A useful case study to develop lattice Boltzmann method performance:

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A useful case study to develop lattice Boltzmann method performance:

Gravity effects on slip velocity and temperature profiles of an air flow inside a microchannel under a constant heat flux boundary condition

Annunziata D’Orazio

^a

, Arash Karimipour

^b,^⇑

aDipartimento di Ingegneria Astronautica, Elettrica ed Energetica, Sapienza Università di Roma, Via Eudossiana 18, Roma 00184, Italy

bDepartment of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

a r t i c l e i n f o

Article history:

Received 10 October 2017

Received in revised form 25 February 2018 Accepted 5 March 2019

Keywords:

Slip velocity Gravity

Lattice Boltzmann method Heat flux

Knudsen number

a b s t r a c t

Mixed convection heat transfer of air in a 2-D microchannel is investigated numerically by using lattice Boltzmann method. The effects of buoyancy forces on slip velocity and temperature profiles are presented while the microchannel side walls are under a constant heat flux boundary condition. Three states are considered as no gravity,Gr= 100 andGr= 500. At each state, the value of Knudsen number is chosen asKn= 0.005,Kn= 0.01 andKn= 0.02 respectively; while Reynolds number and Prandtl number are kept fixed atRe= 1 andPr= 0.7. Density-momentum and internal energy distribution functions are used in order to simulate the hydrodynamic and thermal domains in LBM approach. Develop the ability of LBM to simulate the constant heat flux boundary condition along the microchannel walls in the presence of slip velocity and buoyancy forces is proved for the first time at present work. The new and interesting results are achieved such as generating a rotational cell through the fluid flow due to buoyancy forces which leads to see the negative slip velocity at these areas.

1. Introduction

Many works have been reported concerned Micro and Nano Electro Mechanical Systems which could be applied in microde- vices such as micropumps, microchannels and so on. The microflow regimes might be different from the macroflow ones.

Therefor a dimensionless number of Knudsen (Kn) was introduced to classify the flows [1–3]. The continuum flow regime can be observed at the state of Kn< 0.001, which means the classic Navier-Stokes equations are being true and can be applied for the flow simulations. Moreover the slip flow regime, transient regime and free molecular regime would be occurred at 0.001 <Kn< 0.1, 0.1 <Kn< 10 and Kn> 10, respectively [4–9]. At theses regimes the particle based methods like lattice Boltzmann method (LBM), Molecular dynamic (MD) and direct simulation of Monte Carlo (DSMC) must be applied and the well-known Navier-Stokes equations would not be responded. However Navier-Stokes equations together with slip velocity and temperate jump boundary conditions are able to be used for the slip flows.

These mentioned boundary conditions imply the fluid molecule placed on the constant or moving wall, would not have the same velocity and the same temperature with those of wall[10–16].

Among all particle based methods, LBM involves the less com- putation costs and also can be used for the all flow regimes.

Besides, it works with less complex math equations; hence many articles can be addressed using LBM. It can be mentioned that lattice Boltzmann method is a relatively new simulation approach consisted collision and streaming stages between fictive particles located on the lattice nodes. Suitable deal with complex bound- aries, using parallel algorithms and easy to use for the multi- phase flows are some other advantages of LBM compared with classic CFD approaches. All these benefits are available only after choosing an appropriate collision operator; so that LBM according to BGK model is considered as the alternative approach in this way which shows suitable accuracy and convergence[17–22].

Different models of LBM were developed in order to increase LBM performance such as for the simulation of thermal domains.

Among them, double population approach used density- momentum distribution function for the hydrodynamic field and internal energy distribution function for the thermal field showed a better manner[23–31]. Simulation of 2-D isothermal pressure driven flow inside a microchannel by LBM was reported by Lim et al.[32]. More works of Niu and Shu and Chew[33–35]can be https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.029

⇑ Corresponding author.

E-mail addresses: annunziata.dorazio@uniroma1.it (A. D’Orazio), arash.

karimipour@uniroma1.it,arash.karimipour@pmc.iaun.ac.ir(A. Karimipour).

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

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referred corresponded lattice Boltzmann method based on new definition of relaxation times in term ofKnand by using diffuse scattering boundary condition model for the slip velocity. Several other studies on thermal lattice Boltzmann method (TLBM) through a microchannel considering viscous heat dissipation term were presented by Tian et al.[36,37]; also other works of LBM were reported at different conditions and for various type of working fluids such as nanofluid[38–48].

It is seen that the influences of buoyancy forces are neglected in the most previous researches which means lack of accuracy in results. Moreover heat flux boundary condition along the microchannel walls, in the presence of gravity, has not been reported by LBM (To the best author’s knowledge). Hence present article aims to investigate this subject to develop the LBM performance through such physical problems.

2. Problem statement

Mixed convection heat transfer of air in a 2-D microchannel is investigated numerically by using lattice Boltzmann method (LBM) involved BGK model. Microchannel length is ten times longer than its width (AR = L/H = 10) and its horizontal walls are under a constant heat flux ofq⁰⁰₀. The cold air at Tienters to the microchannel from the left side with a constant velocity of ui

(seeFig. 1).

Three states are considered as no gravity,Gr= 100 andGr= 500.

The first one implies to ignore the gravity acceleration while the two other ones represent to invigorate the buoyancy forces. At each state, the value of Knudsen number is chosen asKn= 0.005, Kn= 0.01 andKn= 0.02 respectively; while Reynolds number and Prandtl number are kept fixed at Re= 1 and Pr= 0.7. It is well known that the characteristic length is chosen as the hydraulic length (DH) in a channel flow, which means DH= 2H have a small amount in a micro flow inside a microchannel due to the fact of low value of microchannel width (H). Hence it’s corresponded Rey- nolds number (Re=

q

iuiDH/

l

) will have a low value at the level of one or even less than one. Density-momentum (f) and internal energy distribution functions (g) are used in order to simulate the hydrodynamic and thermal domains in LBM approach.

3. Present work novelty

The effect of gravity was ignored in the most published papers concerned the microflows. However its effect may be important and considerable on the flow domain at a specified rage of Knudsen number; while there is no article concerned microchannel in the presence of constant heat flux especially applying LBM. Some diffi- culties in this way can be classified as: 1-How to consider the gravity effects in slip velocity model of LBM based on density- momentum distribution function, 2-Develop a new model or use a suitable existence model to simulate the heat flux along the microchannel walls in the presence of both slip velocity and buoyancy forces.

Hence develop the ability of LBM to simulate the constant heat flux boundary condition along the microchannel walls in the presence of slip velocity and buoyancy forces is proved for the first time at present work. Moreover the new and interesting results are achieved which they would be presented in the following sections.

4. Formulation

4.1. Lattice Boltzmann method

Boltzmann equation according to the density-momentum and internal energy distribution functions:

@tfþ ðc:rÞf¼XðfÞ ð1Þ

g¼0:5ðcuÞ²f ð2Þ

which lead to the thermal lattice Boltzmann equation:

@tgþ ðc:rÞg¼XðgÞ ð3Þ In BGK model, the collision term can be written as:

XðfÞ ¼ ff^e

s

f

ð4Þ

XðgÞ ¼ gg^e

s

g

fZ¼0:5ðcuÞ²XðfÞ fZ ð5Þ The heat dissipation term of ‘‘f Z” is:

fZ¼fðcuÞ: @½ tuþ ðc:rÞu ð6Þ

s

fand

s

gare introduced as the hydrodynamic and thermal relaxation times.

New distribution functions off_iandg_iare used as:

f

i¼fiþ dt 2

s

f

ðfif^e_iÞ ð7Þ

Fig. 1.Microchannel configuration under the constant heat flux.

Nomenclature

c =(cx, cy) microscopic velocity cs speed of sound DH=2H hydraulic diameter

f, g distribution functions for density-momentum and internal energy

G =(0, G) buoyancy force per unit mass Gr Grashof number

H, L width and length of the microchannel Kn= k/DH Knudsen number

Pr=t^/

a

Prandtl number q heat flux, Wm²

Re=

q

iuiDH/

l

Reynolds number T temperature, k

u=(u, v) macroscopic velocity vector, ms¹ (U,V)=(u/ui, v/ui) dimensionless velocity Us slip velocity

(X,Y)=(X/H, Y/H) dimensionless Cartesian coordinates Greek symbols

h= T/Ti dimensionless temperature t kinematic viscosity, m²m²s¹s¹

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g

i¼g_iþ dt

2

s

^g^ðgⁱ^g^eⁱ^{Þ þ}^dt²^fⁱ^Zⁱ ^ð8Þ

Maxwell-Boltzmann equilibrium distribution functions are shown as f_i^eand g_i^efor momentum and internal energy. Subscript i represents the lattice link number of D2Q9shown inFig. 2 [17].

Zi¼ðciuÞ:Diuand Di¼@tþci:r ð9Þ ci¼cosⁱ₂¹

p

;sinⁱ₂¹

p

c; i¼1;2;3;4

ci¼ ffiffiffi p2

cosh^ðⁱ⁵₂^Þ

p

þ^p₄i

;sinh^ðⁱ⁵₂^Þ

p

þ^p₄i

c; i¼5;6;7;8 c0¼ð0;0Þ

ð10Þ

Now the collision and streaming stages of LBM are presented as:

f

iðxþcidt;tþdtÞ fiðx;tÞ ¼ dt

s

fþ0:5dt fif^e_i

ð11Þ

g

iðxþcidt;tþdtÞ g_iðx;tÞ ¼ dt

s

^gþ0:5dthg_ig^e_ii

s

^g^dt

s

^gþ0:5dtf_iZi ð12Þ f^e_i ¼

x

ⁱ

q

¹þ3ciu

c² þ9ðciuÞ²

2c⁴ 3ðu²þ

v

²^Þ

2c²

" #

ð13Þ

g^e₀¼

x

⁰ ³^q2^eu²þv² c²

h i

g^e₁_;₂_;₃_;₄¼

x

¹

q

ê ^1:5^þ^1:5^ccⁱû² þ4:5^ðcⁱ_cûÞ4²1:5û²_c^þ2^v²

h i

g^e_5;6;7;8¼

x

²

q

ê ³þ6^c_cⁱû2 þ4:5^ðcⁱ_cûÞ4²1:5û²_c^þ2^v²

h i ð14Þ

where c²=1,

q

^e⁼

q

^{RT and} x0=4/9, xi=1/9 for i=1,2,3,4 and xi=1/36 for i=5, 6, 7, 8 in a 2-D simulation. Hence the macroscopic parameters are achieved:

q

^¼^Pif_i

q

^u¼P

icifi

q

^e¼P

igi^dt₂P

ifiZi

ð15Þ

According to the kinetic theory of the microflows, it is reminded that:

Kn¼ ffiffiffiffiffiffi

p

^k

2 r Ma

Re ð16Þ

s

^f¼ ffiffiffiffiffiffi

p

6^k r

DH:Kn ð17Þ

s

g¼

s

^f

Pr ð18Þ

Fig. 2.Lattice of D2Q9.

θ

2 4 6 8 10

Y

0.00 0.25 0.50 0.75 1.00

x^*=0.0127 x^*=0.864

Fig. 4.Comparison of present work results with those of Kavehpour et al.[7].

(Lines: present work; Symbols: those of Kavehpour et al.) U

0.0 0.5 1.0 1.5

Y

0.00 0.25 0.50 0.75 1.00

Lines: present work Hooman & Ejlali, Kn = 0.0 Hooman & Ejlali, Kn = 0.1

Fig. 3.Validation of present work versus those of Hooman & Ejlali[6].

Table 1

Grid independency study at the state of no-gravity,Re= 1,Kn= 0.01 for three different lattice nodes.

35035 40040 45045

Us 0.0779 0.0782 0.0783

Nux (outlet) 11.1020 11.1023 11.1024

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4.2. Boundary conditions

Non-equilibrium bounce back model is applied to determine the unknown values of hydrodynamic distribution functions for the inlet and outlet sections:

f

1¼f₃þ²₃

q

inuin f

5¼f₇þ¹₂ðf4f₂Þ þ¹₆

q

inuin f

8¼f₆¹₂ðf4f₂Þ þ¹₆

q

inuin

ð19Þ

f

3¼f1²₃

q

outuout f

7¼f5¹₂ðf4f2Þ ¹₆

q

outuout¹₂

q

out

v

^out

f

6¼f8þ¹₂ðf4f2Þ ¹₆

q

outuoutþ¹₂

q

out

v

^out

ð20Þ

And for the unknown thermal distribution functions at these areas:

g₅¼⁶^q^e^þ^3dt P

if_iZ_i6ðg₀þg₂þg₃þg₄þg₆þg₇Þ

2þ3uinþ3u²_in ½3:0þ6uinþ3:0u²in₃₆¹ g₁¼⁶^q^e^þ^3dt

P

if_iZ_i6ðg₀þg₂þg₃þg₄þg₆þg₇Þ

2þ3uinþ3u²_in ½1:5þ1:5uinþ3:0u²in¹₉ g8¼⁶^q^e^þ^3dt

P

if_iZ_i6ðg0þg2þg3þg4þg6þg7Þ

2þ3uinþ3u²_in ½3:0þ6uinþ3:0u²in₃₆¹ ð21Þ

g

6¼⁶^ð^g¹^þ^g⁵^þ^g⁸^Þ^3dtP

ið^cixcÞZ_if_i6q^euout 23uoutþ3u²_out

½3:06:0uoutþ6:0

v

^out^þ^3:0u²outþ3:0

v

²out9:0uout

v

^out36¹

g 3¼^6ðg

1þg5þg8Þ3dtP

ið^cixcÞZifi6q^eu^out 23uoutþ3u²_out

½1:51:5uoutþ3:0u²out1:50

v

²out¹₉ g

7¼^6ðg

1þg5þg8Þ3dtP

ið^cixcÞZifi6q^euout 23uoutþ3u²_out

½3:06:0uout6:0

v

^out^þ^3:0u²outþ3:0

v

²outþ9:0uout

v

^out36¹

ð22Þ Moreover the slip velocity along the lower wall can be deter- mined by using the following equations; although the amount of accommodation factor of ‘‘r” should be chosen in a suitable way to achieve accurate results:

f

2¼f4 ð23-1Þ

f

5;6¼rf_7;8þ ð1rÞf_8;7 ð23-2Þ The corresponded equation for the upper wall can be written similarly:

f

7;8¼rf5;6þ ð1rÞf6;5 ð24-1Þ f

4¼f2 ð24-2Þ

Fig. 5.Streamlines and isotherms atKn= 0.02 for the states of no gravity (top),Gr= 100 (middle) andGr= 500 (bottom).

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4.3. Effects of buoyancy forces

The effects of mixed convection heat transfer through a microflow is studied using Boussinesq approximation which leads to have the buoyancy force asG¼bgðTTÞ.

The Boltzmann equation involved the external force of ‘‘F” is presented as follows:

@tfþðc:rÞf¼ ff^e

s

^f ^þ^F ^ð25Þ

where ‘‘F” based on the buoyancy forces can be written as F¼^G:ðcuÞRT f^e; which means,

fðxþcdt;c;tþdtÞ fðx;c;tÞ

¼ dt

2

s

^f ^f^ðx^þ^cdt;^c;^t^þ^dtÞ^f^e^ðx^þ^cdt;^c;^t^þ^dtÞ

dt

2

s

^f ^fðx;^c;^tÞ^f^e^ðx;^c;^tÞ

þdt

2Fðxþcdt;c;tþdtÞ þdt

2Fðx;c;tÞ ð26Þ

f

ðxþcdt;c;tþdtÞ fðx;c;tÞ

¼ dt

s

^f þ0:5dt fðx;c;tÞ f^eðx;c;tÞ

þ

s

fFdt

s

^fþ0:5dt ð27Þ

Applied the expression off_i¼f_iþ0:5dt=

s

fðf_if^e_iÞ 0:5dtF.

Now discretized form of Eq.(27)is illustrated as:

f

iðxþcidt;tþdtÞ f_iðx;tÞ ¼ dt

s

^fþ0:5dt f_if^e_i

þ dt

s

^f

s

fþ0:5dt

3Gðciy

v

^Þ

c² f^e_i

ð28Þ

0.5 0.25 0.05

2 6

10 16 20 24

0.5

0.05 0.3

0.05 0.3 0.5

2 8 12 16 18 24

0.8 0.1

0.8 0.3

0.6

0.3 0.1

0.6 0.5

2 8 12 16 20 22

Fig. 6.Streamlines and isotherms atKn= 0.005 for the states of no gravity (top),Gr= 100 (middle) andGr= 500 (bottom).

X

0 1 2 3 4 5

s

0.0 0.2 0.4 0.6 0.8 1.0

No gravity, kn=0.005, Re=1, Pr=0.7 No gravity, kn=0.010, Re=1, Pr=0.7 No gravity, kn=0.020, Re=1, Pr=0.7

Fig. 7.Slip velocity along the microchannel walls.

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f_i¼

s

ff_iþ0:5dtf^ei

s

^fþ0:5dt þ 0:5dt

s

f

s

^f þ0:5dt

3Gðciy

v

^Þ

c² f^e_i

ð29Þ

And finally the macroscopic parameters, included the gravity effects:

q

¼X

i

f

i ð30-1Þ

u¼^ð1=

q

^Þ^X

i f

icix ð30-2Þ

v

^¼^ð¹⁼

q

^Þ^X

i f

iciyþdt

2G ð30-3Þ

Moreover the hydrodynamic boundary conditions for the inlet involved gravity:

f

1þf₅þf₈¼

q

in ðf0þf₂þf₃þf₄þf₆þf₇Þ ð31-1Þ f

1þf5þf8¼

q

inuinþ ðf3þf6þf7Þ ð31-2Þ f

5f8¼

q

in

v

ⁱⁿ^{þ ðf}²^þ^f⁴^f⁶^þ^f⁷^Þ^dt₂

q

inG ð31-3Þ where

q

in¼f₀þf₂þf₄þ2ðf3þf₆þf₇Þ

1uin ð32Þ

and f

1f

e

1¼f3f

e

3)f1¼f3f

e 3þf

e

1 ð33Þ

Now, from Eq. (13) together with Eqs.(31-2),(31-3)and Eq.

(34):

X U_s

(Lower wall)

0.0 0.2 0.4 0.6 0.8 1.0

No gravity, kn=0.005, Re=1, Pr=0.7 Gr=100, kn=0.005, Re=1, Pr=0.7 Gr=500, kn=0.005, Re=1, Pr=0.7

X

0 1 2 3 4 5

U_s (Upper wall)

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 8.Slip velocity along the microchannel upper and lower walls atKn= 0.005.

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f

1¼f3þ2

3

q

inuin ð34Þ

f

8¼f6f₄f₂ 2 þ1

6

q

inuw1

2

q

in

v

ⁱⁿ^þ^dt₄

q

inG f

5¼f₇þf₄f₂ 2 þ1

6

q

inuinþ1

2

q

in

v

ⁱⁿ^dt₄

q

inG ð35Þ Through a same approach for the outlet hydrodynamic boundary conditions included buoyance forces, we will have:

f

3¼f1²₃

q

outuout

f

7¼f5¹₂ðf4f2Þ ¹₆

q

outuout¹₂

q

out

v

^out^þ¹4dt

q

outG f

6¼f8þ¹₂ðf4f2Þ ¹₆

q

outuoutþ¹₂

q

out

v

^out¹4dt

q

outG

ð36Þ

Slip velocity along walls of the microchannel considering ‘‘G”

can be demonstrated as follows; while it should be necessary to satisfy a constraint presented in Eq.(37)(For example for the lower wall),

f

2þf5þf6¼

q

w

v

^w^{þ ðf}⁴^þ^f⁷^þ^f⁸^Þ^dt₂

q

wG ð37Þ

f

2¼f₄¹₂dt

q

^G

f

5¼rf₇þ ð1rÞf8 f

6¼rf₈þ ð1rÞf7

ð38Þ

The upper and lower walls of the microchannel are imposed to the constant heat flux. General purpose thermal boundary condition (GPTBC) is applied to simulate the thermal boundary condi-

X U

_s

(Lower wall)

0.0 0.2 0.4 0.6 0.8 1.0

X

0 1 2 3 4 5

U

_s

(Upper wall)

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 9.Slip velocity along the microchannel upper and lower walls atKn= 0.02.

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tions along the microchannel walls. That model was presented by D’Orazio et al.[24,25] according to the non-equilibrium bounce back boundary condition of He & Zou [27,28]. In GPTBC, the unknown distribution functions are supposed to have the equilibrium functions with a counter slip thermal energy density of

q

^e⁰^;

Then it’s value is selected in a way to satisfy the constraints.

g

4;7;8¼

q

ð^eþe⁰Þ g^e_4;7;8=

q

^e^and

q

^e⁰¼2

q

^eþ1:5DtX

i

fiZi3K ð39Þ

Kshows the sum of six known distribution functions of ‘‘g” at the neighbor points and e is the imposed thermal energy density.

Now the heat flux in the form of Boltzmann equation and based on the GPTBC model can be written as follows; However more details can be found in Refs.[23–25].

q¼ X

i

cigi

q

^eudt 2

X

i

cifiZi

!

s

g

s

^gþ0:5dt ð40Þ

Eq.(39)is used to determine the unknown distribution functions along these walls. For example for the upper wall:

X

i

ciygi¼0:5dtX

i

ciyfiZiþ

q

^e^N^V^wþ

s

^gþ0:5dt

s

g

qy ð41Þ

which leads to present the following model for the constant heat flux boundary condition along the microchannel walls in LBM configuration:

g

4¼ 1

1

3¹₂^V_c^wþ¹₂^V_c²^w2

2 4

3 5

ðg2þg₅þg₆Þ dt 2

X⁸

i¼1

ciy

cZif_i

q

^e^N^V_c^w

s

^gþ0:5dt

s

^g ^q^c^y

" #

x

⁴ ^1:5^þ^1:5^c⁴_c2^u^wþ4:5ðc4uwÞ²

c⁴ 1:5U²_wþV²_w c²

" #

ð42Þ

g

7;8¼ 1

1

3¹₂^V_c^wþ¹₂^V_c²^w2

2 4

3 5

ðg2þg5þg6Þ dt 2

X⁸

i¼1

ciy

cZifi

q

^e^N^V_c^w

s

gþ0:5dt

s

^g

q_y c

" #

x

⁷;8 3þ6c_7;8uw

c² þ4:5ðc7;8uwÞ²

c⁴ 1:5U²_wþV²_w c²

" #

ð43Þ And finally the local Nusselt number is achieved as follows:

Nux¼qyDH

DTk ¼DHð@T=@yÞw

TwTbalk ð44Þ

5. Grid study and validation

The values of slip velocity and outlet Nusselt number were examined at different lattice numbers such as 35035, 40040 and 45045 by a developed computer code in FORTRAN language to simulate the flow domain. Very little differences between the results corresponded to 40040 and 45045 were observed which could be ignored so that the lattice nodes of 40040 were found suitable for the following computations (seeTable 1).

Fig. 3illustrates the validation of present work versus those of Hooman & Ejlali[6]concerned a flow and heat transfer through a

microchannel at various amounts of Knand the suitable agree- ments can be observed between them. Moreover the non- dimensional temperature profiles ofhare presented inFig. 4versus those of Kavehpour et al.[7]at two different vertical cross sections along the microchannel for the state of Re= 0.01, Twall=10, Tinlet=1 and Knin= 0.01. This figure also implies the desirable accuracy of using developed code for the next simulations.

6. Results and discussions

Mixed convection heat transfer in a microchannel is investigated by using LBM. Aspect ratio of microchannel equals to 10

X U_s

-0.5 0.0 0.5 1.0 1.5

Upper wall, Gr=500, kn=0.005 Lower wall, Gr=500, kn=0.005

X

s

-0.5 0.0 0.5 1.0 1.5

X

0.0 0.5 1.0 1.5 2.0 2.5

U

U_s

-0.5 0.0 0.5 1.0 1.5

Fig. 10.Slip velocity along the microchannel upper and lower walls atGr= 500.

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while its horizontal walls imposed to the constant heat flux ofq⁰⁰₀. The cold air at Tienters to the microchannel from the left side with a constant velocity of ui. Three states are considered such as no gravity,Gr= 100 andGr= 500. At each one, the value of Knudsen number is chosen asKn= 0.005,Kn= 0.01 andKn= 0.02 respectively; while Reynolds number and Prandtl number are kept fixed atRe= 1 andPr= 0.7.

Fig. 5shows the streamlines and isotherms atKn= 0.02 for different values ofGr. At the state of no gravity (Gr= 0), the inlet flow

from the left side would be affected continuously by the imposed heat flux from the upper and lower walls. However streamlines are completely symmetry and smooth along the walls. Higher temperature of air through the microchannel is quite clear; Moreover isotherms are also completely symmetry to the horizontal central line. MoreGrcorresponds to more influences of buoyancy forces which lead to downward streamlines especially around the entrance regions. The symmetry of isotherms is also becoming less along the length; insofar as Gr= 500 while a rotational long cell

U Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.04L, Kn=0.01, No gravity X=0.06L, Kn=0.01, No gravity X=0.08L, Kn=0.01, No gravity

U Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.04L, Kn=0.01, Gr=100 X=0.06L, Kn=0.01, Gr=100 X=0.20L, Kn=0.01, Gr=100

U

0.0 0.5 1.0 1.5 2.0

-1 0 1 2 3 4

Y

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 12.Horizontal velocity profiles at different cross sections of microchannel at Kn= 0.01.

U Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.04L, Kn=0.005, No gravity X=0.06L, Kn=0.005, No gravity X=0.08L, Kn=0.005, No gravity

U Y

0.0 0.2 0.4 0.6 0.8 1.0

U

0.0 0.5 1.0 1.5 2.0

-2 0 2 4

Y

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 11.Horizontal velocity profiles at different cross sections of microchannel at Kn= 0.005.

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dominates the flow at upper half of domain, isotherms would be affected more and show less symmetries. The effect of lower value ofKnon the streamlines and isotherms at various amounts ofGr can be observed inFig. 6. The same variations are seen except to generate the stronger and larger cell due to higher buoyancy forces.

Fig. 7illustrates the non-dimensional slip velocity along the microchannel walls in the absence of buoyancy forces. Usbegins from its highest value at the inlet and then decreases along the walls until where it approaches to a constant amount. It is

observed that largerKnleads to larger Us.Fig. 8implies the influence ofGron slip velocity atKn= 0.005. A noticeable fluctuation is seen at Gr= 500 along the diagrams of lower wall at 0 < X < 1 where a negative slip velocity is achieved along the upper wall;

which means the flow is in opposite direction of X in this area because of the generating the rotational cell.

MoreoverFig. 9presents the slip velocity at higher amount of Knudsen number asKn= 0.02. Although the positive effect ofKn on Uscan also be seen in this figure but plots are affected less by Grcompared with those of Fig. 8which means lower amplitude

θ

2 4 6 8

Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.04L, Kn=0.005, No gravity X=0.08L, Kn=0.005, No gravity X=0.10L, Kn=0.005, No gravity

θ

2 4 6 8

Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.04L, Kn=0.005, Gr=500 X=0.08L, Kn=0.005, Gr=500 X=0.10L, Kn=0.005, Gr=500

Fig. 13.Temperature profiles at different cross sections of microchannel atKn= 0.005.

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of the fluctuation of the lower wall besides less absolute amount of Usfor the upper wall. In the following, the amounts of Usalong the microchannel upper and lower walls at Gr= 500 are shown in Fig. 10for different values ofKn. This figure shows well the negative slip velocity existence along the upper wall at entrance region.

The horizontal dimensionless velocity profiles (U = u/ui) along various vertical cross sections of the microchannel are presented inFig. 11at Kn= 0.005 for different amounts ofGr. At the state of no gravity, the profiles are completely parabolic but with the U_maxa little less than U = 1.5 at horizontal central line (at Y = 0.5)

due to existence the slip velocity adjacent to the walls. MoreGr corresponds to more buoyancy forces which push downward the flow with X; as Umax1.8 at around Y = 0.3 forGr= 100 and also U_max4.5 at around Y = 0.25 forGr= 500. Moreover U_maxin opposite direction of X equals to 2.3 in recent state which clearly shows that how important the gravity effects might be.Fig. 12pre- sents U profiles at different cross sections atKn= 0.01. Less value of Umaxfor no gravity state at Y = 0.5 (due to generating higher slip velocity) and also less absolute amounts of Umaxin the same and opposite directions of X axis (due to generating stronger buoyancy

θ Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.1L, Kn=0.02, No gravity X=0.3L, Kn=0.02, No gravity X=0.5L, Kn=0.02, No gravity

θ

2 4 6 8

Y

0.0 0.2 0.4 0.6 0.8 1.0

X=0.1L, Kn=0.02, Gr=500 X=0.3L, Kn=0.02, Gr=500 X=0.5L, Kn=0.02, Gr=500

Fig. 14.Temperature profiles at different cross sections of microchannel atKn= 0.02.

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forces), are some of other results of this figure in comparison with those ofFig. 11concerned lower amount ofKn.

Dimensionless temperature profiles (h= T/Ti) along different vertical cross sections of the microchannel at Kn= 0.005 and Kn= 0.02 are shown inFig. 13andFig. 14, respectively. These fig- ures also represent the major effects of gravity acceleration on temperate profiles. More focus on thermal properties of domain, Fig. 15is presented to indicate the local Nusselt number along the lower wall at various amounts ofKnand Gr. At the state of no gravity, Nu_x starts from its maximum and then trends to decrease with X so that reaches to a corresponded constant amount along the microchannel wall. It is observed that lowerKn corresponds to higher Nu (see diagrams related to no gravity state of three plots ofFig. 15). This phenomenon can also be seen more clearly in the plots corresponded to Gr= 500 at Kn= 0.005, Kn= 0.01 and Kn= 0.02 compared to one another. Moreover the amplitude and number of fluctuations increase severely with Gr because of the strong cell generated by buoyancy forces.

7. Conclusion

Mixed convection of the air inside a microchannel was studied numerically by lattice Boltzmann method (TLBM-BGK). So that the following points could be addressed in brief:

1. A new case study concerned LBM performance was developed to consider both effects of natural and force convections in a microflow imposed to the heat flux boundary condition.

2. The suitable models for slip velocity and heat flux boundary conditions in terms of hydrodynamic and thermal distribution functions of LBM, were developed in order to consider the both effects of gravity and velocity inlet.

3. The symmetric forms of streamlines and isotherms were being less at higherGrwhile a rotational strong cell dominated the flow at upper half of domain. However these patterns would be affected more byGrat lowerKn.

4. LargerKnled to larger Us; Although the influence ofKnon Us

decreased at higher amounts ofGr. Moreover the negative slip velocity phenomenon due to the buoyance forces, was observed for the first time at present work.

5. At the state of no gravity, the profiles were parabolic but with Umaxa little less than U = 1.5 at horizontal central line due to existence the slip velocity adjacent to the walls. MoreGrpushed downward the flow with X; as Umax1.8 at around Y = 0.3 for Gr= 100 and also Umax4.5 at around Y = 0.25 for Gr= 500.

Moreover U_max in opposite direction of X equaled to2.3 in recent state which clearly showed that how important the gravity effects might be.

6. LowerKncorresponded to higher Nu. Moreover the amplitude and number of fluctuations of Nu, increased severely withGr because of the strong cell generated by buoyancy forces.

7. Eventually it is seen that the influence of gravity would be important at Kn< 0.01; Hence it should be included at mentioned range to simulate a microflow under the constant heat flux.

Conflict of interest

There is no conflict of interest.

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