Wang, D., & Cheng, P. (2019). A solid-liquid local thermal non-equilibrium lattice Boltzmann model for heat transfer in nanofluids.

A solid-liquid local thermal non-equilibrium lattice Boltzmann model for heat transfer in nanofluids. Part I: Model development, shear flow and heat conduction in a nanofluid

Dongmin Wang, Ping Cheng

^⇑

MOE Key Laboratory for Power Machinery and Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

a r t i c l e i n f o

Article history:

Received 27 July 2018 Accepted 9 October 2018

Keywords:

Nanofluid Heat transfer

Local thermal non-equilibrium Solid-liquid lattice Boltzmann method Dynamic viscosity

Thermal conductivity

a b s t r a c t

A solid-liquid local thermal non-equilibrium lattice Boltzmann model for hydrodynamics and heat trans- fer in a nanofluid is developed in this paper. In this proposed model, interactions between fluid and solid nanoparticles, random motion of nanoparticles as well as heat transfer between nanoparticles and the base fluid are taken into consideration. This novel model is applied to two simple nanofluids problems:

isothermal shear flow between two parallel plates moving at different velocities, and heat conduction between two parallel plates at different temperatures. For the problem of shear flow, it is found that nanoparticles random motion causes the instantaneous velocity distribution of a nanofluid to become non-linear in the shear flow between two parallel plates at microscopic level. At macroscopic level, it is found that as nanoparticles volume fraction increases, effects of nanoparticles random motion are enlarged and the time-average velocity distribution of a nanofluid deviates more from a linear distribu- tion in a shear flow. For the problem of heat conduction between parallel plates at different temperatures, it is found that random motion of nanoparticles together with their high thermal diffusivity and thermal conductivity, causing the instantaneous temperature distribution in the nanofluid to become non-linear between two parallel plates at different temperatures. Temperature inside a nanoparticle is shown to be non-uniform, and temperature gradients in the fluid near a nanoparticles are elevated, thus heat transfer rate is enhanced near a nanoparticle. Time-average temperature gradients of a nanofluid at hot/cooled wall are higher than those of a pure fluid in heat conduction between parallel plates at different temper- atures. Comparisons of calculated dynamic viscosity and thermal conductivity based on this novel model are found in good agreement with existing correlation equations. Thus, the accuracy of this novel solid–

liquid local thermal non-equilibrium model for a nanofluid is validated.

Ó2018 Published by Elsevier Ltd.

1. Introduction

By dispersing metal nanoparticles in conventional fluids, a new type of fluid called ‘nanofluid’ with higher thermal conductivity compared with the base fluid, was corned by Choi and Eastaman [1]in 1995. Since that time, many experimental studies on physical properties of nanofluids have been carried out, and theoretical models for predicting these physical properties have been pro- posed. In addition, a number of mathematical models for predict- ing transport phenomena of nanofluids have been proposed, although none of the proposed theoretical models can predict transport phenomena of nanofluids satisfactorily. The followings are a brief review of previous work on these topics.

1.1. Thermal conductivity of nanofluids

In the early days, the thermal conductivity of a nanofluid was estimated from Maxwell model[2], which gave:

knanofluid

kbasefluid¼knanoparticleþ2kbasefluid2

u

^kbasefluidknanoparticle

knanoparticleþ2kbasefluidþ

u

^kbasefluidknanoparticle

ð1aÞ

where

u

is the volume fraction of nanoparticles. Subsequently, Hamilton and Crosser [3] proposed the following expression for thermal conductivity of a nanofluid:

knanofluid

kbasefluid¼knanoparticleþðn1Þkbasefluidðn1Þ

u

^kbasefluidknanoparticle

knanoparticleþðn1Þkbasefluidþ

u

^kbasefluidknanoparticle

ð1bÞ

https://doi.org/10.1016/j.ijheatmasstransfer.2018.10.048 0017-9310/Ó2018 Published by Elsevier Ltd.

⇑Corresponding author.

E-mail address:pingcheng@sjtu.edu.cn(P. Cheng).

Contents lists available atScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t

where n is the empirical shape factor. When n= 3 for spherical nanoparticles, the Hamilton and Crosser’s model given by Eq.(1b) is reduced to the Maxwell model given by Eq.(1a).

In 2000, Xuan and Roetzel [4] pointed out that the effective thermal conductivity of a nanofluid consisted of two parts:

knanofluid¼ksþkd ð2Þ

whereksis the stagnant thermal conductivity given by Eq.(1)and kdis the thermal dispersion effect analogous to the thermal disper- sion effect in porous media[5], which is of the form:

kd¼C

q

^cpnfudpR

u

ð3Þ

where

q

^cp

nfis the heat capacity of a nanofluid,uis the velocity of a nanofluid andRis the radius of a tube, andCis an unknown con- stant to be determined from experimental data.

Most recently, Kumar et al. [6]developed a semi-theoretical model for thermal conductivity of a nanofluid, where an effective thermal conductivityk_effectiveof a nanoparticle is used to take into consideration of effects of nanoparticles’ random motion on heat conduction in nanofluids, which is expressed as follows:

knanofluid

kbasefluid ¼1þ keffectiv^e

u

^rbasefluid

kbasefluidð1

u

Þrnanoparticle ð4Þ

where keffectiv^e represents effective thermal conductivity of a nanoparticle,r_basefluidandrnanoparticleare radius of base fluid molecules and radius of a nanoparticle respectively.

1.2. Dynamic viscosity of nanofluids

Much less theoretical work has been done on dynamic viscosity of nanofluids. In 1906, Einstein[7]proposed a model for effective dynamic viscosity of a dilute suspension containing spherical par- ticles, which is expressed as:

l

nanofluid¼

l

basefluidð1þ2:5

u

Þ ð5Þ

By considering numbers and volume fraction of nanoparticles, Brinkman[8]extended Einstein’s model for predicting dynamic vis- cosity of less dilute suspensions to give:

l

nanofluid¼

l

basefluidð1

u

Þ^2:5 ð6Þ

Eq. (6) has been widely used to predict dynamic viscosity of nanofluids[9].

1.3. Modeling transport phenomena in nanofluids

Both conventional CFD methods[10–13]and lattice Boltzmann methods[14–27]have been used to investigate transport phenom- ena in nanofluids.

1.3.1. Conventional CFD methods[10–13]

In 2003, Khanafer et al.[10]modeled a nanofluid as a pure fluid containing nanoparticles in thermal equilibrium with base fluid and flowing at the same velocity. The thermal conductivity of a nanofluid was calculated from Eq.(2), the dynamic viscosity of a nanofluid was calculated from Eq.(5). Conventional CFD methods were used to solve the classical problem of natural convection in an enclosure. However, Ho et al.[11]as well as Santra et al.[12]

pointed out that different correlations used for viscosity and ther- mal conductivity could lead to different heat transfer results. In 2006, Buongiorno[13]proposed a four-equation model (i.e. conti- nuity equation, volume faction equation, momentum equation and energy equation) for nanofluids. In this model, the continuity and momentum equations are the same as the pure fluid, with dynamic viscosity of a nanofluid given by correlation equations. Effects of

nanoparticles’ random motion and thermophoresis on transporta- tion of nanoparticles were taken into consideration by modifying volume fraction equation and energy equation of pure fluid, with thermal conductivity of nanofluids given by correlation equations.

1.3.2. Lattice Boltzmann methods[14–27]

In 2004, Xuan and Yao[19]applied the multicomponent lattice Boltzmann method by Shan and Doolen[28]to model nanofluids by considering a nanofluid as a mixture of fluids consisting of a base fluid and nanoparticles as another fluid. Hydrodynamics of the base fluid and of another fluid (nanoparticles) are described by two separate density distribution functionsf^r, where

r

¼1;2 correspond to the base fluid and anther fluid (consisting of nanoparticles). Random motion of nanoparticles was taken into consideration, but it was assumed that nanoparticles and base fluid were under local thermal equilibrium. Subsequently, Wu and Kumar[20]extended such a mixture of fluids model taking into consideration of local thermal non-equilibrium, by introducing two empirical parameters:h_ab (heat transfer coefficient between the base fluid and another fluid) and s(the specific surface area of the base fluid and another fluid inside a lattice). However, the above lattice Boltzmann model needs thermal conductivity and viscosity of nanofluids as input data. For example, Guo et al.[21]

adopted Eq. (5) for dynamic viscosity of nanofluids and Eq. (1) for thermal conductivity of nanofluids. In 2010, Nemati[14]mod- eled a nanofluid as a pure fluid, with hydrodynamics and heat transfer in nanofluids described by one density distribution func- tionfand by one energy distribution functiong, respectively. In this model, Eq. (6) was used for the viscosity of nanofluids and empirical correlations by Chon et al.[29]were used for thermal conductivities of nanofluids. Subsequently, Nemati’s model [14]

was used by a number of other investigators[15–18]to study nat- ural convection of nanofluids.

In 1994, Ladd[22]proposed a lattice Boltzmann model for sim- ulating particulate suspensions, where solid particles were defined as boundary surfaces and nodes inside the boundary surfaces were fluid nodes. This lattice Boltzmann model was extended by Aidun et al. [23] by treating the nodes inside particles as solid nodes underisothermalconditions. Recently, Stratford et al.[24]as well as Jansen and Harting [25]used Ladd’s lattice Boltzmann model to simulate emulsions containing nanoparticles under isothermal conditions. To consider random motion of particles, Jiang [26]

applied random forces on particles in Aidun’s model [23]under isothermal conditions. To account for heat transfer in solid-fluid suspensions, Khiabani et al.[27]added the macro energy equation to solve temperature field in Aidun’s model[23], assuming parti- cles at constant temperature.

It is relevant to point out that microscopic heat transfer mech- anism could not be obtained from homogenous model, nor mixture of fluids model or other early solid-liquid models, in which tem- perature of a nanoparticle had been assumed to be constant. In this paper, we have extended the solid-liquid two-phase lattice Boltz- mann model by Aidun et al.[23], taking into consideration random motion of nanoparticles, as well as the local non-thermal equilib- rium between nanoparticle and the surrounding fluid. It should be pointed out that this novel LB model does not need correlations of physical properties of nanofluids as input data. In this paper, the following two simple problems are solved by this novel LB model:

(i) isothermal shear flow of a nanofluid between two parallel plates moving at different velocities, and (ii) heat conduction in an Al2O3- water nanofluid between two parallel plates at different tempera- tures. The instantaneous velocity/temperature distributions in the nanofluid and temperature distribution inside a nanoparticle are investigated from a microscopic view. It will be shown macroscop- ically that dynamic viscosity and thermal conductivity of

nanofluids predicted by this proposed model are in good agree- ment with existing correlations.

2. Solid-liquid local thermal non-equilibrium lattice Boltzmann model for a nanofluid

2.1. Continuum assumption and unit conversion

In the proposed model, the base fluid surrounding nanoparticles is regarded as a continuum. To certify the rationality of this assumption, the Knudsen numberKnis calculated from[13]:

Kn¼k

D ð7Þ

wherekis the water molecule mean free path, andDis the nanopar- ticle diameter. As the mean free path of water moleculekis of the order of 0.3 nm. For nanoparticles with a diameterDin the range of 10–100 nm, the Knudsen number is very smallKn< 0.03. Thus, the continuum assumption is valid for the base fluid of water. Another assumption is made that the thermal boundary resistance between a solid nanoparticle and the surrounding liquid is ignored, due to thermal boundary resistance has nearly no effects at low heat flux in conduction or convection heat transfer of nanofluids.

In the present model, the 2D grid schematic of a nanoparticle dispersing in base fluid is shown in Fig. 1. By referring to the method on converting lattice unit to real physical unit [30,31], the length of lattice spaceDx¼1 is equal to the length ofDx⁰in real physical unit, and the conversion is expressed as:D^x0¼^t_t^0cc^s0_s. By set- ting the kinematic viscosity

t

used in simulation as 0.15 in lattice unit, and adopting the real sound speed (c⁰_s¼1482:3 m=s) as well as real kinematic viscosity (

t

⁰¼1:010⁶m²=s) of water at 20°C, the lattice spaceD^x¼1 in lattice unit is corresponding to 2.6 nm in real physical unit. It is noted thatcs¼1= ffiffiffi

p3

in a D2Q9 model[32]. So, if the diameter of a nanoparticle is set asDin lattice unit (as shown inFig. 1) in the simulation process, the diameter of a nanoparticle is equal toD2:6 nm in real physical unit. The vol- ume fraction (u) of nanoparticles is calculated as:

u

¼^np_HW^ðD=2Þ2, wherenis the total number of nanoparticles,HandWrepresent the height and width of the simulation domain.

2.2. Hydrodynamics of the base fluid

In the present lattice Boltzmann model for nanofluids, the hydro- dynamics of the base fluid is treated the same as a conventional fluid which is described by the density distribution function. The density

distribution functionfi(x,t) is governed by the following lattice Boltzmann equation with BGK collision operator[32,33]:

f_iðxþeiDt;tþD^tÞ f_iðx;tÞ ¼ 1

s

^{fiðx;tÞ f}

eq i ðx;tÞ

þD^tFi ð8Þ whereD^tis the time spacing andD^tis usually taken as 1.0 in LBM,

s

is the dimensionless collision-relation time andD^tFirepresent the force term on fluid particles.

For a 2D configuration, the D2Q9 model[32]is applied for lat- tice velocities, where the index numberiin Eq.(8)is given as the values of i= 0, 1,. . ., 8 with respect to 9 lattice velocities in a two-dimensional lattice.f0(x,t) represents the resting particles of a lattice at positionxand at timet,fi(x,t) (i= 1, 2, 3, 4) represents the fluid particles moving in the orthogonal directions of a lattice with a lattice velocity ofeiat positionxand at timet, andfi(x,t) (i= 5, 6, 7, 8) represents the fluid particles moving in the diagonal directions of a lattice with a lattice velocity ofeiat positionxand at timet. The 9 lattice velocitieseiis given as follows, respectively:

ei¼ 0;0 ð Þ;i¼0

ccos^pði1₂ ^Þ;sin^pði1₂ ^Þ

;i¼1;2;3;4 ffiffiffi2

p ccos^pð2i₄^9Þ;sin^pð2i₄^9Þ

;i¼5;6;7;8 8>

>>

>>

<

>>

>>

>:

ð9Þ

wherec¼D^x=D^tis the reference lattice speed, in whichD^{xis the} lattice spacing andD^xis usually taken as 1.0 in LBM. The equilib- rium density distribution functionf^eq_i ðx;tÞin Eq.(8)is given as:

f^eq_i ðx;tÞ ¼

q

lwi 1þeiu

c²_s þðeiuÞ² 2c⁴_s uu

2c²_s

" #

ð10Þ where

q

lis the macro density of liquid andu^⁄is the equilibrium velocity of liquid at positionxand at timet. The weight coefficient wi is given as 4/9 (at i¼0), 1/9 (at i¼1;2;3;4), and 1/36 (at i¼5;6;7;8). Note that the sound speedc²_s¼c²=3¼1=3 in a D2Q9 model. The equilibrium velocity of liquidu^⁄used in Eq.(10)is given by:

uðx;tÞ ¼ P

if_iðx;tÞei

q

lðx;tÞ ð11Þ

The relations of mesoscopic physics quantities ½

s

;f_iðx;tÞ;ei with kinematic viscosity of a fluid

t

, macro density of a fluid

q

lðx;tÞ, and macro velocity of a fluidu xð ;tÞin lattice Boltzmann method are defined as:

t

¼ð

s

1=2Þc²_sDt ð12Þ

q

lðx;tÞ ¼X

i

fiðx;tÞ ð13Þ

u x;ð tÞ ¼ P

if_iðx;tÞei

q

lðx;tÞ þ dtF

2

q

lðx;tÞ ð14Þ

2.3. Dynamics of nanoparticles in a nanofluid

2.3.1. Boundary between a solid nanoparticle and base fluid The dynamics of a nanoparticle is affected by the base fluid adjacent to the boundary of the nanoparticle, and these fluid par- ticles are described by the density distribution functions that are calculated in a modified half-way bounce back scheme from Adiun et al.[23]:

fiðx;tþDtÞ ¼fiðx;t_þÞ 6

q

lwiðUPeiÞ ð15Þ

fiðx;t_þÞ ¼fiðx;tÞ f_iðx;tÞ f^eq_i ðx;tÞ

s

^{þDtFi ð16Þ}

Fig. 1.The 2D grid schematic of a nanoparticle in base fluid.

where i represents the direction pointing to the boundary of a nanoparticle,i means the opposite direction of i, UP represents the velocity of a nanoparticle, andf_iðx;t_þÞrepresents the density distribution function at position ofxand at time oftafter collision step.

When a solid nanoparticle moves in the base fluid, some solid nodes in the nanoparticle will turn to new fluid nodes behind the moving solid particle, and the way on extrapolating macro density

q

newðx;tÞof such new fluid nodes is given by Adiun et al.[23]as follow:

q

newðxa;tÞ ¼ 1 Na

X

a

q

aðxa;tÞ ð17Þ

where xa represents the position of fluid nodes adjacent to the boundary of a nanoparticle,

q

aðxa;tÞrepresents the macro density of fluid nodes adjacent to the boundary of a nanoparticle, andNa

represents the total number of fluid nodes adjacent to the boundary of a nanoparticle.

2.3.2. Impinging force (FI) on a solid nanoparticle

All the fluid particles adjacent to the boundary of a nanoparticle will impinge on the nanoparticle and result in impinging force on a solid nanoparticle. The impinging forceFIis calculated in the same way as given by Adiun et al.[23]is:

FI¼X

a

X

i

2f_iðxa;t_þÞ 6

q

awiðUPeiÞ

½ ei ð18Þ

2.3.3. Momentum exchanging force (FM) on a solid nanoparticle On the one hand, a moving nanoparticle in base fluid will cover fluid nodes in front of it and turn these fluid nodes into solid nodes of the nanoparticle at timet, so the momentum of these fluid nodes at timet-D^twill be taken in by the nanoparticle at timet. On the other hand, as some solid nodes of a nanoparticle turn into new fluid nodes behind a moving nanoparticle at timet, so the momen- tum of these solid nodes at timet-D^twill be lost by the nanoparti- cle at timet. Such momentum exchanges between a nanoparticle and its surrounding fluid will result in a momentum exchanging force FM on a moving nanoparticle at time t. The momentum exchanging forceFMat timetis expressed as[34]:

FM¼X

C

q

CuCX

U

q

UuU ð19Þ

whereCrefer to the fluid nodes that will be covered by nanoparti- cles at timet,

q

Candu_Crepresent the corresponding macro density and macro velocity of the fluid nodes.Urefers to the solid nodes in nanoparticles that will be uncovered by nanoparticles at timet,

q

U

and uU represent the corresponding macro density and macro velocity of the solid nodes.

2.3.4. Brownian force (FB) on a solid nanoparticle

The random motion of a nanoparticle follows the fluctuation- dissipation theorem[35], a Brownian forceFBdirectly acting on a nanoparticle is introduced to stimulate random motion of the nanoparticle, the method of which has been successfully used by Yu et al.[36]:

FB¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2nkBT D^t r

G ð20Þ

wheren is the hydrodynamic friction coefficient,kBis the Boltz- mann constant,Tis temperature, and Gis the standard Gaussian random vector.

2.3.5. Repulsion force (FR) between two solid nanoparticles

When two nanoparticles come too close to each other, there is a repulsion forceFRexists between these two nanoparticles to pre- vent them from overlapping with each other. As the distance between surfaces of the two nanoparticles gets more than a critical cut off distancehc, this repulsion forceFRvanishes[34]. In the pre- sent model, the repulsion forceF_Rbetween a pair of nanoparticles is defined similar to that in[34]as follow:

FR¼ 1

2

l

^{jU12j r12 R1þ}_h^R2

3=2

F0þ h R1þR2

F1

"

R1þR2

hc

₃₌₂

F0þ hc

R1þR2

F1

#

C;h<hcFR¼0;h>hc

ð21Þ whereFRrepresents repulsion force on nanoparticle indexed as 1, and jU12j represents the magnitude of the relative velocity of nanoparticle index as 2 to that of a nanoparticle indexed as 1,r12

is the unit direction vector pointing from nanoparticle 1 toward nanoparticle 2,

l

is the dynamic viscosity of fluids andhrepresents the distance between the surface of two nanoparticles. It is noted that hc represents the cut off distance between surfaces of two nanoparticles, R1andR2 represent the radius of nanoparticles, in the present modelhc= 2D^xandR1=R2. The constantF0¼^3ppffiffi₂

4 and F1¼^231p ffiffi

p2

80 ,Cis a constant coefficient that can be adjusted to control the repulsive force at appropriate value for the sake of algorithm stability in simulation. When the distance between the surface of two nanoparticles (h) become too small (h< 0.1Dx), a Hookean repulsion force is also applied on nanoparticles to prevent the breakdown of calculations, which was introduced by Yoshi and Sun[34].

2.3.6. Movement of a nanoparticle in a nanofluid

The movement of a nanoparticle in a nanofluid can be predicted by Newton’s equation of motion:

FP¼MP

dUP

dt ð22aÞ

whereMpis the mass of a nanoparticle given by MP¼

q

P

p

^D₂

2

ð22bÞ withDand

q

Pbeing the diameter and density of the nanoparticle.FP

is the total force acting on a nanoparticle given by

FP¼FIþFMþFBþFR ð22cÞ where FI is impinging force given by Eq. (18), FM is momentum exchanging forcegiven by Eq. (19),FB isBrownian force given by Eq. (20)and F_R is repulsion force given by Eq.(21). The rotation motion of nanoparticles is ignored in the present model because the nanoparticle is assumed to be of spherical shape[34]in the pre- sent model. The velocity of a nanoparticle UPcan be determined from Eq.(22a)together with Eqs.(22b)and(22c)subject to the ini- tial conditionUP= 0 at t = 0.

2.4. Heat transfer between nanoparticles and base fluid in a nanofluid Heat transfer in base fluid and solid nanoparticles can be described by another particle distribution function,i.e.the temper- ature distribution functiongi(x,t). The governing lattice Boltzmann equation forgi(x,t) is similar to that of density distribution func- tion, and it is expressed as follow:

giðxþeiDt;tþDtÞ giðx;tÞ ¼ 1

s

^{T giðx;tÞ g}

eq i ðx;tÞ

ð23aÞ

where

s

^T is the dimensionless collision-relation time for tempera- ture and the equilibrium temperature distribution functiongieq

(x,t) is given as:

g^eq_i ðx;tÞ ¼Twi 1þeiu

c²_s þðeiuÞ² 2c⁴_s uu

2c²_s

" #

ð23bÞ A D2Q9 model has also been applied for the temperature distribu- tion function, thus the thermal diffusivities of base fluid

a

land solid nanoparticles

a

Pas well as macro temperatureT(x,t) are calculated as follow:

a

l¼ð

s

Tl1=2Þc²sD^t ð24aÞ

a

P¼ð

s

TP1=2Þc²sD^t ð24bÞ

Tðx;tÞ ¼X

i

g_iðx;tÞ ð25Þ

where

s

Tland

s

TPare the corresponding dimensionless collision- relation time for base fluid and solid nanoparticles when calculating their temperature distribution functions by Eq.(23a).

To ensure heat transfer and temperature are continuous at the interface of a solid nanoparticle and its surrounding fluid, the con- jugate boundary condition by Li et al. [37] must be imposed as follow:

giðxl;tþDtÞ ¼1

c

1þ

c

^{giðxl;tþÞ þ}

2

c

1þ

c

^{giðxs;tþÞ ð26aÞ}

giðxs;tþDtÞ ¼ 1

c

1þ

c

^{giðxs;tþÞ þ}

2

1þ

c

^{giðxl;tþÞ ð26bÞ}

wherexlandxsrepresent the position of fluid nodes and position of solid nodes. And

c

is the thermal mass ratio of a nanoparticle and the surrounding fluid, its relation with thermal diffusivity of base fluid

a

l, thermal diffusivity of solid nanoparticle

a

P, thermal conduc- tivity of base fluidk_land thermal conductivity of solid nanoparticle kPis given as:^a_a^l

P¼

c

k^kP^l.

3. Shear flow between parallel plates

We now apply the proposed lattice Boltzmann model presented in Section2to anisothermalshear flow of a nanofluid between par- allel plates at length ofL, separated by a distanceHas shown in Fig. 2. The two parallel plates are moving in opposite directions with the velocity of

v

⁰. In this study, we setH =350,L=Hand v0= 0.001 for computations. Periodic boundary conditions are applied at both upper and lower boundaries to simulate two infi- nitely long plates. The velocity of the base fluiducan be obtained from Eq. (14), and the velocity of solid nanoparticles can be obtained from Eq.(22). The hydrodynamic boundary conditions between nanoparticles and base fluid are given by Eqs.(15)and (16).

3.1. Velocity distribution in a nanofluid between parallel plates Fig. 3(a) shows microscopic effects of nanoparticles volume fraction on the instantaneousycomponent velocityuy(x,0,t= 7.09 10⁵) profiles foru= 0.09 (red dashed lines), anduy(x,0,t= 5.54 10⁶) profiles for u = 0.01 (blue dashed lines) in nanofluids (D= 35 nm) between parallel plates. It can be seen that the instan- taneousycomponent velocity varies from0.001 of the left plate’s velocity (atx= 0) to 0.001 of the right plate’s moving velocity (at x=H) both in a pure fluid and in nanofluids. The y component velocity in the pure fluid is linear (a black solid straight line), but it deviates from linear distribution in a nanofluid (dashed lines) due to nanoparticles random motion. At high volume fraction of u= 0.09 (the red dashed line), the fluctuation inuy(x,0,t= 7.09 10⁵) is more violent than thatuy(x,0,t= 5.5410⁶) at low volume fraction ofu= 0.01 (the blue dashed line). It is also noted that there happened to be two nanoparticles (indicated by red circles) in the case ofu= 0.09 but there are no nanoparticles present in the case ofu= 0.01 as shown inFig. 3(a). Theuy(x, 0,t= 7.0910⁵) in a nanoparticle is flat, whileuy(x,0,t= 7.0910⁵) anduy(x,0,t= 5.54

Fig. 2.Shear flow of a nanofluid between two parallel plates moving in opposite directions.

10⁶) in base fluid is fluctuating due to random motion of the nanoparticle.

Fig. 3(b) shows microscopic effects of nanoparticles’ size on the instantaneousycomponent velocityuy(x,0,t= 7.0910⁵) profiles

for D= 35 nm (indicated by red dashed lines) and u_y(x,0,t= 4.60 10⁵) profiles forD= 18 nm (indicated by blue dashed lines) in nanofluids (u= 0.09) between parallel plates. It can be seen that theu_y(x,0,t= 4.6010⁵) in a nanofluid with small size of nanopar- ticles (D= 18 nm) is closer to linear distribution thanuy(x,0,t= 7.09 10⁵) of a nanofluid with larger size of nanoparticles (D= 35 nm).

It is also noted that there happens to have 8 small nanoparticles at that instant of time (indicated by blue circles) in a nanofluid with small size of nanoparticles (D= 18 nm) while there happens to have only 2 larger nanoparticles at that instant of time (with D= 35 nm) (indicated by red circles) as shown inFig. 3(b). But lar- ger nanoparticles can disturb instantaneousuy(x,0,t) in base fluid more significantly than smaller nanoparticles do.

It should be noted that the instantaneous u_y(x,y,t) profile between parallel plates depends not only in time but also in space due to the random motion of nanoparticles. So the time-average velocity u_yð Þx distribution can be obtained by an integration of the instantaneousuy(x,y,t) distribution distributions with respect to space and time as follows:

uyð Þ ¼x 1 t1t0

ð Þ1 L

Z t₁ t0

Z L=2

L=2uyðx;y;tÞdydt ð27Þ The above double integral is first averaged in the whole simulation domain at certain instant of time, then averaged from time step t0= 6.0910⁵ to t1= 7.0910⁵ for the volume fraction of u = 0.09 and from time stept0= 5.3510⁶to t1= 5.5510⁶for the volume fraction ofu= 0.01 after steady state is reached.

Macroscopic effects of nanoparticles volume fraction on the

u_yð Þx profiles between parallel plates are presented inFig. 4 in nanofluids (D= 35 nm) at different volume fractions (u = 0.09 and u= 0.01).Fig. 4 shows that theuyð Þx profiles in nanofluids (denoted by dashed lines) deviate from linear distribution in pure water (denoted by black solid lines). As volume fraction increases fromu= 0.01 tou= 0.09, theuyð Þxfluctuates more violently and deviates more from linear distribution. This indicates that the increasing volume fraction has enlarged the disturbing effect of nanoparticles random motion onuyð Þx profile in a nanofluid.

3.2. Determination of effective dynamic viscosity of nanofluids We now compute the effective dynamic viscosity

l

nanofluidof the nanofluid by

s

x;y¼

l

nanofluid

c

_ ð28Þ

(a) Effects of volume fraction of nanoparticles

(b) Effects of nanoparticles’ diameter

Fig. 3.Instantaneousuy(x,0,t) profiles between parallel plates in pure fluid and in nanofluids: (a) effects of volume fractions and (b) effects of nanoparticles’

diameters.

Fig. 4.Average velocityuyð Þx profiles between parallel plates in pure fluid and in nanofluids at different volume fractions (u= 0.01 andu= 0.09) at steady state.

Fig. 5.Temporal varied lnanofluid with simulation time steps (_c¼610⁶, D= 35 nm,u= 0.01).

where

c

_ represents shear rate which is calculated as:

c

_ ¼2j

v

^0j

H ð29Þ

Based on a lattice Boltzmann model, Mei et al.[38]showed that the shear stress (

s

^xy) acting on fluid particles (at position ofxand at time oft) can be calculated as follows:

s

^{xy¼ 1}₂¹

s

X

i

f_iðx;tÞ f^eq_i ðx;tÞ

ei;xei;y ð30Þ

wheree_i;xande_i;yare Cartesian components of the lattice velocityei

in thex direction andy directions, respectively. It is noted that f_iðx;tÞcan be obtained from collision step and propagation step of Eq.(8).

Due to the random motion of the nanoparticles the calculated dynamic viscosity

l

nanofluid of a nanofluid (D= 35 nm and u = 0.01) from Eq.(28)fluctuates with simulation timetas shown in Fig. 6.Frequency distribution of the variedlnanofluidof a nanofluid from time stept= 1.010⁶to time stept= 5.510⁶.

Fig. 7.Comparison of Brinkman’s correlation[8]with LB simulated results.

Fig. 8.Heat conduction in a nanofluid between two parallel infinite plates with different temperatures.

Fig. 5. The frequency distribution of the temporal varied dynamic viscosities

l

nanofluidfrom time step t= 1.010⁶ to time step t= 5.510⁶is plotted inFig. 6. A Gaussian fitting curve expressed

asy¼182:6eð^{x0:15710:01343}Þ²is used to fit the frequency distribution in

Fig. 6, wherexequals to effective dynamic viscosity andyequals to the frequency count at a certainx. From the Gaussian fitting curve and its expression, it is shown that

l

nanofluid= 0.1571 corresponding to the peak of the Gaussian fitting curve.

The dynamic viscosities of a nanofluid (D= 35 nm) at different volume fractions (u = 0.01–0.09) are obtained from the above method. Brinkman’s model [8], the most widely used model for viscosity of nanofluids, is used to validate the dynamic viscosity determined from the present simulation. The LB simulated results for viscosity and the predicted results from Brinkman’s model given by Eq.(6)are presented inFig. 7. It can be seen that both sim- ulated results and Brinkman’s correlation show that the dynamic viscosity of a nanofluid increases as the volume fraction of nanoparticles increases. It is also shown that the dynamic viscosity of the nanofluid without nanoparticles’ random motion in the pre- sent calculation agree well with the Brinkman’s correlation given by Eq.(6). In addition, at low volume fraction (<6%) of nanoparti- cles, nanoparticles’ random motion do not affect the effective dynamic viscosity of the nanofluid. However, as the volume frac- tion of nanoparticles increase (>6%), nanoparticles’ random motion become strong enough to increase dynamic viscosities of a nanofluid.

4. Heat conduction between parallel plates

Next, we consider heat conduction of a nanofluid (consisting of Al2O3nanoparticles in water) between two parallel plates at differ- ent temperatures as shown inFig. 8. The left plate is at lower wall temperature ofT(0) =Tc= 0 while the right plate is at a high wall temperature of T(H) = TH= 1. Periodic boundary conditions are applied at upper and lower boundaries to simulate that the two plates are of infinite length. The temperatures ofT(x,y,t) in the base fluid and in solid nanoparticles are obtained from Eqs.(23)–(25), with boundary condition given by Eq.(26a)and Eq.(26b), where the ratio of thermal conductivity of solid nanoparticles to the base fluid must be specified. For this purpose, we set^knanoparticle

kbasefluid ¼65 corre-

sponding to the ratio of thermal conductivity of Al2O3

(40 W=ðm KÞ) to the thermal conductivity of water (0:613 W=ðm KÞ). The diameter of nanoparticles is set as D= 10

(a) Density distribution

(b) Temperature distribution

Fig. 9.Density distribution and temperature contours in an Al2O3-water nanofluid between two plates at different temperatures at time stept= 9.910⁶.

(a) t=9.9×10

⁶

( b ) t=1.0×10

⁷

Fig. 10.Temperature contours in an Al2O3nanoparticle and in its surrounding water between two parallel plates at two different times.

(i.e. D¼26 nm) and numbers of nanoparticles are 16–80 in the simulation process, corresponding tou= 0.01–0.05 for the same size of the computation domain.

4.1. Temperature distributions in nanoparticles and in surrounding base fluid

Density and temperature contours in the problem of heat con- duction of an Al2O3-water nanofluid (u= 0.01,D= 26 nm) between two plates att= 9.910⁶are presented inFig. 9.Fig. 9(a) shows that solid nanoparticles distribute randomly in the nanofluid, and Fig. 9(b) shows that some isotherms in the Al2O3-water nanofluid are warped between differentially heated parallel plates.

Temperature contours in an Al₂O₃ nanoparticle and its sur- rounding base water (indicated by a black square in Fig. 9) at t= 9.910⁶and att= 1.010⁷are presented inFig. 10, where it is shown that the temperature distribution inside an Al2O3

nanoparticle is non-uniform.Fig. 10(a) shows that temperature contours become warped in the nanoparticle due to high thermal conductivity of the Al2O3nanoparticle. The temperature contours in the base fluid near left/right side of a nanoparticle are much denser than that in other places, meaning that temperature gradi- ents in the fluid are relatively high near the left/right side of a nanoparticle. So, heat conduction in the base fluid near the left/

right side of a nanoparticle will be intensified, and this helps con- duction heat transfer from the right hot wall to the left cooled wall.

Fig. 10(b) shows that as time step varies from t= 9.910⁶ to t= 1.010⁷, the nanoparticle moves from the left of the black square to the top right corner of the black square, and the temper- ature distribution inside a nanoparticle also varies with time.

Enlarged view of microscopic effects of nanoparticles on instan- taneousT(x,0,t= 5.5810⁶) profiles in an Al2O3-water nanofluid (u= 0.05) between two plates are shown in Fig. 11. A glance at Fig. 11shows that the temperature profile in pure water (black line) is linear along the midline (y= 0), and the instantaneousT (x,0,t= 5.58 10⁶) in a water-Al2O3 nanofluid (red dashed line) fluctuates around linear distribution due to the presences of nanoparticles. A careful check onFig. 11about the instantaneous T(x,0,t= 5.5810⁶) profile in a water-Al2O3nanofluid (red dashed line) also shows that temperature distributions in a solid nanopar- ticle and in the base fluid are different. Instantaneous temperature distribution in a solid nanoparticle is much more flat than that in the surrounding base fluid, due to the high thermal diffusivity and high thermal conductivity of a solid nanoparticle.

Macroscopic effects of nanoparticles on the time-average distri- butionT xð Þand on^@T_@xare shown inFig. 12. The time-average tem- perature distributionT xð Þin an Al₂O₃-water nanofluid (u= 0.05) between two plates can be obtained in the same manner with Fig. 11.InstantaneousT(x,0,t= 5.5810⁶) profile between two parallel plates in

pure water and in a Al2O3-water nanofluid (u= 0.05).

(a) Profiles of time-average temperature ^T( )^x

(b) Profiles of T x

∂

∂

Fig. 12.Time-average temperatureT xð Þ profiles and ^@T_@x profiles between two parallel plates in pure water and in a Al2O3-water nanofluid (u= 0.05).

Fig. 13.Temporal variedknanofluid=kbasefluidin an Al2O3-water nanofluid (D= 26 nm,u

= 0.01) with simulation timest.

Eq.(27), by first averaging in the whole simulation domain in each time steps, then averaging from time step t0= 5.0010⁶ to t1= 5.5810⁶foru= 0.05.Fig. 12(a) shows that theT xð Þprofile in a pure water is linear along the midline (y= 0). The T xð Þin a Al2O3-water nanofluid deviates from linear distribution and is asymmetric in the left cold domain and in the right hot domain, due to random motion of nanoparticles.Fig. 12(b) shows that the

@T@xin a pure water is constant corresponding to linear distribution ofT xð Þ. The^@T_@xin an Al2O3-water nanofluid fluctuates around that of a pure water due to nanoparticles random motion, but becomes much higher than that in a pure water near hot/cooled walls.

4.2. Calculation of thermal conductivity of nanofluids

After the temperature distribution between parallel infinite plates is obtained numerically by the proposed LB model, the ther- mal conductivity (knanofluid) of a nanofluid between two parallel plates can be determined from:

knanofluid¼ q⁰⁰ TLTH

ð Þ=H ð31Þ

whereq⁰⁰represents the heat flux transfer from the right hot plate to the left cooled plate,Hrepresents the distance between the two

parallel plates. The heat fluxq⁰⁰can be calculated in base fluid near the right hot plate according to Fourier’s heat conduction law as follow:

q⁰⁰¼kbasefluid @T

@x

x¼H

ð32Þ From Eq.(31) and (32), the ratio of effective thermal conductivity of a nanofluid to the thermal conductivity of the base fluid can be expressed as:

knanofluid

kbasefluid

¼ H

TLTH

ð Þkbasefluid ^@T_@x x¼H

kbasefluid

¼ H

TLTH

ð Þ

@T

@x

x¼H

ð33Þ The value of knanofluid=kbasefluid in an Al2O3-water nanofluid (D= 26 nm,u= 0.01) would fluctuate after steady state as shown in Fig. 13. The frequency distribution of the temporal varied k_nanofluid=k_basefluid from time stept₀= 3.5010⁶tot₁= 1.0910⁷is plotted in Fig. 14. A Gaussian fitting curve expressed as

y¼288:1eð^x1:0280:0217Þ² is used to fit the frequency distribution in

Fig. 14, wherexequals toknanofluid=kbasefluidandyequals to the fre- quency count at a certainx. From the Gaussian fitting curve and its expression, it is shown thatknanofluid=kbasefluid= 1.028 corresponds to the peak of the Gaussian fitting curve.

The semi-theoretical model given by Kumar et al.[6]is chosen for comparison with simulated results on calculating thermal con- ductivity of Al₂O₃-water nanofluids. Because this model has taken nanoparticles’ random motion into consideration in the parameter of nanoparticles’ effective thermal conductivitykeffectiv^e, and there is only one empirical constant in their model.Fig. 15shows a com- parison of k_nanofluid=k_basefluid in Al₂O₃-water nanofluids at different volume fractions (u= 0.01–0.05) determined from the present sim- ulation, and with those predicted results based on Eq.(4)given by Kumar et al. [6]. The effective thermal conductivity k_effective of a nanoparticle due to random motion in Kumar et al.’s model is asso- ciated with an empirical constant, so we adopted the form of k_effective¼cknanoparticleand the constantc was chosen to be 2.0. The value ofrbasefluidis equal to 0.2 nm in Kumar et al.’s model[6]corre- sponding to the radii of water molecule. It can be seen fromFig. 15 that theknanofluid=kbasefluid>1 in the whole range of volume fraction, which means that adding Al2O3nanoparticles into base water can enhance thermal conductivity of the base water. In addition, as the volume fraction of nanoparticles increases, the value of knanofluid=kbasefluidalso increases.Fig. 15shows that the thermal con- ductivity enhancement in a nanofluid predicted by the present lat- tice Boltzmann model agrees well with the prediction given by Kumar’s model[6].

Fig. 14.Frequency distribution of the variedknanofluid=kbasefluidof an Al2O3-water nanofluid (D= 26 nm,u= 0.01) from time stept= 3.510⁶to time stept= 1.0910⁷.

Fig. 15.Comparison of Kumar’s correlation for prediction^k_k^nanofluid

basefluid[6]of Al2O3-water nanofluids with the present simulation results.

5. Conclusions

In this paper, we have developed a solid-liquid local thermal non-equilibrium lattice Boltzmann model for hydrodynamics and heat transfer in nanofluids. Isothermal shear flow of nanofluids between two parallel plates as well as heat conduction in Al2O3- water nanofluids between two parallel plates at different temper- atures are simulated by this model. The following conclusions can be drawn from this paper:

1. For the problem of shear flow of nanofluids, random motion of nanoparticles disturbs velocity field, resulting in non-linear instantaneous velocity distribution between two parallel plates.

Larger nanoparticles size or higher nanoparticles volume frac- tion would enlarge such effects. The time-average velocity dis- tribution in a nanofluid between two parallel plates fluctuate around a linear distribution in the shear flow, and such fluctu- ations are more violent at higher volume fractions.

2. For the problem of heat conduction of Al2O3-water nanofluids between parallel plates at different temperatures, temperature distribution inside a nanoparticle is non-uniform and the tem- perature gradients in the base fluid surrounding a nanoparticle are elevated. Random motion of nanoparticles together with their high thermal diffusivity and high thermal conductivity can disturb temperature field in the base fluid, resulting in non-linear instantaneous temperature distributions in the base fluid between parallel plates. Because nanoparticles random motions intensify with temperature, the time-average temper- ature distribution in a nanofluid become asymmetric between parallel plates at different temperatures. The time-average tem- perature gradients in a nanofluid near hot/cooled plates are slightly higher than that in pure water.

3. Macroscopically, the present solid-liquid local thermal non- equilibrium lattice Boltzmann model is verified by calculating dynamic viscosities and thermal conductivities of nanofluids based on the proposed model in comparison with existing cor- relations. The calculated dynamic viscosities based on the pre- sent LB model agree well with Brinkman’s formula while calculated thermal conductivities by the present LB model agree well with Kumar’s semi-theoretical model. Thus, the cor- rectness and accuracy of this proposed model are validated.

Conflict of interest

The author declare that there is no conflict of interest.

Acknowledgement

This study was supported by the National Natural Science Foun- dation of China under Grant No. 51420105009.

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