Keywords

1 Opening Remarks

Light travels at different speeds along and against the water flow, theoretically predicted by Fresnel [1] and experimentally verified by Fizeau [2]. This momentous discovery, generally referred to as Fizeau drag, has been well explained by relativistic kinematics. Similar effects have also been revealed in other moving [3, 4] or spatiotemporal [5, 6] media. Recently, two experimental studies have reported plasmonic Fizeau drag by the flow of electrons [7, 8], which results from the nonlinear kinematics of drifting Dirac electrons.

On the other hand, diffusion systems can also exhibit wavelike behaviors [9,10,11,12,13,14], which provides the possibility to realize diffusive Fizeau drag. However, unlike the dragging of photons and polaritons by the momentum interaction (Fig. 15.1a, b), it is intrinsically challenging to drag the macroscopic heat by the biased advection [16, 17] due to the absence of macroscopic heat momentum (Fig. 15.1c). Therefore, the forward and backward propagating speeds of temperature fields are always identical. Nevertheless, the amplitudes of temperature fields are different in opposite directions due to the dissipative property of heat transfer [18, 19]. Therefore, it is still an extremely challenging problem to realize diffusive Fizeau drag.

Fig. 15.1
figure 1

Adapted from Ref. [15]

Origin of diffusive Fizeau drag. Fizeau drag of a light and b polariton by the momentum interaction. c Failure of a direct thermal analog due to the lack of macroscopic heat momentum. d Fizeau drag of heat in a spatiotemporal thermal metamaterial by thermal Willis coupling. The red arrows contain the information on wave number and amplitude, indicating the forward and backward cases with (a), (b), (d) different wave numbers, and (c) different amplitudes.

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2 Theoretical Foundation

We construct a spatiotemporal thermal metamaterial with space-related inhomogeneity and time-related advection to uncover diffusive Fizeau drag in heat transfer (Fig. 15.1d). Since the characteristic length of spatiotemporal modulation is much smaller than the wavelength of wavelike temperature fields, the proposed structure can be regarded as a metamaterial. Neither periodic inhomogeneity nor vertical advection alone contributes to the horizontal nonreciprocity, but their synergistic effect can give rise to diffusive Fizeau drag. The underlying mechanism lies in the coupling between heat flux and temperature change rate, which can be regarded as the thermal counterpart of Willis coupling in mechanical waves [20,21,22,23,24]. Therefore, the present nonreciprocity is distinctly different from the synthetic-motion-induced nonreciprocity [25, 26].

We first explain why the direct scheme presented in Fig. 15.1c fails. Heat transfer in porous media is described by \(\rho _0 \partial _t T+\boldsymbol{\nabla }\left( \phi \rho _a \boldsymbol{u}T-\kappa _0 \boldsymbol{\nabla }T\right) =0\), where \(\rho _0\) (or \(\rho _a\)) is the product of mass density and heat capacity of the porous medium (or fluid), \(\kappa _0\) is the thermal conductivity of the porous medium, \(\phi \) is the porosity, and \(\boldsymbol{u}\) is the velocity of the fluid with the horizontal and vertical components of \(u_x\) and \(u_y\), respectively. We consider a wavelike temperature field described by \(T=Ae^{i(\beta x-\omega t)}+T_r\), where \(\beta \) and \(\omega \) are the wave number and angular frequency, respectively. Here, we use “wavelike” because heat transfer is essentially governed by a diffusive equation rather than a wave equation. We set the temperature field amplitude of A as 1 and the balanced temperature of \(T_r\) as 0 for brevity. We apply a periodic source with a temperature of \(T(x=0)=e^{-i\omega t}\), thus leading to a real \(\omega \) and a complex \(\beta \). The imaginary part of \(\beta \) reflects the spatial decay rate of wavelike temperature fields. We focus on the real part of \(\beta \) because the propagating speed of wavelike temperature fields can be calculated by \(v=\omega /\textrm{Re}[\beta ]\). The substitution of \(T=e^{i(\beta x-\omega t)}\) with a preset real \(\omega \) into the governing equation of heat transfer yields

$$\begin{aligned} \beta _{f,\,b}=\pm \frac{\sqrt{2}\gamma }{4\kappa _0}+i \frac{-8\phi \rho _a u_x \omega \rho _0 \gamma _0\pm \sqrt{2} \gamma \left( 2\phi ^2_{a}{^{2}} u_x^2+\gamma ^2\right) }{16\omega \rho _0 \kappa _0^2}, \end{aligned}$$
(15.1)

where \(\beta _f\) and \(\beta _b\) are, respectively, the forward and backward wavenumbers with a definition of \(\gamma =\sqrt{-\phi ^2 \rho _a^2 u_x^2+\sqrt{\phi ^4 \rho _a^4 u_x^4+16\omega ^2 \rho _0^2 \kappa _0^2}}\). Since a nonzero \(u_x\) cannot generate different \(|\textrm{Re}[\beta ]|\) in opposite directions, the forward and backward propagating speeds of temperature fields are identical, i.e., no diffusive Fizeau drag.

To achieve diffusive Fizeau drag, we introduce spatially-periodic inhomogeneity to the porous medium,

$$\begin{aligned} \rho (\xi )&=\rho _0(1+\Delta _\rho \cos (G\xi +\theta )),\end{aligned}$$
(15.2a)
$$\begin{aligned} \kappa (\xi )&=\kappa _0(1+\Delta _\kappa \cos (G\xi )), \end{aligned}$$
(15.2b)

where \(\Delta _\rho \) and \(\Delta _\kappa \) are the modulation amplitudes, \(G=2\pi /d\) is the modulation wave number, d is the horizontal modulation wavelength, \(\xi =x+\zeta y\) is the generalized coordinate with a definition of \(\zeta =d/h\), h is the vertical height, and \(\theta \) is the modulation phase difference. To exclude the captivation that the horizontal advection can generate nonreciprocal amplitudes of temperature fields, as described by the imaginary part of Eq. (15.1), we consider the upward advection with a speed of \(u_y\), which does not contribute to the horizontal nonreciprocity. The governing equation of heat transfer in spatiotemporal thermal metamaterials can be expressed as

$$\begin{aligned} \overline{\rho }(\xi ) \frac{\partial T}{\partial t}+\phi u_y \frac{\partial T}{\partial y}+\frac{\partial }{\partial x} \left( -D_0 \overline{\kappa }(\xi ) \frac{\partial T}{\partial x}\right) +\frac{\partial }{\partial y} \left( -D_0 \overline{\kappa }(\xi ) \frac{\partial T}{\partial y}\right) =0, \end{aligned}$$
(15.3)

with definitions of \(\overline{\rho }(\xi )=\rho (\xi )/\rho _0\), \(\overline{\kappa }(\xi )=\kappa (\xi )/\kappa _0\), \(\epsilon =\rho _a/\rho _0\), and \(D_0=\kappa _0/\rho _0\).

We further consider a wavelike temperature field with a spatially-periodic modulation,

$$\begin{aligned} T=F(\xi ) e^{i(\beta x-\omega t)} =\left( \sum _s F_s e^{isG\xi }\right) e^{i(\beta x-\omega t)} , \end{aligned}$$
(15.4)

where \(F(\xi )\) is a Bloch modulation function with parameters of \(s=0,\,\pm 1,\,\pm 2,\,\ldots ,\,\pm \infty \) and \(F_0=1\). We can treat \(e^{i(\beta x-\omega t)}\) as the temperature field envelope and \(F(\xi )\) as local inhomogeneity. The substitution of Eq. (15.4) into Eq. (15.3) yields a series of component equations related to the order of s. For accuracy, we consider \(s=0,\,\pm 1,\,\pm 2,\,\cdots ,\,\pm 10\) and \(F_{|s|>10}=0\) to obtain twenty-one equations with twenty-one unknown numbers including \(\beta \) and \(F_{|s|\le 10}\), so \(\beta \) can be numerically calculated.

The properties of spatiotemporal modulation are reflected in three crucial dimensionless parameters of \(2\pi \Gamma =\phi u_y d/D_0\), \(\Lambda =\Delta _\rho \cos \theta /\Delta _\kappa \), and \(\zeta =d/h\). The parameter of \(2\pi \Gamma \) is similar to the Peclet number, which can describe the ratio of advection to diffusion. The parameters of \(\Lambda \) and \(\zeta \) reflect the influences of modulation amplitude and wavelength, respectively. We define the speed ratio as \(\eta =|v_f/v_b |=|\textrm{Re}[\beta _b ]/\textrm{Re}[\beta _f ]|\) to discuss the degree of nonreciprocity, where \(v_f\) and \(v_b\) are the forward and backward propagating speeds of temperature fields, respectively.

We first discuss \(\Lambda \) when \(\zeta =0.2\) (Fig. 15.2a). Since \(2\pi \Gamma =0\) and \(2\pi \Gamma \rightarrow \infty \) always yield \(\eta =1\), it is necessary to introduce the vertical advection, but not the larger, the better. Meanwhile, a speed difference still exists when \(\Lambda =0\) (i.e., \(\Delta _\rho =0\)), so it is unnecessary to modulate \(\rho \) and \(\kappa \) simultaneously. We find two types of curves in Fig. 15.2a. Type I features that \(\eta \) is always larger than 1 (the top three curves). Type II features that \(\eta \) is first larger and then smaller than 1 (the bottom three curves). The transition between types I and II is at the critical point of \(\Lambda =1\) (the third curve from the top), where the modulations in Eqs. (15.2a) and (15.2b) do not affect the effective thermal diffusivity in the vertical direction. When we change \(\zeta \) from 0.2 to 1 (Fig. 15.2b) and 2 (Fig. 15.2c), type III curves appear, with \(\eta \) always smaller than 1. These three types indicate that nonreciprocal speeds can be flexibly manipulated.

Fig. 15.2
figure 2

Adapted from Ref. [15]

Numerical results of the speed ratio of \(\eta =|v_f/v_b|\) as a function of \(2\pi \Gamma =\phi u_y d/D_0\). \(\Lambda =\Delta _\rho \cos \theta /\Delta _\kappa \) is tuned by a–c \(\Delta _\rho \) or (d) \(\theta \). Except the parameters presented in a–f, the others are \(\phi =0.1\), \(\epsilon =1\), \(D_0=5\times 10^{-5}\) m\(^2\)/s, \(d=0.02\) m, and \(\omega =\pi /10\) rad/s for a–f; \(\Delta _\rho =0.7\) for (d); \(\Delta _\rho =0.6\) for (f); \(\Delta _\kappa =0.5\) for (a)-(e); \(\Delta _\kappa =0.9\) for (f); \(\theta =0\) for (a)–(d) and (f); and \(\theta =\pi /2\) for (e).

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We further discuss \(\theta \) when \(\zeta =1\) (Fig. 15.2d), so \(\Lambda =\Delta _\rho \cos \theta /\Delta _\kappa \) can be both positive and negative. The critical point of \(\Lambda =1\) still determines the transition between types I and II. Moreover, since \(\theta =\pi /2\) always leads to \(\Lambda =0\), the curves in Fig. 15.2e are almost overlapped. We also discuss the thermal diffusivity of \(D=\kappa /\rho \) (Fig. 15.2f), where \(\kappa \) is the balanced value of the periodic thermal conductivity and \(\rho \) is the balanced value of the periodic product of mass density and heat capacity. The peaks of \(\eta \) appear at almost the same value of \(2\pi \Gamma \). Meanwhile, the peak of \(\eta \) gets larger as the thermal diffusivity decreases, which does not mean that the smaller the thermal diffusivity is, the better. We do not discuss the small thermal diffusivity because the system becomes insulated.

Fig. 15.3
figure 3

Adapted from Ref. [15]

Simulation results of diffusive Fizeau drag. a Thermal dispersion. b Wave number difference \(\Delta \textrm{Re}[\beta ]=\textrm{Re}[\beta _f]+\textrm{Re}[\beta _b]\) as a function of \(\omega \). c Time difference per unit of distance \(\Delta t/|x|=\Delta \textrm{Re}[\beta ]/\omega \) as a function of \(\omega \). Evolution of \(T^*\) when (d1)–(d4) \(2\pi \Gamma =0\) or (e1)–(e4) \(2\pi \Gamma =8\), corresponding to \(u_y=0\) or \(u_y=0.2\) m/s, respectively. Parameters: \(\phi =0.1\), \(\epsilon =1\), \(D_0=5\times 10^{-5}\) m\(^2\)/s, \(\Delta _\rho =0.9\), \(\Delta _\kappa =0.9\), \(\theta =\pi \), \(d=0.02\) m, \(h=0.02\) m, and \(t_0=20\) s. The simulation length is \(30d=0.6\) m. The left and right boundaries are insulated. The upper and lower boundaries are set with periodic conditions. Sim.: Simulation; and Num.: Numerical.

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We further plot the thermal dispersion in Fig. 15.3a. The thermal dispersion curve is symmetric when \(2\pi \Gamma =0\), but becomes asymmetric when \(2\pi \Gamma =8\), which is the proof of diffusive Fizeau drag. We also plot the wavenumber difference \(\Delta \textrm{Re}[\beta ]=\textrm{Re}[\beta _f]+\textrm{Re}[\beta _b]\) in Fig. 15.3b, demonstrating linear responses to \(\omega \). More intuitively, a speed difference leads to a time difference of temperature field evolution at two symmetric positions of x and \(C-x\) to reach the same phases. The forward phase at x is \(\textrm{Re}[\beta _f]x-\omega t_f\), and the backward phase at \(-x\) is \(-\textrm{Re}[\beta _b]x-\omega t_b\). The same phases correspond to a time difference of \(\Delta t=t_f-t_b\), which can be calculated by

$$\begin{aligned} \Delta t=\Delta \textrm{Re}[\beta ]|x|/\omega . \end{aligned}$$
(15.5)

Since \(\Delta t\) increases linearly with |x|, we focus on the parameter of \(\Delta t/|x|=\Delta \textrm{Re}[\beta ]/\omega \) in Fig. 15.3c, which is almost invariant as \(\omega \) changes.

3 Finite-Element Simulation

Finite-element simulations are also performed with COMSOL Multiphysics. For brevity, we define a dimensionless temperature of \(T^*=(T-T_r )/A\) and a dimensionless time of \(t^*=t/t_0\), where \(t_0\) is the time periodicity of the temperature source. When \(2\pi \Gamma =0\) (Fig. 15.3d1), the forward and backward cases are identical at \(y=0\), but a slight difference appears at \(y=\pm h/4\) (Fig. 15.3d2, d3) due to the local inhomogeneity described by the \(F(\xi )\) in Eq. (15.4). As long as we discuss the average temperature in the vertical direction, the effect of local inhomogeneity can be excluded, so the forward and backward cases become identical again (Fig. 15.3d4). We further set \(2\pi \Gamma =8\), and the simulation results demonstrate a time difference of \(\Delta t^*=0.14\), which can be observed locally (Fig. 15.3e1–e3) and globally (Fig. 15.3e4). The numerical results predict a time difference of \(\Delta t^*=0.15\), indicating that the numerical calculations are convincing. Meanwhile, we plot the numerical results with dotted curves, which agree well with the simulation results.

Fig. 15.4
figure 4

Adapted from Ref. [15]

Influences of inhomogeneity on thermal Willis coupling. The left column shows different kinds of inhomogeneity. The right column shows the evolution of \(T^*\). The parameters and boundary conditions are the same as those in Fig. 15.3.

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We analytically homogenize the governing equation to reveal the underlying mechanism of diffusive Fizeau drag. We find two high-order terms of \(\partial ^2_t\) and \(\partial _t \partial _x\) in the homogenized equation. This situation is similar to the properties of Willis metamaterials that result from the homogenization of inhomogeneous media [20,21,22,23,24]. The modified constitutive relation describing the heat flux of J can be approximately expressed as \(\tau \partial _t J+J=-\kappa _e \partial _x T_0+\sigma _2 \partial _t T_0\), where \(\tau \), \(\kappa _e\), \(\sigma _2\), and \(T_0\) are the homogenized parameters. Besides the temperature gradient of \(\partial _x T_0\), the horizontal heat flux is also coupled with the temperature change rate of \(\partial _t T_0\), which can be referred to as the thermal Willis term. Moreover, the thermal Willis term can lead to nonreciprocal \(|\textrm{Re}[\beta ]|\), but cannot generate nonreciprocal \(|\textrm{Im}[\beta ]|\). This property indicates an obvious speed difference but no amplitude difference in opposite directions, which agrees with the simulation results in Fig. 15.3e1–e4.

Inhomogeneity is crucial to thermal Willis coupling. It will disappear when we consider only the horizontal inhomogeneity (Fig. 15.4a1, a2) or only the vertical inhomogeneity (Fig. 15.4b1, b2). We further change the modulations from cosine functions to square wave functions denoted by \(\Pi \). When the periodicity of inhomogeneity is the same as that in Fig. 15.3e and \(2\pi \Gamma =8\), a time difference of \(\Delta t^*=0.32\) can be observed (Fig. 15.4c1, c2). Therefore, the square wave modulation is more efficient than the cosine modulation (\(\Delta t^*=0.14\)). We further reduce the modulation wavelength by a factor of five (Fig. 15.4d1). When \(2\pi \Gamma =8\), a time difference of \(\Delta t^*=0.04\) appears, but it is far smaller than that of \(\Delta t^*=0.32\) in Fig. 15.4c2. Therefore, the more homogeneous parameters yield weaker thermal Willis coupling, which is consistent with the current understanding in mechanical waves [20,21,22,23,24].

Fig. 15.5
figure 5

Adapted from Ref. [15]

Experimental suggestions. a Three-dimensional and b two-dimensional diagrams of a three-layer pipe. Temperature evolution when (c1)–(c4) \(\Omega =0\) or (d1)–(d4) \(\Omega =2\pi \) rad/s. The parameters of the center layer are \(\rho _0=2\times 10^6\) J m\(^{-3}\) K\(^{-1}\), \(\kappa _0=200\) W m\(^{-1}\) K\(^{-1}\), \(\Delta _\rho =0.9\), \(\Delta _\kappa =0.9\), and \(\theta =\pi \). Those of the inner and outer layers are \(\rho =2\times 10^6\) J m\(^{-3}\) K\(^{-1}\) and \(\kappa =10\) W m\(^{-1}\) K\(^{-1}\). The other parameters are \(d=20\) mm, \(r_1=2.43\) mm, \(r_2=2.93\) mm, \(r_3=3.43\) mm, \(r_4=3.93\) mm, and \(t_0=20\) s. The simulation length is \(15d=300\) mm. The left and right boundaries are insulated.

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4 Experimental Suggestion

For experimental suggestions, we design a three-dimensional structure without fluids, i.e., a three-layer solid pipe (Fig. 15.5a, b). The inner and outer layers are homogeneous with the same angular velocities of \(\Omega \). The center layer is stationary with spatially-periodic parameters of \(\rho (\xi ')=\rho _0(1+\Delta _\rho \cos (G\xi '+\theta ))\) and \(\kappa (\xi ')=\kappa _0(1+\Delta _\kappa \cos (G\xi '))\), where \(\xi '=x+\alpha /G\) is the generalized coordinate in three dimensions with definitions of \(\cos \alpha =z/\sqrt{y^2+z^2}\) and \(\sin \alpha =y/\sqrt{y^2+z^2}\). The two rotating layers can provide the surface advection to the center layer, which has a similar effect as the bulk advection. The simulation results without and with angular rotation are presented in Fig. 15.5c1–c4, d1–d4, respectively. The detecting locations are in the inner, center, outer, and all layers, respectively. With proper angular rotation, a time difference between the evolution of the forward and backward temperature field indicates diffusive Fizeau drag.

5 Conclusion

We conclude the distinctive features of diffusive Fizeau drag. (I) As described by Eq. (15.1), only the biased advection cannot realize diffusive Fizeau drag. (II) Diffusive Fizeau drag in spatiotemporal thermal metamaterials results from thermal Willis coupling between heat flux and temperature change rate. (III) Diffusive Fizeau drag is unexpected because the vertical advection can generally not induce horizontal nonreciprocity. (IV) Three curves in Fig. 15.2 indicate that diffusive Fizeau drag can be flexibly controlled.

We have revealed diffusive Fizeau drag in a spatiotemporal thermal metamaterial, featuring a speed difference of temperature field propagation in opposite directions. Spatial or temporal modulation alone cannot realize the horizontal nonreciprocity, so spatiotemporal modulation necessarily introduces the high-order coupling, referred to as thermal Willis coupling, between heat flux and temperature change rate. Diffusive Fizeau drag has also been visualized by observing the time difference of temperature field evolution at two symmetric positions. These results suggest a distinct mechanism to achieve nonreciprocal diffusion [25,26,27,28] by thermal Willis coupling and also have potential applications for controlling nonequilibrium heat and mass transfer [30, 31].

6 Exercise and Solution

Exercise

  1. 1.

    Derive Eq. (15.1).

Solution

  1. 1.

    Heat transfer in a two-dimensional homogeneous porous medium is governed by

    $$\begin{aligned} \rho _0 \frac{\partial T}{\partial t}+\boldsymbol{\nabla }\cdot \left( \phi \rho _a uT-\kappa _0 \boldsymbol{\nabla }T\right) =0, \end{aligned}$$
    (15.6)

    with definitions of \(\rho _0=\phi \rho _a+(1-\phi ) \rho _{s0}\) and \(\kappa _0=\phi \kappa _a+(1-\phi ) \kappa _{s0}\). \(\rho _a\) (or \(\rho _{s0}\)) is the product of mass density and heat capacity of the fluid (or solid). \(\kappa _a\) (or \(\kappa _{s0}\)) is the thermal conductivity of the fluid (or solid). The substitution of a wavelike temperature field described by \(T=e^{i(\beta x-\omega t)}\) into Eq. (15.6) yields

    $$\begin{aligned} -i\omega \rho _0+i\beta \phi \rho _a u_x+\beta ^2 \kappa _0=0. \end{aligned}$$
    (15.7)

    Since we apply a periodic source with a temperature of \(T(x=0)=e^{(-i\omega t)}\), \(\omega \) is real and \(\beta \) is complex. We can take \(\beta =p+iq\), with p and q being two real numbers, so Eq. (15.7) can be rewritten as

    $$\begin{aligned} -i\omega \rho _0+i(p+iq)\phi \rho _a u_x+(p+iq)^2 \kappa _0=0, \end{aligned}$$
    (15.8)

    which can be further decomposed into two equations according to its real and imaginary parts,

    $$\begin{aligned} -q\phi \rho _a u_x+(p^2-q^2 ) \kappa _0=0,\end{aligned}$$
    (15.9a)
    $$\begin{aligned} -\omega \rho _0+p\phi \rho _a u_x+2pq\kappa _0=0. \end{aligned}$$
    (15.9b)

    The solution to Eqs. (15.9a) and (15.9b) is just Eq. (15.1).