Effective medium theory for thermal scattering off rotating structures

Jiaxin Li; Jiaxin Li; Ying Li; Ying Li; Ying Li; Ying Li; Wuyi Wang; Wuyi Wang; Longqiu Li; Cheng-Wei Qiu; Cheng-Wei Qiu

doi:10.1364/OE.399799

1. Introduction

Effective medium theory has been developed and used in a wide range of problems in physics and engineering for over a century. It is able to define averages for predicting material performance when a field in a material medium is to be considered [1]. Effective medium theory can be used to calculate various effective property parameters of composite materials, including the effective electric permittivity [2–4], magnetic permeability [3,4], elastic moduli [5], electric conductivity [6] and thermal conductivity [7], etc. Recently, effective medium theory serves as a tool for designing metamaterials or meta-devices [8–10], which are designed with a variety of novel functions by the ordered arrangement of natural materials. As an example, the concept of neutral inclusion using coated spheres [11,12] is applied to realize the transparency in scattering of electromagnetic waves [13], elastic waves [14] and heat fluxes [15,16]. Besides the transformation theory [17–24], effective medium theory also plays an important role in the design of thermal materials with composite structures such as the doublet structure [10], alternating layer structure [25–29], bi-layer shell [30–32], core-shell structure [33,34] and periodic particle structure [35,36]. Thanks to the judicious design of these structures, thermal metamaterials are capable of achieving functions like thermal cloaking [37,38], concentrating [39], bending [29] and camouflage [23,24,30].

However, existing EMT is mostly limited to static medium, and the presentence and influence of moving parts in a medium has not been thoroughly studied. The introduction of moving inclusions to effective medium theory has great prospects in practical applications, because adding a degree of freedom of movement can greatly enhance the functional diversity and flexibility of device performance. Metamaterials with spinning components in wave systems have been proposed recently [40,41], but thermal metamaterials including spinning parts have not been fully studied so far. For heat transfer, it is well known that the thermal convection in moving matter is essentially different from thermal conduction [42,43]. It is still elusive as for how to “translate” the advection effect to the effective thermal conductivity. Here, a theoretical framework is developed to study the heat transfer through concentric structures in mechanical rotation at arbitrary speeds. We show that the effects of a scatterer with rotating structures on the temperature field can be represented as a complex effective thermal conductivity. We further demonstrate the equivalence between the effective thermal conductivity and the anti-symmetric thermal conductivity tensor in terms of its exterior influence on background temperature field. In our rotating structures, the convective effect is reflected in the asymmetry of the thermal conductivity tensor, which is obviously different from natural materials with symmetric conductivity tensor. Our work expands effective medium theory by uniting the effects of both conduction and convection, which may provide theoretical basis for the design of functional materials and structures.

2. Complex effective thermal conductivity for rotating structures

2.1 Thermal scattering off an isotropic object

Distinguished from the scattering off objects for heat diffusion waves generated by time-harmonic heat sources [44,45] or in other wave systems, the scattering discussed in this manuscript mainly refers to the heat transfer effect of an object on the surrounded temperature field or heat fluxes under static temperature boundary conditions. First, we define the effective thermal conductivity of an object in a pure conductive case. Consider the two-dimensional system of an unknown object with radius R_out and isotropic effective thermal conductivity κ* at the center of a square background with side length L and thermal conductivity κ_b. The left and right boundaries are maintained at constant temperatures T_L and T_R with T_R − T_L = AL, where A is the temperature gradient. The upper and lower boundaries are thermally insulated. In a polar coordinate system (r,θ), the steady-state temperature fields on the background T_b(r,θ) and on the object T_c(r,θ) are solved from the heat transfer equation and the matching conditions

(1)$$\begin{array}{cc} \frac{{{\partial ^2}{T_\textrm{b}}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T_\textrm{b}}}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{T_\textrm{b}}}}{{\partial {\theta ^2}}} = 0,\textrm{ }r \ge {R_{out}}\textrm{ }&{ {{T_\textrm{b}}} |_{r = {R_{out}}}} = { {{T_\textrm{c}}} |_{r = {R_{out}}}}\\ \frac{{{\partial ^2}{T_\textrm{c}}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T_\textrm{c}}}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{T_\textrm{c}}}}{{\partial {\theta ^2}}} = \textrm{0},\textrm{ }r \le {R_{out}}\textrm{ }&{\kappa _\textrm{b}}{\left. {\frac{{\partial {T_\textrm{b}}}}{{\partial r}}} \right|_{r = {R_{out}}}} = {\kappa ^\ast }{\left. {\frac{{\partial {T_\textrm{c}}}}{{\partial r}}} \right|_{r = {R_{out}}}} \end{array}$$

The Laplace’s equations can be solved by a variable separation T(r,θ) = F(r)G(θ), where G(θ) is a periodic function. Assuming that the background width L is large, we can set G(θ) = eⁱ^θ, where higher-order dependence on θ is neglected. The actual field requires that

(2)$${T_{\textrm{b,c}}}({r,\theta } )= {T_{\textrm{const}}} + \frac{1}{2}[{{F_{\textrm{b,c}}}(r ){e^{i\theta }} + \overline {{F_{\textrm{b,c}}}(r )} {e^{ - i\theta }}} ]$$

where T_const is a constant determined by the left and right temperature boundaries. An overline takes complex conjugation. Substituting Eq. (2) into the heat transfer equation, and focusing on the r-dependent parts, we have the general solutions

(3)$${F_\textrm{b}}(r )= Ar + {z_\textrm{b}}{r^{ - 1}},\textrm{ }{F_\textrm{c}}(r )= {z_\textrm{c}}r$$

where z_b and z_c are constants solved from the matching conditions. Thus

(4)$${z_\textrm{b}} = A{R_{out}}^2\frac{{{\kappa _\textrm{b}} - {\kappa ^\ast }}}{{{\kappa _\textrm{b}} + {\kappa ^\ast }}}$$

Note that when κ* < κ_b (κ* > κ_b), z_b > 0 (z_b < 0) and the object attracts (repels) the heat fluxes. On the other hand, κ* can be inferred by its influence on external temperature field which is characterized as z_b

(5)$${\kappa ^\ast } = {\kappa _\textrm{b}}\frac{{A{R_{out}}^2 - {z_\textrm{b}}}}{{A{R_{out}}^2 + {z_\textrm{b}}}} = {\kappa _\textrm{b}}\alpha ({{R_{out}}} )$$

where α(R_out) describes the thermal scattering effect on the outer boundary (r = R_out) of the object. We use α(R_out) to conveniently denote the relation between the effective thermal conductivity κ* and the background thermal conductivity κ_b.

2.2 Complex effective thermal conductivity of a rotating object

As shown in Fig. 1(a), for a solid object with thermal conductivity κ₀ and diffusivity D₀, if it is rotating at angular speed Ω₀, the heat transfer equation on it is

(6)$$\frac{{{\partial ^2}{T_\textrm{0}}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T_0}}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{T_0}}}{{\partial {\theta ^2}}} = \frac{{{\mathrm{\Omega }_0}}}{{{D_0}}}\frac{{\partial {T_0}}}{{\partial \theta }}$$

Substituting the variable separated solution Eq. (2) into Eq. (6) gives

(7)$${r^2}{F_0}^{\prime \prime }(r )+ r{F_0}^\prime (r )- \left( {i\frac{{{\mathrm{\Omega }_0}}}{{{D_0}}}{r^2} + 1} \right){F_0}(r )= 0$$

Fig. 1. Heat transfer through a circular object. (a) Schematic. (b) Temperature field for the thermal conduction through a stationary object. (c) Temperature field for the thermal convection through a rotating object. (d) The modulus, real part, imaginary part, and (e) the argument of the effective thermal conductivity of the rotating object at various rotation speeds.

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The variable change x = (iΩ₀/D₀)^1/2r can be introduced to rewrite Eq. (7) as a first-order modified Bessel equation ${x^2}f^{\prime\prime}(x )+ xf^{\prime}(x )- ({{x^2} + 1} )f(x )= 0$. Its solution is f(x) = c₁I₁(x) + c₂K₁(x), where c₁ and c₂ are constants. I₁(x) and K₁(x) are the first-order modified Bessel function of the first and second kind, respectively. They have the limiting properties$\mathop {\lim }\limits_{x \to 0} {I_1}(x )= {x / 2}$, $\mathop {\lim }\limits_{x \to 0} {K_1}(x )= {\textrm{1} / {({\textrm{4}x} )}}$, $\mathop {\lim }\limits_{x \to \infty } {I_1}^\prime (x )= \mathop {\lim }\limits_{x \to \infty } {I_1}(x )$, $\mathop {\lim }\limits_{x \to \infty } {K_1}^\prime (x )= \mathop {\lim }\limits_{x \to \infty } {K_1}(x )$. Therefore, it is reasonable to define

(8)$$M({\mathrm{\Omega },r} )= \left\{ {\begin{array}{cc} {{I_1}\left( {\sqrt {i\mathrm{\Omega }/D} r} \right)}&{\mathrm{\Omega } \ne 0}\\ r&{\mathrm{\Omega } = 0} \end{array}} \right.,\textrm{ }N({\mathrm{\Omega },r} )= \left\{ {\begin{array}{cc} {{K_1}\left( {\sqrt {i\mathrm{\Omega }/D} r} \right)}&{\mathrm{\Omega } \ne 0}\\ {1/r}&{\mathrm{\Omega } = 0} \end{array}} \right.$$

We remove the subscripts of Ω and D for generality. Using functions M(Ω,r) and N(Ω,r), we are able to give a unified expression of the temperature field on an object that can be either rotating or at rest

(9)$$T\left( {r,\theta } \right) = {T_{\textrm{const}}} + \frac{1}{2}\left[ {F\left( r \right){e^{i\theta }} + \overline {F\left( r \right)} {e^{ - i\theta }}} \right],\textrm{ }F\left( r \right) = {c_{1}}M\left( {\mathrm{\Omega },r} \right) + {c_{2}}N\left( {\mathrm{\Omega },r} \right)$$

For any value of Ω, N(Ω,r) approaches infinity as r → 0. If the object occupies the point r = 0 as in the present case, the constant c₂ must be zero to avoid singularity. Thus, here we have ${F_0}(r )= {z_0}M({{\mathrm{\Omega }_0},r} )$, where c₁ is rewritten as z₀ for consistency. Note that z₀ could be a complex number. Putting this solid object with radius R₁ at the center of the background κ_b [Fig. 1(a)], z₀ can be solved from the matching conditions

(10)$${ {{T_\textrm{b}}} |_{r = {R_1}}} = { {{T_0}} |_{r = {R_1}}},\textrm{ }{\kappa _\textrm{b}}{\left. {\frac{{\partial {T_\textrm{b}}}}{{\partial r}}} \right|_{r = {R_1}}} = {\kappa _0}{\left. {\frac{{\partial {T_0}}}{{\partial r}}} \right|_{r = {R_1}}}$$

Thereby, based on Eq. (5), we obtain the complex effective thermal conductivity of a rotating object as following

(11)$${\kappa ^\ast }_0 = {\kappa _\textrm{b}}\alpha ({{R_1}} )= {\kappa _0}{R_1}\frac{{M^{\prime}({{\mathrm{\Omega }_0},{R_1}} )}}{{M({{\mathrm{\Omega }_0},{R_1}} )}}$$

where the prime only takes derivative to r. It is a valid definition since κ*₀ is only a function of conditions inside the central region (r ≤ R₁), independent of the background (κ_b) or the external temperature gradient field (A). Note that κ*₀ is reduced to κ₀ at zero rotation speed.

For the set up as shown in Fig. 1(a), the temperature field is numerically simulated using COMSOL Multiphysics for both Ω₀ = 0 [Fig. 1(b)] and Ω₀ = 0.1 rad s⁻¹ [Fig. 1(c)]. The advection effect is significant for the mechanical rotation as shown in Fig. 1(c). For the numerical simulations, the following parameters are set: L = 20 cm, R₁ = 4 cm, T_L = 273 K, T_R = 323 K, and other boundaries with the thermal insulation condition. The inclusion material is set to be steel with thermal conductivity κ₀ = 50 W m⁻¹ K⁻¹ and thermal diffusivity D₀ = 13.3 mm² s⁻¹.

We thus show that rotating an object effectively changes its thermal conductivity from a real number to a complex number. The effect on the thermal conduction surrounding it is two-fold. First, the modulus |κ*₀| increases with the rotation speed [Fig. 1(d)], so a rotating object effectively enhances thermal conduction. The effect has been utilized to realize the thermal analogue of zero-index material [43]. Second, the argument Arg(κ*₀) is nonzero at nonzero rotation speeds [Fig. 1(e)], so the temperature field T_b(r,θ) is always “rotated” due to the nonzero Arg(Ar + z_b/r). This advection effect clearly distinguishes the heat transfer through a moving matter from that with a motionless material, since it changes the direction of the surrounding heat transfer by breaking the reciprocity. From the property of the first-order modified Bessel function, it follows that

(12)$$\mathop {\lim }\limits_{{\mathrm{\Omega }_0} \to \infty } {{M^{\prime}({{\mathrm{\Omega }_0},{R_1}} )} / {M({{\mathrm{\Omega }_0},{R_1}} )}} = {e^{ {\pm} i\frac{\pi }{4}}}\sqrt {|{{\mathrm{\Omega }_0}} |/{D_0}} $$

For common materials the diffusivity is always small: D ∼ 10⁻⁶ to 10⁻⁵ m² s⁻¹. Therefore, it is expected that the effective thermal conductivity quickly becomes proportional to the square root of the rotation speed [see Fig. 1(f)], and its argument quickly approaches ±π/4, as confirmed in Fig. 1(e).

2.3 Complex effective thermal conductivity of a core-shell structure

Based on the above results, in the following we study the general case of a core-shell structure as in Fig. 2. The shell can have multiple layers. For k = 1, 2, … n, where n is the number of layers, the k-th layer has thermal conductivity κ_k, diffusivity D_k, interior radius R_k, exterior radius R_k₊₁, and rotation speed Ω_k. From above, we have known that κ*₀ is independent of the exterior and completely determines the exterior temperature field, and so is the effective thermal conductivity of the core-shell structure κ*_0∼_n. Assuming that the effective thermal conductivity κ*_0∼(n−1) of a (n−1)-layer core-shell structure is already known, we can conveniently replace the complicated equations of inner 0∼(n−1) layers with a stationary one and solve the following heat transfer equations:

(13)$$\begin{array}{l} \frac{{{\partial ^2}{T_\textrm{b}}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T_\textrm{b}}}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{T_\textrm{b}}}}{{\partial {\theta ^2}}} = 0,\textrm{ }r \ge {R_{n + 1}}\\ \frac{{{\partial ^2}{T_n}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T_n}}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{T_n}}}{{\partial {\theta ^2}}} = \frac{{{\mathrm{\Omega }_n}}}{{{D_n}}}\frac{{\partial {T_n}}}{{\partial \theta }},\textrm{ }{R_n} \le r \le {R_{n + 1}}\\ \frac{{{\partial ^2}{T^{\ast }}_{0 \sim ({n - 1} )}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T^{\ast }}_{0 \sim ({n - 1} )}}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{T^{\ast }}_{0 \sim ({n - 1} )}}}{{\partial {\theta ^2}}} = 0,\textrm{ }r \le {R_n} \end{array}$$

where T*_0∼(n−1) represents an effective temperature distribution caused by κ*_0∼(n−1), n = 1,2,…

Fig. 2. Schematic of a core (blue) surrounded by a shell with n layers.

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Focusing on the r-dependent parts, the general solutions are

(14)$$\begin{array}{l} {F_\textrm{b}}(r )= Ar + {z_\textrm{b}}{r^{ - 1}},\textrm{ }r \ge {R_{n + 1}}\\ {F_n}(r )= {z_{2n - 1}}M({{\mathrm{\Omega }_n},r} )+ {z_{2n}}N({{\mathrm{\Omega }_n},r} ),\textrm{ }{R_n} \le r \le {R_{n + 1}}\\ {F^{\ast }}_{0 \sim ({n - 1} )}(r )= {z^{\ast }}_{0 \sim ({n - 1} )}r,\textrm{ }r \le {R_n} \end{array}$$

where we have used the boundary conditions on the background and the nonsingularity condition at r = 0. The remaining four magnitudes (complex numbers) z_b, z_2n-1, z_2n, and z*_0∼(n−1) can be solved from the matching conditions for both temperature and heat fluxes at r = R_n and r = R_n₊₁. After cancelling z_2n-1 and z_2n, the matching conditions give

(15)$$({A{R_{n + 1}} + {z_\textrm{b}}{R_{n + 1}}^{ - 1}} )\left[ {\begin{array}{c} 1\\ {{\kappa_\textrm{b}}\alpha ({{R_{n + 1}}} )} \end{array}} \right]\textrm{ = }{z^{\ast }}_{0 \sim ({n - 1} )}{R_n}{t_n}^{ - 1}\left[ {\begin{array}{cc} {{q_n}}&{ - {\kappa_n}^{ - 1}{p_n}}\\ {{\kappa_n}{s_n}}&{ - {r_n}} \end{array}} \right]\left[ {\begin{array}{c} 1\\ {{\kappa^{\ast }}_{0 \sim ({n - 1} )}} \end{array}} \right]$$

where we define the five cross-products

(16)$$\begin{array}{l} {p_k} = M({{\mathrm{\Omega }_k},{R_{k + 1}}} )N({{\mathrm{\Omega }_k},{R_k}} )- M({{\mathrm{\Omega }_k},{R_k}} )N({{\mathrm{\Omega }_k},{R_{k + 1}}} )\\ {q_k} = {R_k}[{M({{\mathrm{\Omega }_k},{R_{k + 1}}} )N^{\prime}({{\mathrm{\Omega }_k},{R_k}} )- M^{\prime}({{\mathrm{\Omega }_k},{R_k}} )N({{\mathrm{\Omega }_k},{R_{k + 1}}} )} ]\\ {r_k} = {R_{k + 1}}[{M^{\prime}({{\mathrm{\Omega }_k},{R_{k + 1}}} )N({{\mathrm{\Omega }_k},{R_k}} )- M({{\mathrm{\Omega }_k},{R_k}} )N^{\prime}({{\mathrm{\Omega }_k},{R_{k + 1}}} )} ]\\ {s_k} = {R_k}{R_{k + 1}}[{M^{\prime}({{\mathrm{\Omega }_k},{R_{k + 1}}} )N^{\prime}({{\mathrm{\Omega }_k},{R_k}} )- M^{\prime}({{\mathrm{\Omega }_k},{R_k}} )N^{\prime}({{\mathrm{\Omega }_k},{R_{k + 1}}} )} ]\\ {t_k} = {R_k}[{M({{\mathrm{\Omega }_k},{R_k}} )N^{\prime}({{\mathrm{\Omega }_k},{R_k}} )- M^{\prime}({{\mathrm{\Omega }_k},{R_k}} )N({{\mathrm{\Omega }_k},{R_k}} )} ]\end{array}$$

We focus on the temperature field on the background which is determined by z_b or α(R_n₊₁). Then we can obtain the effective thermal conductivity of the core-shell structure using κ*_0∼_n = κ_bα(R_n₊₁), that is

(17)$$\kappa _{0 \sim n}^\ast{=} {\kappa _n}\frac{{{\kappa _n}{s_n} - {\kappa ^{\ast }}_{0 \sim ({n - 1} )}{r_n}}}{{{\kappa _n}{q_n} - {\kappa ^{\ast }}_{0 \sim ({n - 1} )}{p_n}}}$$

Equation (17) is our central result which directly illustrates how the property of the object is modified by adding the shell. Since M(Ω,r) and N(Ω,r) are complex functions for nonzero Ω, the imaginary part of the effective thermal conductivity is also modified. Therefore, there exists some special combinations of rotation speeds (Ω₀, Ω₁,…, Ω_n) that make the imaginary part Im(κ*_0∼_n) = 0, which means that the structure behaves as normal solid materials, for the rotating effect on background temperature field is elimated. Note that when n = 2, Ω_1,2= 0 and κ₁→ 0, Eq. (17) reduces to κ_0∼2 = κ₂(R₃² − R₂²)/(R₃² + R₂²), which agrees with the result of the bi-layer thermal cloak [30]. When n = 2, Ω₁→ ∞ and Ω₂= 0, Eq. (17) reduces to κ_0∼2 = κ₂(R₃² + R₂²)/(R₃² − R₂²), which is consistent with the conclusion of the thermal zero-index cloak [43].

3. Effective thermal conductivity tensor for rotating structures

3.1 Thermal scattering off an anisotropic object

If an object with radius R_out has anisotropic thermal conductivity tensor

(18)$$[{{\kappa^\ast }} ]= \left[ {\begin{array}{cc} {{\kappa^{rr}}}&{{\kappa^{r\theta }}}\\ {{\kappa^{\theta r}}}&{{\kappa^{\theta \theta }}} \end{array}} \right]$$

then the steady-state temperature field on the object T_c(r,θ) satisfies the following heat transfer equation

(19)$${\kappa ^{rr}}\left( {\frac{{{\partial^2}{T_\textrm{c}}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {T_\textrm{c}}}}{{\partial r}}} \right) + ({{\kappa^{r\theta }} + {\kappa^{\theta r}}} )\frac{1}{r}\frac{{{\partial ^2}{T_\textrm{c}}}}{{\partial r\partial \theta }} + {\kappa ^{\theta \theta }}\frac{1}{{{r^2}}}\frac{{{\partial ^2}{T_\textrm{c}}}}{{\partial {\theta ^2}}} = 0,\textrm{ }r \le {R_{out}}$$

Substituting the variable separated solution Eq. (2) into Eq. (19) gives

(20)$${\kappa ^{rr}}[{{r^2}{F_\textrm{c}}^{\prime \prime }(r )+ r{F_\textrm{c}}^\prime (r )} ]+ i({{\kappa^{r\theta }} + {\kappa^{\theta r}}} )r{F_\textrm{c}}^\prime (r )- {\kappa ^{\theta \theta }}{F_\textrm{c}}(r )= 0$$

The general solution of Eq. (20) is

(21)$${F_\textrm{c}}(r )= {c_1}{r^u} + {c_2}{r^{ - \bar{u}}},\textrm{ }u = \sqrt {\frac{{{\kappa ^{\theta \theta }}}}{{{\kappa ^{rr}}}} - {{\left( {\frac{{{\kappa^{r\theta }} + {\kappa^{\theta r}}}}{{2{\kappa^{rr}}}}} \right)}^2}} - i\frac{{{\kappa ^{r\theta }} + {\kappa ^{\theta r}}}}{{2{\kappa ^{rr}}}}$$

where c₁ and c₂ are constants. At r = 0 the temperature should be finite, thus c₂ = 0. For consistency, we rewrite c₁ as z_c and get ${F_\textrm{c}}(r )= {z_\textrm{c}}{r^u}$.

Consider that the object [κ*] is embedded in a square background (L × L) with thermal conductivity κ_b, where a temperature gradient (A) is launched in the x direction by maintaining its left and right boundary at constant temperatures T_L and T_R with T_R − T_L = AL (the upper and lower boundaries are thermally insulated). The r-depended part of the temperature field on the background is still F_b(r) = Ar + z_b/r. Solving the matching conditions

(22)$${ {{T_\textrm{b}}} |_{r = {R_{out}}}} = { {{T_\textrm{c}}} |_{r = {R_{out}}}},\textrm{ }{\kappa _\textrm{b}}{\left. {\frac{{\partial {T_\textrm{b}}}}{{\partial r}}} \right|_{r = {R_{out}}}} = {\left. {\left( {{\kappa^{rr}}\frac{{\partial {T_\textrm{c}}}}{{\partial r}} + \frac{{{\kappa^{r\theta }}}}{r}\frac{{\partial {T_\textrm{c}}}}{{\partial \theta }}} \right)} \right|_{r = {R_{out}}}}$$

Cancelling z_c gives

(23)$$\; {\kappa ^{rr}}u + i{\kappa ^{r\theta }} = {\kappa _\textrm{b}}\frac{{A{R_{out}}^2 - {z_\textrm{b}}}}{{A{R_{out}}^2 + {z_\textrm{b}}}} = {\kappa _b}\alpha ({{R_{out}}} )= {\kappa ^\ast }$$

where we identify the same α(R_out) as in Eq. (5). Therefore, the same z_b can be obtained if the object is replaced with another one whose thermal conductivity is isotropic and equals

(24)$${\kappa ^\ast } = \sqrt {{\kappa ^{\theta \theta }}{\kappa ^{rr}} - {{\left( {\frac{{{\kappa^{r\theta }} + {\kappa^{\theta r}}}}{2}} \right)}^2}} + i\frac{{{\kappa ^{r\theta }} - {\kappa ^{\theta r}}}}{2}$$

For normal solid materials, the Onsager reciprocity requires that the tensor [κ*] is symmetric (κ^rθ = κ^θr), then the effective thermal conductivity κ* = (det[κ*])^1/2 is real. However, when the tensor [κ*] is asymmetric (κ^rθ ≠ κ^θr), the effective thermal conductivity κ* is a complex number, which is inapplicable for natural materials but can be used to equivalently describe the scattering effect of a rotating object.

3.2 Effective thermal conductivity tensor of a core-shell structure

From the effective thermal conductivity for a general tensor with elements (κ^rr,κ^rθ,κ^θr and κ^θθ) in Eq. (24), we know that a complex effective thermal conductivity can be associated with an asymmetric thermal conductivity tensor. There are various combinations of the four elements to give a same effective complex κ*. If we take a special case that the thermal conductivity tensor is anti-symmetric (κ^rr = κ^θθ and κ^rθ = −κ^θr), Eqs. (18) and(24) can be simplified as

(25)$$[{{\kappa^\ast }} ]= \left[ {\begin{array}{cc} {{\kappa^{rr}}}&{{\kappa^{r\theta }}}\\ { - {\kappa^{r\theta }}}&{{\kappa^{rr}}} \end{array}} \right],\textrm{ }{\kappa ^\ast } = {\kappa ^{rr}} + i{\kappa ^{r\theta }}$$

We notice that the complex κ* is equivalent to an anti-symmetric tensor

(26)$$[{{\kappa^\ast }} ]= \left[ {\begin{array}{cc} {\textrm{Re}({{\kappa^\ast }} )}&{\textrm{Im}({{\kappa^\ast }} )}\\ { - \textrm{Im}({{\kappa^\ast }} )}&{\textrm{Re}({{\kappa^\ast }} )} \end{array}} \right] = |{{\kappa^\ast }} |\left[ {\begin{array}{cc} {\cos [{\textrm{Arg}({{\kappa^\ast }} )} ]}&{\sin [{\textrm{Arg}({{\kappa^\ast }} )} ]}\\ { - \sin [{\textrm{Arg}({{\kappa^\ast }} )} ]}&{\cos [{\textrm{Arg}({{\kappa^\ast }} )} ]} \end{array}} \right] = |{{\kappa^\ast }} |R[{\textrm{Arg}({{\kappa^\ast }} )} ]$$

where R(ϕ) is the rotation tensor of angle ϕ. We confirm that the heat transfer through a rotating object breaks reciprocity.

In order to show the equivalence between the complex κ* and the anti-symmetric tensor [κ*], we perform COMSOL simulations with a core-shell structure, and compare its scattering effect to that of its corresponding effective anti-symmetric tensor [κ*_0∼1] (see Fig. 3). For results in Fig. 3, R₁ = 4 cm, R₂ = 5 cm, κ₀= κ₁ = 50 W m⁻¹ K⁻¹, κ_b = 110 W m⁻¹ K⁻¹ and D₀ = D₁ = 13.3 mm² s⁻¹. Other setups remain the same as Fig. 1(a). We summarize in Table 1 the rotation speeds Ω₀ and Ω₁, the effective thermal conductivity of the core-shell structure κ*_0∼1 (written as a complex number) and [κ*_0∼1] (written as an anti-symmetric tensor) and the relative magnitude of the external scattered field z_b/(AR₂²). The results of case I, II and III of the model in Fig. 3(a) [Fig. 3(b)] are depicted in Figs. 3(d), 3(g) and 3(j) [Figs. 3(e), 3(h) and 3(k)], respectively. And the temperature profiles on the red line in Fig. 3(c) are illustrated in Figs. 3(f), 3(i) and 3(l). We see that the external scattering fields in Figs. 3(e), 3(h) and 3(k) are the same as those in Figs. 3(d), 3(g) and 3(j). Temperature profiles in Figs. 4(f), 4(i) and 4(l) also show good agreements.

Fig. 3. Verification of the effective thermal conductivity tensor of a core-shell structure. (a) A core-shell structure with rotating movement. (b) A homogeneous circular object with thermal conductivity tensor [κ*_0∼1]. (c) A transverse line on which the temperature profiles are measured. (d) Temperature field for heat transfer through the model in (a). (e) Temperature field for heat transfer through the model in (b). Except for the translucent region, the field in (e) is the same as that in (d). (f) Temperature profiles on the transverse line in both (d) and (e). The same is true for (g)-(l).

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Table 1. Parameters for Fig. 3.

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3.3 Effective thermal conductivity tensor of a multi-layer shell

According to Eqs. (11) and (17), we can get the effective thermal conductivity κ*₀, κ*_0∼1, …, κ*_0∼_n of a core-shell structure, which can only characterize the scattering effects on the external background, but not to calculate the effective thermal conductivity of a multi-layer shell (e.g. [κ*_1∼2]). This is because the complex thermal conductivity is actually an equivalent description of a special thermal conductivity tensor (an anti-symmetric tensor). It is only valid when we focus on the external scattering effect, but we cannot assume that the effective thermal conductivity is an anti-symmetric tensor when we are concerned with both the external and internal effects of a shell. Actually, the effective thermal conductivity of a multi-layer shell should be expressed as a general tensor

(27)$$[{{\kappa^\ast }_{1 \sim n}} ]= \left[ {\begin{array}{cc} {{\kappa^{rr}}_{1 \sim n}}&{{\kappa^{r\theta }}_{1 \sim n}}\\ {{\kappa^{\theta r}}_{1 \sim n}}&{{\kappa^{\theta \theta }}_{1 \sim n}} \end{array}} \right]$$

Use T*_1∼n as the effective temperature field on the n-layer shell with conductivity tensor [κ*_1∼n]. The heat transfer equation becomes

(28)$${\kappa ^{rr}}_{1 \sim n}\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial {T^{\ast }}_{1 \sim n}}}{{\partial r}}} \right) + ({{\kappa^{r\theta }}_{1 \sim n} + {\kappa^{\theta r}}_{1 \sim n}} )\frac{1}{r}\frac{{{\partial ^2}{T^{\ast }}_{1 \sim n}}}{{\partial r\partial \theta }} + {\kappa ^{\theta \theta }}_{1 \sim n}\frac{1}{{{r^2}}}\frac{{{\partial ^2}{T^{\ast }}_{1 \sim n}}}{{\partial {\theta ^2}}} = 0,\textrm{ }{R_1} \le r \le {R_{n + 1}}$$

Using the similar method as Eqs. (19)-(21), focusing on the r-dependent part, we obtain the general solution

(29)$${F^{\ast }}_{1 \sim n}(r )= {z_1}{r^{{u_{_{1 \sim n}}}}} + {z_2}{r^{ - \overline {{u_{_{1 \sim n}}}} }}, \textrm{ }{u_{1 \sim n}} = \sqrt {\frac{{{\kappa ^{\theta \theta }}_{1 \sim n}}}{{{\kappa ^{rr}}_{1 \sim n}}} - {{\left( {\frac{{{\kappa^{r\theta }}_{1 \sim n} + {\kappa^{\theta r}}_{1 \sim n}}}{{2{\kappa^{rr}}_{1 \sim n}}}} \right)}^2}} - i\frac{{{\kappa ^{r\theta }}_{1 \sim n} + {\kappa ^{\theta r}}_{1 \sim n}}}{{2{\kappa ^{rr}}_{1 \sim n}}}$$

Since the r-depended parts of the temperature field on the outside background κ_b and the inside core κ_c are still F_b(r) = Ar + z_b/r and F_c(r) = z_cr, respectively, the matching conditions can be reduced to

(30)$$({A{R_{n + 1}} + {z_\textrm{b}}{R_{n + 1}}^{ - \textrm{1}}} )\left[ {\begin{array}{c} 1\\ {{\kappa_\textrm{b}}\alpha ({{R_{n + 1}}} )} \end{array}} \right] = \frac{{{z_\textrm{c}}{R_1}}}{{X + \bar{X}}}\left[ {\begin{array}{cc} {\bar{X}Y + X{{\bar{Y}}^{ - 1}}}&{{\kappa_\textrm{c}}^{ - 1}({Y - {{\bar{Y}}^{ - 1}}} )}\\ {{\kappa_\textrm{c}}X\bar{X}({Y - {{\bar{Y}}^{ - 1}}} )}&{XY + \bar{X}{{\bar{Y}}^{ - 1}}} \end{array}} \right]\left[ {\begin{array}{c} 1\\ {{\kappa_\textrm{c}}} \end{array}} \right]$$

where $X = {{({{\kappa^{rr}}_{1 \sim n}u + i{\kappa^{r\theta }}_{1 \sim n}} )} / {{\kappa _\textrm{c}}}}$, $Y = {({{{{R_{n + 1}}} / {{R_1}}}} )^{{u_{\textrm{1} \sim n}}}}$.

On the other hand, recall the thermal scattering of a structure with a stationary core κ_c and n rotating layers κ₁, κ₂,…,κ_n. According to Eq. (17), we deduce the matching conditions after cancelling intermediate variables as following

(31)$$({A{R_{n + 1}} + {z_\textrm{b}}{R_{n + 1}}^{ - 1}} )\left[ {\begin{array}{c} 1\\ {{\kappa_\textrm{b}}\alpha ({{R_{n + 1}}} )} \end{array}} \right] = \mathop \prod \limits_{k = 1}^n \frac{{{z_\textrm{c}}{R_1}}}{{{t_k}}}\left[ {\begin{array}{cc} {{q_k}}&{ - {\kappa_k}^{ - 1}{p_k}}\\ {{\kappa_k}{s_k}}&{ - {r_k}} \end{array}} \right]\left[ {\begin{array}{c} 1\\ {{\kappa_\textrm{c}}} \end{array}} \right]$$

To calculate the effective thermal conductivity tensor of this n-layer shell [κ*_1∼n], comparing Eq. (30) with Eq. (31), we can solve for X and Y. The solutions are

(32)$$\begin{array}{l} X = {{\left[ {({\overline \lambda \mu - \lambda \overline \mu } )\mp \sqrt {{{({\overline \lambda \mu - \lambda \overline \mu } )}^2} + 4({\lambda \overline \lambda - 1} )({\mu \overline \mu - 1} )} } \right]} / {[{2({\lambda \overline \lambda - 1} )} ]}}\\ Y = {{\left[ {({\overline \lambda \mu + \lambda \overline \mu + 2} )\pm \sqrt {{{({\overline \lambda \mu - \lambda \overline \mu } )}^2} + 4({\lambda \overline \lambda - 1} )({\mu \overline \mu - 1} )} } \right]} / {[{2({\overline \lambda + \overline \mu } )} ]}} \end{array}$$

where λ and µ are calculated by

(33)$$\left[ {\begin{array}{c} \lambda \\ {{\kappa_\textrm{c}}\mu } \end{array}} \right] = \mathop \prod \limits_{k = 1}^n \frac{1}{{{t_k}}}\left[ {\begin{array}{cc} {{q_k}}&{ - {\kappa_k}^{ - 1}{p_k}}\\ {{\kappa_k}{s_k}}&{ - {r_k}} \end{array}} \right]\left[ {\begin{array}{c} 1\\ {{\kappa_\textrm{c}}} \end{array}} \right]$$

Therefore, we obtain the formula for calculating [κ*_1∼n]

(34)$$\begin{array}{cc} {\kappa ^{rr}}_{1 \sim n} = {\kappa _\textrm{c}}\frac{{X + \bar{X}}}{{\textrm{ln}Y\bar{Y}}}\textrm{ln}({{R_{n + 1}}/{R_1}} )\textrm{ }&{\kappa ^{r\theta }}_{1 \sim n} = i{\kappa _\textrm{c}}\frac{{\bar{X}\textrm{ln}Y - X\textrm{ln}\bar{Y}}}{{\textrm{ln}Y\bar{Y}}}\\ {\kappa ^{\theta r}}_{1 \sim n} = i{\kappa _\textrm{c}}\frac{{X\textrm{ln}Y - \bar{X}\textrm{ln}\bar{Y}}}{{\textrm{ln}Y\bar{Y}}}\textrm{ }&{\kappa ^{\theta \theta }}_{1 \sim n} = {\kappa _\textrm{c}}\frac{{X + \bar{X}}}{{\textrm{ln}Y\bar{Y}}}\frac{{({\textrm{ln}Y} )({\textrm{ln}\bar{Y}} )}}{{\textrm{ln}({{R_{n + 1}}/{R_1}} )}} \end{array}$$

[κ*_1∼n] is only a function of conditions of the multi-layer shell as well as the core (r ≤ R_n₊₁), independent of the background (κ_b) or the external field (A). Note that [κ*_1∼n] is depended on the core κ_c, that is, the thermal scattering effect of a shell is influenced by the material inside it.

Fig. 4. Verification of the effective thermal conductivity tensor of a shell. (a) A core surrounded by a shell with two rotating layers. (b) A core surrounded by a homogeneous shell with thermal conductivity tensor [κ*_1∼2]. (c) A transverse line on which the temperature profiles are measured. (d) Temperature field for heat transfer through the model in (a). (e) Temperature field for heat transfer through the model in (b). Except for the translucent region, the field in (e) is the same as that in (d). (f) Temperature profiles on the transverse line in both (d) and (e). The same is true for (g)-(l).

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To verify the validity of Eq. (34), we perform COMSOL simulations with a bi-layer shell (n = 2) by comparing its scattering effect to that of its corresponding effective thermal conductivity tensor [κ*_1∼2] (see Fig. 4). For results in Fig. 4, R₁ = 4 cm, R₂ = 5 cm, R₃ = 6 cm, κ_c= κ₁ = κ₂ = 50 W m⁻¹ K⁻¹, κ_b = 110 W m⁻¹ K⁻¹ and D₁ = D₂ = 13.3 mm² s⁻¹. We summarize in Table 2 the rotation speeds Ω₁ and Ω₂, the effective thermal conductivity of the shell [κ*_1∼2] (written as an asymmetric tensor), the relative magnitude of the external scattered field z_b/(AR₃²) and the internal scattered field z_c/A. The results of case i, ii and iii of the model in Fig. 4(a) [Fig. 4(b)] are depicted in Figs. 4(d), 4(g) and 4(j) [Figs. 4(e), 4(h) and 4(k)], respectively. The temperature profiles plotted in Figs. 4(f), 4(i) and 4(l) demonstrate good coincidence of the fields on the red line. Except for the translucent region, the fields in Figs. 4(e), 4(h) and 4(k) are the same as those in Figs. 4(d), 4(g) and 4(j). Thus the validity of the effective thermal conductivity tensor is verified.

Table 2. Parameters for Fig. 4.

View Table | View all tables in this article

4. Conclusion

In summary, the effective thermal conductivity of a rotating object is derived strictly based on its thermal scattering effect on the temperature field outside. Following the idea, we propose to add a concentric shell with n layers to surround the central object and obtain the effective thermal conductivity of the core-shell structure κ*_0∼_n. The effective thermal conductivity of a core-shell structure is a complex number, which is equivalent to an anti-symmetric tensor, if we are only concern about the external influence on background temperature field. But if we are interested in the temperature fields not only outside but also inside the object, another effective thermal conductivity could be defined and calculated only for the shell, say [κ*_1∼_n]. It is usually an asymmetric tensor, but not an anti-symmetric one that can be represented with a complex number like κ*_0∼_n. Interestingly, although it is defined based on the effects of the shell, the value of κ*_1∼_n still depends on the material of the interior object. Our results provide a new perspective towards thermal convection based on effective parameters, indicating broad possibilities to manipulate heat transfer with unprecedented freedom. Mechanical transmission mechanism can be used to make the rotational symmetric structures spin in practical applications. An example of the way to get the layers to spin and control the speeds is to combine the layers with bevel gears which are driven by variable speed motors. The degree of freedom of mechanical motion can be used to design thermal devices with various functions, including adjustable thermal conductivity, controlled temperature field distribution, tunable scattering cancellation or thermal transparency, active thermal cloaking or disguising, and so on.

Funding

Ministry of Education - Singapore (R-263-000-E19-114); PetroChina Innovation Foundation (2018D-5007-0305); National Natural Science Foundation of China (51675122, 51822503); China Scholarship Council.

Disclosures

The authors declare no conflicts of interest.

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	I	II	III
(−Ω₀,Ω₁) (rad s⁻¹)	(0, 0.04767)	(0.1944, 0.04767)	(10, 0.04767)
κ*_0∼1 (W m⁻¹ K⁻¹)	55.65 + 65.66i	110	206.4 + 15.33i
[κ*_0∼1] (W m⁻¹ K⁻¹)	$[\begin{array}{cc} 55.65 & 65.66 \\ - 65.66 & 55.65 \end{array}]$	$[\begin{array}{cc} 110 & 0 \\ 0 & 110 \end{array}]$	$[\begin{array}{cc} 206.4 & 15.33 \\ - 15.33 & 206.4 \end{array}]$
z_b/(AR₂²)	0.1478 − 0.4549i	0	−0.3064 − 0.0336i

	i	ii	iii
(−Ω₁,Ω₂) (rad s⁻¹)	(0.1666, 0)	(0.1666, 0.06031)	(0.1666, 0.1)
[κ*_1∼2] (W m⁻¹ K⁻¹)	$[\begin{array}{cc} 54.46 & 87.17 \\ 289.9 & 785.1 \end{array}]$	$[\begin{array}{cc} 47.61 & 97.20 \\ 97.20 & 511.8 \end{array}]$	$[\begin{array}{cc} 48.59 & 118.1 \\ - 28.12 & 296.6 \end{array}]$
z_b/(AR₃²)	−0.1707 + 0.2778i	0	−0.0418 − 0.2495i
z_c/A	0.1581 + 0.5829i	0.4836 + 0.5264i	0.6419 + 0.3239i

	I	II	III
(−Ω₀,Ω₁) (rad s⁻¹)	(0, 0.04767)	(0.1944, 0.04767)	(10, 0.04767)
κ*_0∼1 (W m⁻¹ K⁻¹)	55.65 + 65.66i	110	206.4 + 15.33i
[κ*_0∼1] (W m⁻¹ K⁻¹)	$[\begin{array}{cc} 55.65 & 65.66 \\ - 65.66 & 55.65 \end{array}]$	$[\begin{array}{cc} 110 & 0 \\ 0 & 110 \end{array}]$	$[\begin{array}{cc} 206.4 & 15.33 \\ - 15.33 & 206.4 \end{array}]$
z_b/(AR₂²)	0.1478 − 0.4549i	0	−0.3064 − 0.0336i

	i	ii	iii
(−Ω₁,Ω₂) (rad s⁻¹)	(0.1666, 0)	(0.1666, 0.06031)	(0.1666, 0.1)
[κ*_1∼2] (W m⁻¹ K⁻¹)	$[\begin{array}{cc} 54.46 & 87.17 \\ 289.9 & 785.1 \end{array}]$	$[\begin{array}{cc} 47.61 & 97.20 \\ 97.20 & 511.8 \end{array}]$	$[\begin{array}{cc} 48.59 & 118.1 \\ - 28.12 & 296.6 \end{array}]$
z_b/(AR₃²)	−0.1707 + 0.2778i	0	−0.0418 − 0.2495i
z_c/A	0.1581 + 0.5829i	0.4836 + 0.5264i	0.6419 + 0.3239i

Effective medium theory for thermal scattering off rotating structures

Abstract

1. Introduction

2. Complex effective thermal conductivity for rotating structures

2.1 Thermal scattering off an isotropic object

2.2 Complex effective thermal conductivity of a rotating object

2.3 Complex effective thermal conductivity of a core-shell structure

3. Effective thermal conductivity tensor for rotating structures

3.1 Thermal scattering off an anisotropic object

3.2 Effective thermal conductivity tensor of a core-shell structure

3.3 Effective thermal conductivity tensor of a multi-layer shell

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (4)

Tables (2)

Equations (34)

Optics Express