#### Coupled transport

In a nanofluid system, normally, there are two or more transport processes that occur simultaneously. Examples are the heat conduction in dispersed phase, heat conduction in continuous phase, mass transport, and chemical reactions either among the nanoparticles or between the nanoparticles and the base fluid. These processes may couple (interfere) and cause new induced effects of flows occurring without or against its primary thermodynamic driving force, which may be a gradient of temperature, or chemical potential, or reaction affinity. Two classical examples of coupled transport are the Soret effect (also known as thermodiffusion or thermophoresis) in which directed motion of particles or macromolecules is driven by thermal gradient and the Dufour effect that is an induced heat flow caused by the concentration gradient.

While the coupled transport is well recognized to be very important in thermodynamics [

68], it has not been well appreciated yet in the nanofluid society. The first attempts of examining the effect of coupled transport on nanofluid heat conduction have been recently made in some studies [

1,

2,

9,

18], which are briefly outlined here. With the coupling between the heat conduction in the fluid and particle phases denoted by

*β* and

*σ-* phases, respectively, the temperature

*T* obeys the following energy equations [

1,

2]

${\gamma}_{\beta}\frac{\partial {T}_{\beta}}{\partial t}={k}_{\beta \beta}\Delta {T}_{\beta}+{k}_{\beta \sigma}\Delta {T}_{\sigma}+h{a}_{\upsilon}\left({T}_{\sigma}-{T}_{\beta}\right)$

(8)

and

${\gamma}_{\sigma}\frac{\partial {T}_{\sigma}}{\partial t}={k}_{\sigma \sigma}\Delta {T}_{\sigma}+{k}_{\sigma \beta}\Delta {T}_{\beta}-h{a}_{\upsilon}\left({T}_{\sigma}-{T}_{\beta}\right)$

(9)

where *T* is the temperature; subscripts *β* and *σ* refer to the *β* and *σ-* phases, respectively. *γ*_{
β
}= (1 - *φ*)(*ρc*)*β* and *γ*_{
σ
}= *φ*(*ρc*)_{
σ
}are the effective thermal capacities of *β* and *σ-* phases, respectively, with *ρ* and *c* as the density and the specific heat. *φ* is the volume fraction of the *σ-* phase. *h* and *a*_{
υ
}come from modeling of the interfacial flux and are the film heat transfer coefficient and the interfacial area per unit volume, respectively. *k*_{
ββ
}and *k*_{
σσ
}are the effective thermal conductivities of the *β* and *σ-* phases, respectively; *k*_{
βσ
}and *k*_{
σβ
}are the coupling (cross) effective thermal conductivities between the two phases.

Rewriting Equations (

8) and (

9) in their operator form, we obtain

$\left[\begin{array}{cc}{\gamma}_{\beta}\frac{\partial}{\partial t}-{k}_{\beta \beta}\Delta +h& -{k}_{\beta \sigma}\Delta -h{a}_{\upsilon}\\ -{k}_{\sigma \beta}\Delta -h{a}_{\upsilon}& {\gamma}_{\sigma}\frac{\partial}{\partial t}-{k}_{\sigma \sigma}\Delta +h{a}_{\upsilon}\end{array}\right]\left[\begin{array}{c}{T}_{\beta}\\ {T}_{\sigma}\end{array}\right]=0$

(10)

An uncoupled form can then be obtained by evaluating the operator determinant such that

$\left[\left({\gamma}_{\beta}\frac{\partial}{\partial t}-{k}_{\beta \beta}\Delta +h{a}_{\upsilon}\right)\left({\gamma}_{\sigma}\frac{\partial}{\partial t}-{k}_{\sigma \sigma}\Delta +h{a}_{\upsilon}\right)-{\left({k}_{\beta \sigma}\Delta -h{a}_{\upsilon}\right)}^{2}\right]{\u3008{T}_{i}\u3009}^{i}=0$

(11)

where the index

*i* can take

*β* or

*σ*. Its explicit form reads, after dividing by

*ha*_{υ}(

*γ*_{
β
}+

*γ*_{
σ
})

$\frac{\partial {T}_{i}}{\partial t}+{\tau}_{q}\frac{{\partial}^{2}{T}_{i}}{\partial {t}^{2}}=\alpha \Delta {T}_{i}+\alpha {\tau}_{T}\frac{\partial}{\partial t}\left(\Delta {T}_{i}\right)+\frac{\alpha}{k}\left[F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}\right]$

(12)

where

$\begin{array}{l}{\tau}_{q}=\frac{{\gamma}_{\beta}{\gamma}_{\sigma}}{h{a}_{\upsilon}({\gamma}_{\beta}+{\gamma}_{\sigma})},\phantom{\rule{0.5em}{0ex}}{\tau}_{T}=\frac{{\gamma}_{\beta}{k}_{\sigma \sigma}+{\gamma}_{\sigma}{k}_{\beta \beta}}{h{a}_{\upsilon}({k}_{\beta \beta}+{k}_{\sigma \sigma}+{k}_{\beta \sigma}+{k}_{\sigma \beta})},\\ k={k}_{\beta \beta}+{k}_{\sigma \sigma}+{k}_{\beta \sigma}+{k}_{\sigma \beta},\phantom{\rule{0.5em}{0ex}}\alpha =\frac{{k}_{\beta \beta}+{k}_{\sigma \sigma}+{k}_{\beta \sigma}+{k}_{\sigma \beta}}{{\gamma}_{\beta}+{\gamma}_{\sigma}},\\ F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}=\frac{{k}_{\beta \sigma}{k}_{\sigma \beta}-{k}_{\beta \beta}{k}_{\sigma \sigma}}{h{a}_{\upsilon}}{\Delta}^{2}{T}_{i}.\end{array}$

(13)

Equation (12) is not a classical heat-conduction equation, but can be regarded as a dual-phase-lagging (DPL) heat-conduction equation with ((*k*_{
βσ
}*k*_{
σβ
}- *k*_{
ββ
}*k*_{
σσ
})/(*ha*_{
υ
}))Δ^{2}*T*_{
i
}as the DPL source-related term $F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}$ and with *τ*_{
q
}and *τ*_{
T
}as the phase lags of the heat flux and the temperature gradient, respectively [2, 18, 69]. Here, *F*(r, *t*) is the volumetric heat source. *k*, *ρc*, and *α* are the effective thermal conductivity, capacity and diffusivity of nanofluids, respectively.

The computations of

*k*_{
ββ
},

*k*_{
σσ
},

*k*_{
βσ
}, and

*k*_{
σβ
}are available in [

27,

28] for some typical nanofluids. The coupled-transport contribution to the nanofluid thermal conductivity, the term (

*k*_{
βσ
}+

*k*_{
σβ
}), can be as high as 10% of the of the overall thermal conductivity [

27,

28]. The more striking effect of the coupled transport on nanofluid heat conduction can be found by considering

$\frac{{\tau}_{T}}{{\tau}_{q}}=1+\frac{{\gamma}_{\beta}^{2}{k}_{\sigma \sigma}+{\gamma}_{\sigma}^{2}{k}_{\beta \beta}-2{\gamma}_{\beta}{\gamma}_{\sigma}{k}_{\beta \sigma}}{{\gamma}_{\beta}{\gamma}_{\sigma}({k}_{\beta \beta}+{k}_{\sigma \sigma}+{k}_{\beta \sigma}+{k}_{\sigma \beta})},$

(14)

which is smaller than 1 when

${\gamma}_{\beta}^{2}{k}_{\sigma \sigma}+{\gamma}_{\sigma}^{2}{k}_{\beta \beta}-2{\gamma}_{\beta}{\gamma}_{\sigma}{k}_{\beta \sigma}={\left({\gamma}_{\beta}\sqrt{{k}_{\sigma \sigma}}-{\gamma}_{\sigma}\sqrt{{k}_{\beta \beta}}\right)}^{2}+2{\gamma}_{\beta}{\gamma}_{\sigma}(\sqrt{{k}_{\beta \beta}{k}_{\sigma \sigma}}-{k}_{\beta \sigma})<0.$

(15)

Therefore, by the condition for the existence of thermal waves that requires *τ*_{
T
}/*τ*_{
q
}*<* 1 [18, 70], thermal waves may be present in nanofluid heat conduction.

Note also that, for heat conduction in nanofluids, there is a time-dependent source term *F*(r, *t*) in the DPL heat conduction (Equations (12) and (13)). Therefore, the resonance can also occur. When *k*_{
βσ
}= *k*_{
σβ
}= 0 so that *τ*_{
T
}/*τ*_{
q
}is always larger than 1, thermal waves and resonance would not appear. Therefore, the coupled transport could change the nature of heat conduction in nanofluids from a diffusion process to a wave process, thus having a significant effect on nanofluid heat conduction.

Therefore, the cross coupling between the heat conduction in the fluid and particle manifests itself as thermal waves at the macroscale. Depending on factors such as material properties of nanoparticles and base fluids, nanoparticles' geometrical structure and their distribution in the base fluids, and interfacial properties and dynamic processes on particle-fluid interfaces, the cross-coupling-induced thermal waves may either enhance or counteract with the molecular-dynamics-driven heat diffusion. Consequently, the heat conduction may be enhanced or weakened by the presence of nanoparticles. This explains the thermal conductivity data that fall outside the H-S bounds (Figures 1 and 2).

If the coupled transport between heat conduction and particle diffusion is considered, then the temperature

*T* and particle volume fraction

*φ* satisfy the following equations of energy and mass conservation:

${\gamma}_{\beta}\frac{\partial {T}_{\beta}}{\partial t}={k}_{\beta \beta}\Delta {T}_{\beta}+{k}_{\beta \sigma}\Delta {T}_{\sigma}+{\rho}_{\sigma}{k}_{\beta \text{m}}\Delta \phi +h{a}_{\upsilon}\left({T}_{\sigma}-{T}_{\beta}\right),$

(16)

${\gamma}_{\sigma}\frac{\partial {T}_{\sigma}}{\partial t}={k}_{\sigma \sigma}\Delta {T}_{\sigma}+{k}_{\sigma \beta}\Delta {T}_{\beta}+{\rho}_{\sigma}{k}_{\sigma \text{m}}\Delta \phi -h{a}_{\upsilon}\left({T}_{\sigma}-{T}_{\beta}\right),$

(17)

and

${\rho}_{\sigma}\frac{\partial \phi}{\partial t}={\rho}_{\sigma}{D}_{\sigma \sigma}\Delta \phi +{D}_{\text{m}\beta}\Delta {T}_{\beta}+{D}_{m\sigma}\Delta {T}_{\sigma}+{D}_{\text{mT}}\left({T}_{\beta}-{T}_{\sigma}\right),$

(18)

where subscripts m and T stand for mass transport and thermal transport, respectively.

*D*_{
σσ
}is the effective diffusion coefficient for nanoparticles.

*k*_{β m},

*k*_{σ m},

*D*_{mβ},

*D*_{mσ}, and

*D*_{mT} are five transport coefficients for coupled heat and mass transport. By following a similar procedure as that of developing Equation (

12), an uncoupled form with

*u* (

*T*_{
β
},

*T*_{
σ
}, or

*φ*) as the sole unknown variable is obtained,

$\frac{\partial \left(\Delta u\right)}{\partial t}+{\tau}_{q}\frac{{\partial}^{2}\left(\Delta u\right)}{\partial {t}^{2}}=\alpha \Delta \left(\Delta u\right)+\alpha {\tau}_{T}\frac{\partial}{\partial t}\left(\Delta \left(\Delta u\right)\right)+\frac{\alpha}{k}\left[F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}\right]$

(19)

where

${\tau}_{q}=\frac{{\gamma}_{\beta}{k}_{\sigma \sigma}+{\gamma}_{\sigma}{k}_{\beta \beta}+{\gamma}_{\beta}{\gamma}_{\sigma}{D}_{\sigma \sigma}}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)},$

(20)

$\begin{array}{l}k=\frac{{\gamma}_{\beta}+{\gamma}_{\sigma}}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}\\ \times \left\{{k}_{\beta \text{m}}\left[{D}_{\text{mT}}\left({k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)-h{a}_{\upsilon}\left({D}_{\text{m}\beta}+{D}_{\text{mT}}\right)\right]-{k}_{\sigma \text{m}}\left[{D}_{\text{mT}}\left({k}_{\beta \beta}+{k}_{\beta \sigma}\right)+h{a}_{\upsilon}\left({D}_{\text{m}\sigma}+{D}_{\text{m}\beta}\right)\right]\right\}\\ \times \frac{h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}\end{array}$

(21)

$\alpha =\frac{k}{{\gamma}_{\beta}+{\gamma}_{\sigma}}$

(22)

$\begin{array}{l}\frac{1}{{\tau}_{T}}=\frac{{k}_{\beta \text{m}}\left[{D}_{\text{mT}}\left({k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)-h{a}_{\upsilon}\left({D}_{\text{m}\beta}+{D}_{\text{mT}}\right)\right]-{k}_{\sigma \text{m}}\left[{D}_{\text{mT}}\left({k}_{\beta \beta}+{k}_{\beta \sigma}\right)+h{a}_{\upsilon}\left({D}_{\text{m}\sigma}+{D}_{\text{m}\beta}\right)\right]}{{D}_{\sigma \sigma}\left({\gamma}_{\beta}{k}_{\sigma \sigma}-{\gamma}_{\sigma}{k}_{\beta \beta}\right)+{k}_{\beta \beta}{k}_{\sigma \sigma}-{k}_{\beta \sigma}{k}_{\sigma \beta}-{\gamma}_{\sigma}{k}_{\beta \text{m}}{D}_{\text{m}\beta}-{\gamma}_{\beta}{k}_{\sigma \text{m}}{D}_{\text{m}\sigma}}\\ \times \frac{h{a}_{\upsilon}{D}_{\sigma \sigma}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)}{{D}_{\sigma \sigma}\left({\gamma}_{\beta}{k}_{\sigma \sigma}-{\gamma}_{\sigma}{k}_{\beta \beta}\right)+{k}_{\beta \beta}{k}_{\sigma \sigma}-{k}_{\beta \sigma}{k}_{\sigma \beta}-{\gamma}_{\sigma}{k}_{\beta \text{m}}{D}_{\text{m}\beta}-{\gamma}_{\beta}{k}_{\sigma \text{m}}{D}_{\text{m}\sigma}}\end{array}$

(23)

$\begin{array}{l}F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}=\\ \frac{h{a}_{\upsilon}{\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}^{2}}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}\frac{{\partial}^{2}u}{\partial {t}^{2}}\\ +\frac{{\gamma}_{\beta}{\gamma}_{\sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}\frac{{\partial}^{3}u}{\partial {t}^{3}}\\ +\frac{\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)\left[{k}_{\beta \text{m}}\left({k}_{\sigma \sigma}{D}_{\text{m}\beta}-{k}_{\sigma \beta}{D}_{\text{m}\sigma}\right)+{k}_{\sigma \text{m}}\left({k}_{\beta \beta}{D}_{\text{m}\sigma}-{k}_{\beta \sigma}{D}_{\text{m}\beta}\right)\right]}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}{\Delta}^{3}u\\ -\frac{{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)\left({k}_{\beta \beta}{k}_{\sigma \sigma}-{k}_{\beta \sigma}{k}_{\sigma \beta}\right)}{{D}_{\text{mT}}\left({\gamma}_{\sigma}{k}_{\beta \text{m}}-{\gamma}_{\beta}{k}_{\sigma \text{m}}\right)+h{a}_{\upsilon}\left({k}_{\beta \beta}+{k}_{\beta \sigma}+{k}_{\sigma \beta}+{k}_{\sigma \sigma}\right)+h{a}_{\upsilon}{D}_{\sigma \sigma}\left({\gamma}_{\beta}+{\gamma}_{\sigma}\right)}{\Delta}^{3}u\end{array}$

(24)

This can be regarded as a DPL heat-conduction equation regarding Δ*u* with *τ*_{
q
}*, τ*_{
T
}, and $F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}$ as the phase lags of the heat flux and the temperature gradient, and the source-related term, respectively. Therefore, the coupled heat and mass transport is capable of varying not only thermal conductivity from that in Equation (13) to the one in Equation (21) but also the nature of heat conduction from that in Equation (12) to the one in Equation (19). As practical nanofluid system always involves many transport processes simultaneously, the coupled transport could play a significant role. For assessing its effect and understanding heat conduction in nanofluids, future research is in great demand on coupling (cross) transport coefficients that are derivable by approaches like the up-scaling with closures [2, 27, 28], the kinetic theory [71, 72], the time-correlation functions [73, 74], and the experiments based on phenomenological flux relations [68]. While the uncoupled form of conservation equations, such as Equations (12) and (19), is very useful for examining nature of heat transport, its coupled form, such as Equations (8), (9), (16)-(18), is normally more readily to be resolved for the temperature or concentration fields after all the transport coefficients are available.

#### Brownian motion

In nanofluids, nanoparticles randomly move through liquid and possibly collide. Such a Brownian motion was thus proposed to be one of the possible origins for thermal conductivity enhancement because (i) it enables direct particle-particle transport of heat from one to another, and (ii) it induces surrounding fluid flow and thus so-called microconvection. The ratio of the former contribution to the thermal conductivity (

*k*_{BD}) to the base fluid conductivity (

*k*_{f}) is estimated based on the kinetic theory [

75],

$\frac{{k}_{\text{BD}}}{{k}_{\text{f}}}=\frac{{\left(\rho c\right)}_{\text{p}}\phi {k}_{\text{B}}T}{3\pi \mu {d}_{\text{p}}{k}_{\text{f}}}$

(25)

where subscripts p and BD stand for the nanoparticle and the Brownian diffusion, respectively;

*k*_{B} is the Boltzmann's constant (1.38065 × 10

^{-23}J/K); and

*μ* is the fluid viscosity. The kinetic theory also gives an upper limit for the ratio of the latter's contribution to the thermal conductivity (

*k*_{BC}) to the base fluid conductivity (

*k*_{f}) [

76],

$\frac{{k}_{\text{BC}}}{{k}_{\text{f}}}=\frac{{k}_{\text{B}}T}{3\pi \mu {d}_{\text{p}}{\alpha}_{\text{f}}}$

(26)

where subscript BC refers to the Brownian-motion-induced convection, and *α*_{f} is the thermal diffusivity of the base fluid.

Consider a 1% volume fraction of *d*_{p} = 10 nm copper nanoparticle in water suspension at *T* = 300 K. (*ρc*)_{P} = 8900 kg/m^{3} × 0.386 kJ/(kg K) = 3435.4 kJ/(m^{3} K), μ = 0.798 × 10^{-3}kg/(ms), *k*_{f} = 0.615 W/(mK), and *α*_{f} = 1.478 × 10^{-7} m^{2}/s. These yield *k*_{BD}/*k*_{f} = 3.076 × 10^{-6} and *k*_{BC}/*k*_{f} = 3.726 × 10^{-4}. Therefore, both contributions are negligibly small.

Although the direct contribution of particle Brownian motion to the nanofluid conductivity is negligible, its indirect effect could be significant because it plays an important role in processes of particle aggregating and coupled transport.