since the numerators of Eq. (16.18) are approximately zero (i.e., \(\sigma _s\approx 0\)), the denominators of Eq. (16.18) should also be approximately zero to ensure that Eq. (16.18) is nonzero,

    $$\begin{aligned} \nonumber&-\cos \gamma _1\sin \gamma _1+\cos ^2\gamma _1\tan \theta _i=\\&-\cos \gamma _2\sin \gamma _2+\cos ^2\gamma _2\tan \theta _r\approx 0. \end{aligned}$$
    (16.19)

    The physical meaning is that the convective velocities along the y axis are nonzero. Solving Eq. (16.19), we can then derive

    $$\begin{aligned} \tan \theta _i\approx \frac{\cos \gamma _1\sin \gamma _1}{\cos ^2\gamma _1}=\tan \gamma _1,\end{aligned}$$
    (16.20a)
    $$\begin{aligned} \tan \theta _r\approx \frac{\cos \gamma _2\sin \gamma _2}{\cos ^2\gamma _2}=\tan \gamma _2. \end{aligned}$$
    (16.20b)