# Convective Heat Transfer in Porous Materials

Peng Jin<EMAIL_ADDRESS>Department of Physics, State Key Laboratory
of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures
(MOE), Fudan University, Shanghai 200438, China  Gaole Dai  School of
Sciences, Nantong University, Nantong 226019, China  Fubao Yang  Department of
Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro
and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China

###### Abstract

Thermal convection stands out as an exceptionally efficient thermal transport
mechanism, distinctly separate from conduction and radiation. Yet, the
inherently elusive nature of fluid motion poses challenges in accurately
controlling convective heat flow. While recent innovations have harnessed
thermal convection to achieve effective thermal conductivity, fusing thermal
convection in liquids and thermal conduction in solids together to form hybrid
thermal metamaterials is still challenging. In this review, we introduce the
latest progress in convective heat transfer. Leveraging the right porous
materials as a medium allows for a harmonious balance and synergy between
convection and conduction, establishing stable heat and fluid flows. This
paves the way for the innovative advancements in transformation thermotics.
These findings demonstrate the remarkable tunability of convective heat
transport in complex multicomponent thermal metamaterials.

Keywords Convective heat transfer · Porous materials · Hybrid metamaterials

##  I Introduction

Over the past decade, the emergence of thermal metamaterials (PJ-DaiIJHMT20, ;
PJ-rmp, ; PJ-click, ; PJ-research, ; PJ-am, ; PJ-zhounc, ; PJ-XuSCPMA20, ; PJ-
Huang20, ; PJ-WangPRE20, ; PJ-XuPRAP20, ; PJ-XuESEE20, ; PJ-HuangESEE20, ; PJ-
XuEPJB20, ; PJ-SuEPL20, ; PJ-WangPRAP20, ; PJ-XuIJHMT20, ; PJ-XuAPL20, ; PJ-
XuEPL20, ; PJ-XuCPLEL20, ; PJ-WangICHMT20, ; PJ-HuangPhysics20, ; PJ-
YangJAP20, ; PJ-JinIJHMT20, ; PJ-XuEPL20-1, ; PJ-XuPRE20, ; PJ-WangiScience20,
; PJ-YangPRAP20, ; PJ-XuCPL20, ; PJ-XuIJHMT21, ; PJ-DaiJNU21, ; PJ-
DaiiScience21, ; PJ-WangEPL21, ; PJ-XuEPL21, ; PJ-LiuJAP21, ; PJ-WangATE21, ;
PJ-WangPRAP21, ; PJ-XuPRE21, ; PJ-YangPR21, ; PJ-JinIJHMT21, ; PJ-ZhangTSEP21,
; PJ-TianIJHMT21, ; PJ-XuAPL21, ) and transformation thermotics (PJ-
FanAPL2008, ; PJ-XuEPL21-1, ; PJ-LiuJAP21-1, ; PJ-LeiEPL21, ; PJ-QuEPL21, ;
PJ-GaoNM21, ; PJ-ZhangATS22, ; PJ-HuangAMT22, ; PJ-ZhuangSCPMA22, ; PJ-
DaiPRAP22, ; PJ-XuPRL22, ; PJ-LinSCPMA22, ; PJ-ZhangCPL22, ; PJ-WangCPB22, ;
PJ-YangPRAP22, ; PJ-LiPF22, ; PJ-ZhuangPRE22, ; PJ-JinPNAS23, ; PJ-XuBook23, ;
PJ-XuPRL22-1, ; PJ-YaoISci22, ; PJ-ZhangPRD22, ; PJ-ZhouEPL23, ; PJ-
ZhangPRA23, ; PJ-LeiIJHMT23, ) has greatly broadened the horizons of heat
manipulation (PJ-ZhuangIJMSD23, ; PJ-XuNSR23, ; PJ-LeiMTP23, ; PJ-ZhangCPL23,
; PJ-YangPRA23, ; PJ-ZhangNRP23, ; PJ-XuPANS23, ; PJ-DaiPR23, ; PJ-LiJAP10, ;
PJ-QiuEPL13, ; PJ-QiuIJHT14, ; PJ-QiuAIPAdv.15, ; PJ-Tan20, ; PJ-Tan16, ; PJ-
Tan15, ; PJ-Tan11, ; PJ-Tan4, ; PJ-ShenAPL16, ; PJ-ShenPRL16, ; PJ-XinPA17, ;
PJ-HuangFP17, ; PJ-HuangPB17, ; PJ-XinFP17, ; PJ-YangAPL17, ; PJ-XuEPJB17, ).
This expansion has proven invaluable in a variety of applications (PJ-
WangJAP17, ; PJ-JiPA18, ; PJ-MengCPB18, ; PJ-ShangIJHMT18, ; PJ-JiCTP18, ; PJ-
DaiEPJB18, ; PJ-DaiPRE18, ; PJ-WangJAP18, ; PJ-ShangJHT18, ; PJ-WangIJTS18, ;
PJ-JiIJMPB18, ; PJ-XuEPJB18, ; PJ-XuJAP18, ; PJ-XuPLA18, ; PJ-XuPRE18, ; PJ-
DaiJAP18, ; PJ-HuangPP18, ; PJ-ShangAPL18, ; PJ-YangJAP19, ; PJ-XuEPJB19, ;
PJ-XuPRE19, ; PJ-WangPRA19, ; PJ-XuPRA19a, ; PJ-XuEPL19, ; PJ-YangPRE19, ; PJ-
XuPRA19, ; PJ-YangEPL19, ; PJ-ZhouESEE19, ; PJ-XuPRAP19, ; PJ-XuEPJB19-1, ;
PJ-YangEPL19-1, ; PJ-XuPRE19-1, ; PJ-HuangESEE19, ), from thermal cloaking and
camouflage (PJ-XuSCPMA20, ; PJ-Huang20, ; PJ-WangPRE20, ; PJ-XuPRAP20, ; PJ-
XuESEE20, ; PJ-HuangESEE20, ; PJ-XuEPJB20, ; PJ-SuEPL20, ; PJ-WangPRAP20, ;
PJ-XuIJHMT20, ; PJ-XuAPL20, ; PJ-XuEPL20, ; PJ-XuCPLEL20, ; PJ-WangICHMT20, ;
PJ-HuangPhysics20, ; PJ-YangJAP20, ; PJ-JinIJHMT20, ; PJ-XuEPL20-1, ; PJ-
XuPRE20, ; PJ-WangiScience20, ; PJ-YangPRAP20, ; PJ-XuCPL20, ; PJ-XuIJHMT21, ;
PJ-DaiJNU21, ; PJ-WangEPL21, ; PJ-XuEPL21, ; PJ-LiuJAP21, ; PJ-WangATE21, ;
PJ-WangPRAP21, ; PJ-XuPRE21, ; PJ-ZhuangIJMSD23, ; PJ-XuNSR23, ; PJ-LeiMTP23,
; PJ-ZhangCPL23, ; PJ-YangPRA23, ; PJ-ZhangNRP23, ; PJ-YangPRE19, ; PJ-
XuPRA19, ) to heat management in microchips PJ-YaoISci22 ; PJ-ZhouEPL23 ; PJ-
LeiIJHMT23 , energy conservation in everyday life (PJ-JiCTP18, ; PJ-DaiEPJB18,
; PJ-DaiPRE18, ; PJ-WangJAP18, ; PJ-ShangJHT18, ; PJ-WangIJTS18, ), and
thermoregulation in biological cells PJ-ZhangPRA23 ; PJ-ZhangNRP23 . Yet, much
of the progress in this field has been concentrated on conductive thermal
metamaterials (PJ-FanAPL2008, ; PJ-LiJAP10, ; PJ-QiuEPL13, ; PJ-QiuIJHT14, ;
PJ-QiuAIPAdv.15, ; PJ-Tan16, ; PJ-ShenPRL16, ). These materials primarily rely
on diffusive or effective heat conduction, which is constrained by Onsager’s
reciprocity. Such reliance places constraints on the versatility of heat
manipulation. Additionally, traditional thermal metamaterials lack the
flexibility to adjust their functions based on given temperature conditions
(PJ-Tan16, ; PJ-ShenPRL16, ), depriving them of the adaptive control often
required.

Thermal convection (PJ-DaiPRE18, ; PJ-DaiPRE23, ; PJ-DaiJAP18, ; PJ-
YangESEE19, ; PJ-JinPNAS23, ; PJ-XuSCPMA20, ; PJ-XuIJHMT20, ), with its
distinct nature, plays a crucial role as a mechanism for thermal transport.
Historically, its role was often overshadowed in the realms of thermal
metamaterials and transformation thermotics. Only recently has the theory of
transformation thermotics expanded its scope to include thermal convection
(PJ-DaiPRE18, ; PJ-DaiPRE23, ; PJ-DaiJAP18, ; PJ-XuIJHMT20, ), necessitating
the creation of a novel theoretical framework. Yet, merging thermal convection
in liquids with thermal conduction in solids to create hybrid thermal
metamaterials presents a significant challenge. This is because these distinct
paths of heat transport must align and collaborate harmoniously to generate
stable heat and liquid flows that fulfill the requirements of the underlying
thermotic transformation. Designing such hybrid materials proves more
intricate than traditional all-solid thermal metamaterials. While there have
been efforts to integrate thermal convection to achieve remarkable thermal
conductivity levels (exemplifying synthetic Onsager reciprocity) PJ-31 ; PJ-32
; PJ-33 , there is a pressing need to develop thermal metamaterials that can
simultaneously control both conductive and convective heat flows beyond the
bounds of Onsager reciprocity.

In this review, we present the latest advancements in heat transfer of porous
materials. We commence by elaborating on the foundational principles of
transformation thermotics, addressing both steady-state and transient-state
challenges of convective heat transfer in porous mediums PJ-DaiPRE18 ; PJ-
DaiPRE23 ; PJ-DaiJAP18 . This potent theory paves the way for the conceptual
design of innovative thermal devices, including illusion and camouflage
mechanisms PJ-YangESEE19 ; PJ-XuSCPMA20 . Further, we clarify the emergence of
experimental platforms for the realization of continuous switch between
thermal cloak and thermal concentration PJ-JinPNAS23 , which reveals the
significant tunability of the hybrid thermal metamaterial. Finally, we
envision that such porous mediums could serve as the ideal physical platform
for achieving robust thermal-protected transport with topological features.

##  II Steady-state Transformation Thermo-hydrodynamics

When addressing heat transfer in fluids, we begin by adjusting the heat
conduction equation for incompressible flow, excluding heat sources and
disregarding the viscous dissipation term PJ-Landau , as

$\rho C_{p}\nabla\cdot(\vec{v}T)=\nabla\cdot(\eta\nabla T),$ (1)

where $\rho$, $C_{p}$, $\eta$, and $\vec{v}$ are respectively the density,
specific heat at constant pressure, thermal conductivity, and the velocity of
the fluid. As is known, $\rho C_{p}\nabla\cdot(\vec{v}T)$ is the term due to
advection. Equation (1) represents the convection-diffusion equation. For the
sake of clarity, we assume a laminar, Newtonian flow and consider the density
to be unaffected by temperature variations.

For the coordinate transformation $\\{x_{i}\\}\rightarrow\\{y_{j}\\}$ and the
associated Jacobian matrix
$\textbf{J}=\frac{\partial(y_{1},y_{2},y_{3})}{\partial(x_{1},x_{2},x_{3})},$
we can write PJ-JRCI2013

$\rho C_{p}\sum\limits_{j}\frac{\partial}{\partial
y_{i}}\left(\frac{1}{\det\textbf{J}}\sum\limits_{i}J_{ij}^{\intercal}v_{i}T\right)=\sum\limits_{ijkl}\frac{\partial}{\partial
y_{k}}\left(\frac{1}{\det\textbf{J}}J_{ki}\eta_{ij}J_{jl}^{\intercal}\frac{\partial
T}{\partial y_{l}}\right).$ (2)

Let $\vec{v}^{\prime}=\frac{\textbf{J}\vec{v}}{\det\textbf{J}}$ and
$\eta^{\prime}=\frac{\textbf{J}{\eta}\textbf{J}^{\intercal}}{\det\textbf{J}},$
and we achieve

$\rho
C_{p}\left[\nabla^{\prime}\cdot(\vec{v}^{\prime}T)\right]=\nabla^{\prime}\cdot(\eta^{\prime}\nabla^{\prime}T).$
(3)

Eqs. $\left(\ref{PJ-heat transfer}\right)$ and $\left(\ref{PJ-heat transfer
transformed}\right)$ have the consistent form, and thermal convection are
included in transformation thermotics.

Following this, we conceptualize and determine the velocity distribution
$\vec{v}^{\prime}(\vec{r},t)$ and the anisotropic thermal conductivity
$\eta^{\prime}$ of the liquid medium. Typically, to fully characterize the
state of fluids, we require knowledge of the velocity $\vec{v}$ and two
additional thermodynamic quantities, such as $\rho$ and pressure $p$. These
parameters are ascertained using Eq. $\left(\ref{PJ-heat transfer}\right)$ in
conjunction with the Navier-Stokes equations and the continuity equation PJ-
Landau

$(\vec{v}\cdot\nabla)\vec{v}=-\frac{1}{\rho}\nabla
p+\frac{\beta}{\rho}\nabla\cdot\nabla\vec{v},$ (4) $\nabla\cdot\vec{v}=0.$ (5)

Here, $\beta$ denotes the dynamic viscosity. For clarity, we take
$\vec{v}(\vec{r},t)=\vec{v}(\vec{r})$ and $\rho(\vec{r},t)\equiv\rho$.
However, Eq. $\left(\ref{PJ-conservation law}\right)$ retains its form under
coordinate transformation whereas Eq. $\left(\ref{navier stokes}\right)$
typically does not. But we can overlook nonlinear term
$(\vec{v}\cdot\nabla)\vec{v}$ when Reynolds number Re is small (akin to the
elastic equation in PJ-NJP2006 ). Experimentally, inducing anisotropy in
$\eta^{\prime}$ for fluids poses challenges, even though this has been
effectively achieved for heat conduction in solids. Encouragingly, recent
progress in velocity control, as highlighted in PJ-PRL2011 , spurs us to
simultaneously explore heat transfer and velocity management in porous media.

In fully-filled porous media, we give equations for steady flow as PJ-Bear1 ;
PJ-Bear2

$\rho_{f}C_{p,f}(\vec{v}\cdot\nabla T)=\nabla\cdot(\eta_{m}\nabla T),$ (6)
$\nabla p+\frac{\beta}{k}\vec{v}=0,$ (7) $\nabla\cdot\vec{v}=0,$ (8)

where $k$ denotes the permeability and $\eta_{m}$ is the effective thermal
conductivity of the porous media. Meanwhile, $\rho_{f}$ and $C_{p,f}$ are the
density and specific heat at constant pressure of fluid material,
respectively. By taking the volume average of solid and liquid components PJ-
Bear2 , the effective conductivity $\eta_{m}$ is given by

$\eta_{m}=(1-\phi)\eta_{s}+\phi\eta_{f}.$ (9)

where $\phi$ represents the porosity and $\eta_{f}$ and $\eta_{s}$ are the
thermal conductivity of fluid and solid material of porous media,
respectively. In Eq. $\left(\ref{PJ-heat transfer porous}\right),$ the local
thermal equilibrium of fluids and solid materials is assumed, indicating that
they possess the same temperature at the contact point. Meanwhile, we assume
$\nabla\cdot(\vec{v}T)=\vec{v}\cdot\nabla T$, given by Eq. $\left(\ref{PJ-
conservation law 2}\right).$ Eq. $\left(\ref{PJ-darcy's law}\right)$ denotes
the Darcy’s law, in case of small-enough Re and $k$. From
$\lambda=-\frac{k}{\beta}$ and
$\vec{v}^{\prime}=\textbf{J}\vec{v}/(\det\textbf{J})$, we easily rewrite Eq.
$\left(\ref{PJ-darcy's law}\right)$ under transformation
$\\{x_{i}\\}\rightarrow\\{y_{j}\\}$,

$v^{\prime}_{j}=\sum\limits_{i}J_{ji}v_{i}/(\det\textbf{J})=\sum\limits_{ik}J_{ji}\lambda_{ki}\frac{\partial
p}{\partial
x_{k}}/(\det\textbf{J})=\sum\limits_{ikl}J_{lk}\lambda_{ki}J^{\intercal}_{ij}\frac{\partial
p}{\partial y_{l}}/(\det\textbf{J}),$ (10)

which indicates the relationships:
$\vec{v}^{\prime}=\lambda^{\prime}\nabla^{\prime}p$ and
$\lambda^{\prime}=\frac{\textbf{J}\lambda\textbf{J}^{\intercal}}{\det\textbf{J}}.$

All the Eqs. $\left(\ref{PJ-heat transfer porous}\right)$, $\left(\ref{PJ-
darcy's law}\right)$ and $\left(\ref{PJ-conservation law 2}\right)$ keep
invariant form under any coordinate transformation, and we can obtain the
wanted temperature and velocity distribution without tuning properties of
fluid materials. That is to say, we only need to transform the permeability

$k^{\prime}=\frac{\textbf{J}k\textbf{J}^{\intercal}}{\det\textbf{J}},$ (11)

and the thermal conductivity

$\left\\{\begin{array}[]{ll}\eta^{\prime}_{m}=\dfrac{\textbf{J}\eta_{m}\textbf{J}^{\intercal}}{\det\textbf{J}},\\\
\eta_{f}^{\prime}=\eta_{f},\\\
\eta_{s}^{\prime}=\dfrac{\eta_{m}^{\prime}-\phi\eta_{f}}{1-\phi}.\end{array}\right.$
(12)

Using different spatial transformations, we can manipulate the heat flow as we
wish.

##  III Transient-state Transformation Thermo-hydrodynamics

In this section, we extend the transformation theory to encompass transient
thermal convection in porous media. The Darcy’s law can be generalized as
follows PJ-BurcharthCE95 ; PJ-ZhuTPM2014 ; PJ-ZhuTPM2016 ; PJ-DaiJAP18 :

$\tau\frac{\partial\vec{v}}{\partial t}+\vec{v}=-\dfrac{\beta}{\eta}\nabla p.$
(13)

Here, $\tau$ denotes the characteristic time measuring the velocity varying.
Additionally, $\beta$ signifies the permeability of porous medium, while
$\eta$ stands for the dynamic viscosity. In many scenarios, the relaxation
process within porous media is rapid, resulting in the term
$\tau\frac{\partial\vec{v}}{\partial t}$ being quite negligible PJ-ZhuTPM2016
. Consequently, we can omit this term and continue to operate within the
steady Darcy framework. Furthermore, the continuity equation can be modified
as PJ-Landau

$\frac{\partial(\phi\rho_{f})}{\partial t}+\nabla\cdot(\rho_{f}\vec{v})=0,$
(14)

where $\rho_{f}$ denotes the density of the fluid medium and $\phi$ stands for
the porosity. Then, the heat transfer of incompressible flow in fully-filled
porous media is given by PJ-Bear1 ; PJ-Bear2

$\frac{\partial(\rho C)_{m}T}{\partial
t}+\nabla\cdot(\rho_{f}C_{f}\vec{v}T)=\nabla\cdot(\kappa_{m}\nabla T),$ (15)

where $T$ is temperature, and $\rho_{s}$ is the density of solid in porous
media, respectively. Here, $C_{f}$ and $C_{s}$ are the specific heat of the
fluid and solid porous materials, respectively. The effective product of
density and specific heat of the whole porous material, governed by the
average-volume method PJ-Bear2 ,

$(\rho C)_{m}=(1-\phi)(\rho_{s}C_{s})+\phi(\rho_{f}C_{f}).$ (16)

Similarly, the effective thermal conductivity $\kappa_{m}$ are also the
summation of $\kappa_{f}$ (for fluids) and $\kappa_{s}$ (for solids),

$\kappa_{m}=(1-\phi)\kappa_{s}+\phi\kappa_{f}.$ (17)

Note that the Eq. (15) is the unsteady convection-diffusion equation whose
form-invariance under coordinate transformations are proved PJ-GuenneauAIP15 ;
PJ-JRCI2013 . Additionally, the substitution of Eq. (14) into Eq. (15) leads

$(\rho C)_{m}\frac{\partial T}{\partial t}+\rho_{f}C_{f}(\vec{v}\cdot\nabla
T)=\nabla\cdot(\kappa_{m}\nabla T).$ (18)

Like the steady-state scenarios, we can see all governing equitations satisfy
the transformation theory. PJ-DaiPRE18 . For both these steady and unsteady
situations, the transformation matrix for an isotropic virtual space is as
follows: $\frac{\textbf{J}\textbf{J}^{\intercal}}{\det\textbf{J}},$ where J
signifies the Jacobian matrix mapping from the transformed coordinate to its
original counterpart. It’s also essential to adjust the permeability and heat
conductivity as
$\beta^{\prime}=\dfrac{\textbf{J}\beta\textbf{J}^{\intercal}}{\det\textbf{J}}$
and
$\kappa_{m}^{\prime}=\dfrac{\textbf{J}\kappa_{m}\textbf{J}^{\intercal}}{\det\textbf{J}}$.
Distinctively, for the unsteady scenarios, it becomes necessary to modify both
the porosity and the product of density and specific heat. The transformation
is

$\left\\{\begin{array}[]{ll}\phi^{\prime}=\dfrac{\phi}{\det\textbf{J}}\\\
(\rho_{f}C_{f})^{\prime}=\rho_{f}C_{f}\\\
(\rho_{s}C_{s})^{\prime}=\dfrac{1-\phi}{\det\textbf{J}-\phi}\rho_{s}C_{s}\end{array}\right.$
(19)

Using this approach, we maintain the properties of fluid unchanged, focusing
solely on crafting the required solid metamaterial. It’s noteworthy that
$\rho_{f}$ is not a constant anymore if we further consider its correlation
with temperature variations over time and space. For clarity, we make the
assumption that

$\rho=\rho_{0}[1-\gamma(T-T_{0})],$ (20)

where $\gamma=(\frac{\partial\rho}{\partial T})_{p}/\rho$ represents the
density expansion ratio at a constant pressure. Without loss of generality, we
give the $\gamma>0.$ Therefore, we consider the transformed equations:

$\left\\{\begin{array}[]{ll}\vec{v}^{\prime}=-\dfrac{\beta^{\prime}}{\eta}\nabla
p\\\ \dfrac{\partial(\phi^{\prime}\rho_{f})}{\partial
t}+\nabla\cdot(\rho_{f}\vec{v}^{\prime})=0\\\ (\rho
C)^{\prime}_{m}\dfrac{\partial T}{\partial
t}+\rho_{f}C_{f}(\vec{v}^{\prime}\cdot\nabla
T)=\nabla\cdot(\kappa^{\prime}_{m}\nabla T)\end{array}\right.,$ (21)

where the transformed velocity $\vec{v}^{\prime}$ is
$\textbf{J}\vec{v}/\det\textbf{J}$ PJ-JRCI2013 ; PJ-GuenneauAIP15 ; PJ-
DaiPRE18  and $(\rho
C)^{\prime}_{m}=(1-\phi^{\prime})(\rho_{s}C_{s})^{\prime}+\phi^{\prime}(\rho_{f}C_{f}).$

##  IV Potential Applications

Actually, metamaterials designed by transformation thermotics typically
exhibit characteristics that are anisotropic, inhomogeneous, and at times even
singular. These traits present significant fabrication challenges. For
addressing these complexities in hybrid thermal systems, researchers turn to
effective medium theories and multilayered composite structures to achieve the
desired outcomes. Regrettably, a fitting theory for managing such hybrid
thermal systems has not been developed yet. As a result, there is a pressing
need to devise a theory that streamlines the intricate parameters introduced
by transformation thermotics.

To address this challenge, we draw inspiration from the concept of neutral
inclusion within porous materials. By tailoring two pivotal parameters—thermal
conductivity and permeability—we are able to achieve three distinct types of
thermal illusions: transparency, concentration, and cloaking. To elaborate,
thermal transparency involves creating a core-shell structure to preserve the
temperature, velocity, and heat flux distributions of the background
undisturbed. Notably, this approach obviates the need for anisotropy,
inhomogeneity, and singularity. Similarly, to attain thermal concentration or
cloaking, we fashion an anisotropic shell, eliminating the necessity for
inhomogeneous and singular parameters. As these three functions—transparency,
concentration, and cloaking—maintain the background’s temperature, velocity,
and heat flux distributions undisturbed, we conveniently term them
collectively as thermal illusion.

We assume a steady-state thermal convection-diffusion process in porous
materials with incompressible fluids and neglect the viscous dissipation term.
Therefore, the governing equation is expressed as

$\displaystyle\rho_{f}C_{p,f}(\vec{v}\cdot\nabla
T)=\nabla\cdot(\overset{\leftrightarrow}{\kappa}\cdot\nabla T),$ (22)

where $\rho_{f}$, $C_{p,f}$, and $\vec{v}$ are the density, heat capacity, and
the velocity of the fluid at constant pressure, respectively, and $T$ denotes
the temperature when the porous material reach equilibrium. Moreover,
$\overset{\leftrightarrow}{\kappa}$, representing the average thermal
conductivity tensor of the solid and the fluid material, is defined as
$\overset{\leftrightarrow}{\kappa}=(1-\phi)\overset{\leftrightarrow}{\kappa}_{s}+\phi\overset{\leftrightarrow}{\kappa}_{f}$,
where $\phi$ stands for the porosity of the media.
$\overset{\leftrightarrow}{\kappa}_{s}$ and
$\overset{\leftrightarrow}{\kappa}_{f}$ are the thermal conductivity tensors
of the solid and the fluid material, respectively. In cases where the fluid
exhibits laminar flow at a minimal velocity, the velocity $\vec{v}$ is
described by Darcy’s law,

$\vec{v}=-\left(\overset{\leftrightarrow}{\sigma}/\eta\right)\cdot\nabla p,$
(23)

where $\overset{\leftrightarrow}{\sigma}$ and $\eta$ is the permeability
tensor and dynamic viscosity, respectively. $p$ stands for pressure. In such
case, both Re (Reynolds number) and $\overset{\leftrightarrow}{\sigma}$ are
small enough. The conductive flux $\vec{j}$ is governed by Fourier’s law,

$\vec{j}=-\overset{\leftrightarrow}{\kappa}\cdot\nabla T.$ (24)

For clarity, we consider the steady state

$\displaystyle\nabla\cdot\vec{v}=0,$ (25) $\displaystyle\nabla\cdot\vec{j}=0.$
(26)

We also consider one type of fluid with constant dynamic viscosity $\eta$.
Then, Eqs. (25) and (26) are expressed as

$\displaystyle\nabla\cdot(-\overset{\leftrightarrow}{\sigma}\cdot\nabla p)=0,$
(27) $\displaystyle\nabla\cdot(-\overset{\leftrightarrow}{\kappa}\cdot\nabla
T)=0.$ (28)

Finally, Eqs. (27) and (28) share a comparable mathematical structure.
Therefore, the effective medium theory is capable of addressing both thermal
conductivity and permeability. Using $\tau$, we can harmonize the
representation of $\kappa$ and $\sigma$.

Figure 1: Thermal illusion in porous materials. The scale of the system is
$10^{-5}$ m. a Schematic. The background velocity is along the $x$ direction,
as described by the black flow lines. Region I $(r<r_{1})$ is composed of
isotropic porous media, region II $(r_{1}<r<r_{2})$ is composed of isotropic
media for thermal transparency, and anisotropic media for thermal concentrator
or cloak, and region III $(r>r_{2})$ is composed of isotropic background
porous media. For thermal illusion in porous media, the black lines in region
III with the core-shell structure should be undistorted. b Temperature profile
of a pure background (reference). c Temperature profile of a thermal
transparency. d Temperature profile of a thermal concentrator. e Temperature
profile of a thermal cloak. White lines are the isotherms. Adapted from Ref.
PJ-YangESEE19

We aspire to eliminate the need for anisotropic, inhomogeneous, and singular
parameters. To this end, we turn to the concept of neutral inclusion. This
idea provides a methodology to determine the effective thermal conductivity of
a core-shell configuration. Subsequently, it becomes essential to compute the
effective permeability for the same structure. As depicted in Fig. 1, we
designate the core to be isotropic with parameter $\tau_{1}$, the metashell to
be anisotropic with parameter $\overset{\leftrightarrow}{\tau}_{2}={\rm
diag}(\tau_{rr},\tau_{\theta\theta})$ (the shell becomes isotropic when
$\tau_{rr}=\tau_{\theta\theta}$ ), and the background to be isotropic with
parameter $\tau_{3}$. Consequently, the core-shell structure’s effective
parameter $\tau_{e}$ can be deduced as follows:

$\displaystyle\tau_{e}=c\tau_{rr}\frac{\tau_{1}+c\tau_{rr}+(\tau_{1}-c\tau_{rr})f^{c}}{\tau_{1}+c\tau_{rr}-(\tau_{1}-c\tau_{rr})f^{c}},$
(29)

where $c=\sqrt{\tau_{\theta\theta}/\tau_{rr}}$ denotes the anisotropy of the
shell, and $f=(r_{1}/r_{2})^{2}$ represents the core fraction. To maintain the
heat flux and velocity distributions in the background (region III) as though
the core-shell structure is absent at the center, we define
$\tau_{e}=\tau_{3}$.

Next, we perform finite-element simulations to validate the theory. As shown
in schematic of Fig. 1, we take the pressure source as $\Delta p=400$ Pa and
the heat source $\Delta T=40$ K. The liquid in porous material is set as water
with $\rho_{f}=10^{3}$ kg/m3, $C_{p,f}=4.2\times 10^{3}$ J$\cdot$kg-1K-1, the
dynamic viscosity $\eta=10^{-3}$ Pa$\cdot$s, and $\kappa_{f}=0.6$ Wm-1K-1. The
porosity is $\phi=0.9$. The size parameters are $r_{1}=2\times 10^{-5}$ m and
$r_{2}=3.2\times 10^{-5}$ m. The average thermal conductivity tensors are set
to be $\kappa_{1}=6$ Wm-1K-1, $\overset{\leftrightarrow}{\kappa}_{2}={\rm
diag}(4,4)$ Wm-1K-1, and $\kappa_{3}=\kappa_{e}$ given by Eq. (29). The
thermal conductivity tensor of the solid are calculated as
$\overset{\leftrightarrow}{\kappa}_{s}=(\overset{\leftrightarrow}{\kappa}-\phi\kappa_{f})/(1-\phi)$.
The permeability tensors are set to be $\sigma_{1}=5\times 10^{-12}$ m2,
$\overset{\leftrightarrow}{\sigma}_{2}={\rm diag}(2,2)\times 10^{-12}$ m2 (the
magnitude $10^{-12}$ is common in nature), and $\sigma_{3}=\sigma_{e}$ given
by Eq. (29). In all these cases, we calculate Reynolds numbers ${\rm
Re}=r_{2}\rho_{f}v/\eta<1$ (the maximum value is 0.64) and $\sigma\ll
r_{2}^{2}$, ensuring the applicability of Darcy’s law. The simulation results
are shown in Figs. 1. Figure 1b displays a simulation of a pure background
without the core-shell structure, serving as a reference. Based on the above
core-shell parameters, we then produce a thermal transparency pattern; see
Fig. 1c. For realizing a thermal cloak (or concentrator) pattern, we set
$\tau_{rr}\ll\tau_{\theta\theta}$ (or $\tau_{rr}>\tau_{\theta\theta}$).
Temperature profiles are shown in Fig. 1d and Fig. 1e. Finally, different
thermal patterns are created including thermal transparency, thermal cloak and
thermal concentrator. The performance of thermal illusion are not affected as
long as the permeability and thermal conductivity satisfy Eq. (29).

##  V Laboratory Experiment of Steady-state Transformation Thermo-
hydrodynamics

Figure 2: Liquid-solid hybrid thermal metamaterial. a Illustration of the
metadevice based on the liquid-solid hybrid thermal metamaterial. b Photos of
the top and the bottom of the sample. Scale bar is 6 cm. c The switch between
thermal cloaking and thermal concentration corresponds to a topological switch
in virtual space. d The heat flux amplification factor $\beta$ can be tuned
continuously by the external hydraulic pressure. Meanwhile, the function of
the metadevice is switched. e Measured temperature profile of the thermal
metadevice at $\Delta P$ = 0. White triangles denote the positions with the
temperature of 20.8 ${}^{\circ}C$. f Observed streamlines of the thermal
metadevice at $\Delta P$ $\neq$ 0\. g Measured temperature profile of the
thermal metadevice at $\Delta P$ $\neq$ 0\. Horizontal white triangles denote
the positions with the temperature (from up to down) of 39.0 ${}^{\circ}C$,
39.0 ${}^{\circ}C$, and 38.8 ${}^{\circ}C$, respectively. Adapted from Ref.
PJ-JinPNAS23

In crafting the hybrid thermal metamaterial, we employ fine-designed porous
structures, enabling both thermal convection and conduction to coexist within
the same space, as illustrated in Fig. 2a. This design process is bifurcated
into two stages. Initially, by sculpting the basic unit, we engineer a porous
substance that allows localized, independent modulation of both thermal
conduction and convection attributes. Subsequently, leveraging the advanced
principles of transformation thermotics, we shape the spatial characteristics
of these thermal properties to realize the intended functionalities of the
thermal metadevice.

The local manipulation of both conductive and convective thermal properties is
achieved through the design of basic units. We have two types of units. The
type-I unit comprises a cuboid featuring a hemispherical region filled with
water (see lower-right inset of Fig. 2a). In contrast, the type-II unit is a
cuboid possessing cylindrical five air holes. The effective thermal
conductivity of each unit is given by
$\bm{\kappa}=\left(1-\phi_{l}-\phi_{a}\right)\bm{\kappa}_{\rm
s}+\phi_{l}\bm{\kappa}_{\rm l}+\phi_{a}\bm{\kappa}_{\rm a}$ where
$\bm{\kappa}_{\rm s}$, $\bm{\kappa}_{\rm l}$, and $\bm{\kappa}_{\rm a}$ are
the thermal conductivity of the solid, liquid, and air, respectively.
$\phi_{l}$ and $\phi_{a}$ represent the filling fraction of the liquid and air
region, respectively. Within each unit, the thermal conductivity can be
adjusted based on the selected solid material and the filling fractions
$\phi_{l}$ and $\phi_{a}$. Concurrently, the permeability $\bm{\sigma}$ can be
modulated based on the geometry of the liquid or air region. For instance, in
the type-II units air holes are employed to deflect the liquid flow. The
orientation of such units can be strategically adjusted to modify the
permeability $\bm{\sigma}$.

We craft a metashell design wherein the thermal conductivity
$\bm{\kappa}^{\prime}$ distribution is tailored for thermal cloaking, guided
by our choice of the transformation $\bm{\Xi}$. This transformation correlates
to a virtual space with a hole at the center. The hole in the virtual space is
exactly the origin of the thermal cloaking effect: The heat flows cannot touch
any object in the hole in the virtual space, while in real space, an object in
the core region remains unaffected by the heat flows. Conversely, the liquid
permeability distribution $\bm{\sigma}^{\prime}$ arises from the
transformation of the thermal convection $\bm{\Lambda}$, crafted for thermal
concentration. This transformation maps to a virtual space with no hole. From
the geometric point of view, the virtual space with a hole is topologically
distinct from the virtual spaces with no hole. Consequently, with increasing
hydraulic pressure difference $\Delta P$ = $P_{\rm h}$ \- $P_{\rm l}$ ($P_{\rm
h}$ and $P_{\rm l}$ are the hydraulic pressure at the hot and cold sides of
the metadevice, respectively), thermal convection becomes dominant and the
device function switches from thermal cloaking to thermal concentration.
Meanwhile, the virtual space undergoes topology switch (see Fig. 2c). In
particular, the nontrivial topology within the virtual space for thermal
cloaking indicates that there are some properties robust to external
conditions. These properties are the heat current in the core region. In the
thermal cloaking regime, such a heat current is irrelevant with external
temperature distributions. In contrast, for thermal concentration, the heat
current in the core region is highly sensitive to external temperature
regions. The switch between these two functions reflect the topology change in
the virtual space. To provide a quantitative assessment of our metadevice’s
function, we introduce the heat flux amplification factor $\beta$. This is
determined by the averaged amplitude of the total heat flux in the core region
($\Omega_{1}$) over the same quantity when the system is changed to the
background (henceforth denoted as “the reference”).

We proceed to showcase the transition between thermal cloaking and thermal
concentration through regulated hydrodynamics. Under boundary condition I,
where $\Delta P$ = 0, we evaluate the temperature profile within the
metadevice. As depicted in Fig. 2e, the temperature distribution captured by
the infrared camera displays a perfect pattern of thermal cloaking. Notably,
the core region has a consistent temperature distribution around
$20.8^{\circ}C$. This suggests no conductive heat flow in the core region.
Additionally, the temperature profile in the background region remains mostly
undisturbed. Therefore, it can be interpreted as $\beta<1$, signifying that
the metadevice is operating in the cloaking mode.

Under boundary condition II, we target for the realization of the thermal
concentration. We ensure that thermal convection takes precedence in these
scenarios. We experimentally showcase the performance of thermal
concentration, examining it through the lens of fluid dynamics and temperature
profiling. To visualize the fluid flow, we perforate six holes beneath a
colorant container, a concoction of alkanes and toner. Once the system
stabilizes into a nonequilibrium steady-state, the colorant is methodically
dripped from the metadevice’s left boundary through these holes, ensuring
simultaneous and equidistant distribution. Fig. 2f showcases the six
streamlines. The central four streamlines converge into the core area, and all
the streamlines outside the region $\Omega_{2}$ are only marginally distorted.
This pattern underscores that the core zone experiences a more substantial
flow (or a heightened fluid velocity) compared to the backdrop. It is worth
noting that within the $\Omega_{2}$ region, colorant distribution differs
between the upper and lower sections. This discrepancy arises from the
asymmetrical layout of the type-II basic units, further compounded by the
unequal positioning of the six holes in relation to these sections. Yet, the
fluid flow’s concentration remains distinctly visible in Fig. 2g.
Subsequently, we gauge the temperature profile in the metadevice under
identical conditions. The measured temperature profile (see Fig. 2g) exhibits
several features. First, the overall temperature of the metadevice is higher
than in the cloaking case. Moreover, the temperature gradient is pushed to the
right side of the metadevice. There are visible correlations between the
temperature profile and liquid flow profile, indicating that the thermal
transport is now dominated by the convective heat flow carried by the water.
The convective heat flow in the core region is larger than that in the
background region because of $v_{\Omega_{1}}>v_{\Omega_{3}}$ with
$T_{\Omega_{1}}$ $\approx$ $T_{\Omega_{3}}$ (see the white triangles in Fig.
2g). In this phase, $\beta>1$. Consequently, the metadevice is transitioned
into thermal concentration by increasing the external hydraulic pressure.

##  VI Discussion and Conclusion

In conclusion, we delve into the transformation thermo-hydrodynamics theory
and the corresponding experimental methodologies that probe heat transfer in
porous materials. These specially engineered porous materials exhibit
remarkable adaptability in heat management, all while ensuring the temperature
field in the backdrop remains undisturbed — a feat beyond the capabilities of
traditional thermal metamaterials. Characterized by their unique liquid-solid
amalgamation, these hybrid thermal metamaterials hold immense promise for a
plethora of applications. These include thermal illusions and camouflage,
enhanced cooling and heat regulation in electronic gadgets, sustainable
infrastructure, and sophisticated heat modulation in intelligent materials and
machinery. Delving deeper into these hybrid metamaterials might just unveil
groundbreaking insights into the intricacies of complex systems. (PJ-
ZhouPA2009, ; PJ-ZhangCPL2009, ; PJ-ZhaoJAP2009, ; PJ-TanJPCB2009, ; PJ-
TianJAP2009, ; PJ-GuCTP2009, ; PJ-WangOL2009, ; PJ-GaoJAP2009, ; PJ-
TanJPCB2009-1, ; PJ-LiuEPJAP2009, ; PJ-HanPLA2009, ; PJ-JianJPCC2009, ; PJ-
FanJPCC2009, ; PJ-WangPNAS09, ; PJ-WuEPJAP09, ; PJ-HuangSSC00, ; PJ-
HuangCTP01-2, ; PJ-HuangCTP01-1, ; PJ-PanPB01, ; PJ-HuangPRE01, ; PJ-
HuangJPCM02, ; PJ-HuangPRE02, ; PJ-HuangCTP02, ; PJ-HuangPLA02, ; PJ-
HuangCTP03, ; PJ-HuangPRE03-2, ; PJ-GaoPRE03, ; PJ-HuangJAP03, ; PJ-
HuangPRE03-1, ; PJ-GaoEPJB03, ; PJ-DongJAP04, ; PJ-KoJPCM04, ; PJ-GaoPRB04, ;
PJ-HuangPRE04g, ; PJ-LiuPLA04, ; PJ-DongJAP04-1, ; PJ-HuangCPL04, ; PJ-
HuangPRE04f, ; PJ-KoEPJE04, ; PJ-HuangAPL04, ; PJ-HuangCP04, ; PJ-HuangPRE04e,
; PJ-HuangEL04, ; PJ-HuangJPCB04, ; PJ-HuangJCP04, ; PJ-HuangPRE04d, ; PJ-
HuangPRE04c, ; PJ-HuangJPCM04, ; PJ-HuangPLA04, ), including nonlinear systems
(PJ-ZhangCPB10, ; PJ-LiuCTP10, ; PJ-FanCTP10, ; PJ-GaoPRL10, ; PJ-WangOL10, ;
PJ-BaoJPCM10, ; PJ-GaoPP10, ; PJ-TanSM10, ; PJ-MengJPCB11, ; PJ-SuFOP11, ; PJ-
LiuIJMPB11, ; PJ-ZhaoPNAS11, ; PJ-FanJPDAP11, ; PJ-WangJPCB11, ; PJ-ZhaoFOP12,
; PJ-SongARCS12, ; PJ-Li2012sm, ; PJ-Wang2012cpb, ; PJ-li2012epl, ; PJ-
huang2013cp, ; PJ-wei2013plos, ; PJ-liang2013pre, ; PJ-Fan2013fop, ; PJ-
Meng2013mp, ; PJ-Wang2013ACM, ; PJ-Liu2013ASP, ; PJ-wei2013AnAM, ; PJ-
chen2013prl, ; PJ-Fan2013cpb, ; PJ-Liang2013fopw, ; PJ-Meng2013pre, ; PJ-
Li2013EPJP, ; PJ-QiuEPJB13, ; PJ-QiuPLoS13, ; PJ-QiuFP14, ; PJ-QiuPLA14, ; PJ-
QiuPLoS14, ; PJ-QiuCPB14, ; PJ-QiuJSM14, ; PJ-QiuCTP14, ; PJ-QiuPLoS14-1, ;
PJ-QiuTEPJ-AP14, ; PJ-QiuCTP15, ; PJ-QiuSpringer15, ; PJ-QiuJP.Phys.Chem.B15,
; PJ-QiuPR15, ; PJ-QiuEur.Phys.J.Appl.Phys.15, ; PJ-QiuPLA15, ; PJ-Tan19, ;
PJ-Tan18, ; PJ-Tan17, ; PJ-Tan12, ; PJ-Tan10, ; PJ-Tan9, ; PJ-Tan8, ), soft
matter systems (PJ-HuangPRE04b, ; PJ-HuangPRE04a, ; PJ-HuangJAP05, ; PJ-
XiaoPRB05, ; PJ-HuangAPL05b, ; PJ-HuangOL05, ; PJ-HuangJPCB05, ; PJ-LiuCTP05,
; PJ-HuangPLA05, ; PJ-HuangPRE05b, ; PJ-HuangPRE05a, ; PJ-HuangJOSAB05, ; PJ-
HuangAPL05a, ; PJ-DongEPJB05, ; PJ-HuangJMMM05, ; PJ-HuangJAP06, ; PJ-
TianPRE06, ; PJ-WangCPL06, ; PJ-ShenCPL06, ; PJ-WangJPCB06, ; PJ-CaoJPCB06, ;
PJ-HuangPR06a, ; PJ-XuPLA06, ; PJ-TianCPL06, ; PJ-FanAPL06, ; PJ-XueCPL06, ;
PJ-FanJPCB06, ; PJ-TianPRE07, ; PJ-YangJAP07, ; PJ-TianEPL07, ; PJ-FangCPL07,
; PJ-ChenJPA07, ; PJ-GaoJPCC07, ; PJ-WangAPL07, ; PJ-HuangNY07, ; PJ-ZhuJAP08,
; PJ-YePA081, ; PJ-HuangJRCC08, ; PJ-XuJMR08, ; PJ-JianJRCB08, ; PJ-
ZhangAPL08, ; PJ-GaoAPL2008, ; PJ-FanJAP2008, ; PJ-XiaoJPCB2008, ; PJ-2008, ;
PJ-WangOL2008, ; PJ-Tan3, ; PJ-Tan2, ), and statistical physics PJ-Tan14 ; PJ-
Tan13 ; PJ-Tan7 ; PJ-Tan6 ; PJ-Tan5 .

Let’s delve into the topological transition inherent to the hybrid
metamaterial. From a physics standpoint, the transformation theory serves as a
conduit, bridging geometric transformations with their parametric
counterparts. As a result, the topological attributes of a geometric
transformation find their mirrored representation in material parameters. This
allows us to explore topology via (effective) thermal conductivity. For
instance, when considering a core-shell structure designed for thermal
cloaking, its effective thermal conductivity can be perceived as a topological
invariant — it remains unaltered by the thermal conductivity of the core. This
implies that a geometric transformation with a distinct topological nature can
lead to singular (either zero or infinite) thermal conductivity, effectively
segmenting a region from its surroundings. Hence, transformation thermotics
becomes instrumental in drawing connections between geometric and parametric
topologies. Broadly speaking, situations with zero or infinite thermal
conductivity bear topological significance. This is analogous to the unique
nature of zero-index metamaterials: they facilitate seamless transport without
reflections in real space and correspond to materials showcasing Dirac
dispersions in wave-vector space.

In wrapping up, we foresee a transformative application of our metadevice in
managing localized temperature variations within biological cells or tissues
PJ-TY , all while ensuring the broader environment of the human body remains
unperturbed. The extracellular fluid within organisms, a composite of liquid
and solid-like elements (as illustrated in Fig. 2a), can be conceptualized as
a porous medium. By designing porous metamaterials tailored to a specific
scale, we can effectively model this environment. In a consistent external
setting, discerning the direction of localized heat flow is feasible. Given
our structure’s axisymmetric design, as guided by the transformation theory,
there’s no external deviation. This implies that our model can be oriented
along the direction of this localized heat flow. The adiabatic boundaries
positioned at the top and bottom mirror open boundaries since there’s a null
heat flow vertically. This makes our model an apt representation of real
conditions. By manipulating external hydraulic pressures, our uniquely
designed metamaterials can regulate localized temperature variations in living
cells or tissues. These hot/cold spots typically correspond to temporary
temperature deviations. Harnessing hydraulic pressure control, these
deviations can be mitigated swiftly. For such localized temperature variances,
elevating fluid velocity and heat flow can expedite the shift towards a more
stable temperature environment. Moreover, within a biological context, this
supplemental flux aids in balancing chemical concentrations, such as ATPs and
CO2, further bolstering the restoration of biological functions. Furthermore,
embedding these micro-scale metamaterials within the human body for
therapeutic purposes is a tangible possibility. Our design’s capability to
operate without disrupting the background thermal or fluidic environment is of
paramount significance, especially given its implications for human health.

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