Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection

    Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection M. Sheikholeslami, Ali J. Chamkha PII: DOI: Reference:

S0167-7322(16)32297-8 doi: 10.1016/j.molliq.2016.11.001 MOLLIQ 6541

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

17 August 2016 29 October 2016 1 November 2016

Please cite this article as: M. Sheikholeslami, Ali J. Chamkha, Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.11.001

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ACCEPTED MANUSCRIPT Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection

c

Mechanical Engineering Department, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia

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b

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Department of Mechanical Engineering, Babol University of Technology, Babol, IRAN

Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia

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a

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M. Sheikholeslami a, Ali J. Chamkha b,c, 1

Abstract

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Magnetohydrodynamic nanofluid forced convective heat transfer is investigated considering Marangoni convection. A two-phase model is selected for modeling of a

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nanofluid. The Runge-Kutta integration scheme is utilized to solve this problem.

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Influences of the Marangoni ratio, Schmidt number, Brownian motion parameter, magnetic number and thermophoretic parameter on the hydrothermal characteristics are

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presented. Results depict that the temperature augments with increases of the Schmidt number, Brownian motion, magnetic number and the thermophoretic parameters but it reduces with the rise of the Marangoni ratio. As the Marangoni ratio augments, the hydraulic boundary layer thickness enhances. Keywords: Marangoni convection; Nanofluid; Lorentz force; Magnetohydrodynamic; Two-phase model.

Nomenclature B0



Magnetic induction

C

Similarity independent variable



Dimensional concentration

Dimensionless concentration

1

Corresponding Author: Emails: [email protected] (Ali J. Chamkha) , [email protected], [email protected] (M. Sheikholeslami)

1

ACCEPTED MANUSCRIPT DB

Mass



Surface tension



Thermal diffusivity

diffusivity

Ha

Hartmann

M



Magnetic



ratio



Prandtl number Schmidt

Fluid density

Subscripts

number

C

Solutal quantity

0

at   0

Greek symbols



at   

el

T

Thermal quantity

T

Fluid

MA

Sc

temperature

NU

Pr

Dimensionless

SC

Marangoni

Dynamic viscosity

RI

number r

PT

number

Vertical and

TE

v ,u

D

temperature

horizontal

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components

Electrical conductivity

1. Introduction

Marangoni convection appears due to surface tension gradients [1]. Liquid–liquid interfaces can generate Marangoni boundary layer. Newly, innovative kinds of fluids are required to reach more efficient performance. A nanofluid was proposed as innovative way to enhance heat transfer. Alsabery et al. [2] applied heatline analysis for the simulation of nanofluid conjugate free convection. They concluded that the Nusselt number augments with the rise of the wall thickness. Sheikholeslami and Ganji [3] presented various applications of nanofluids in their review paper. Nasrin et al. [4] 2

ACCEPTED MANUSCRIPT analyzed nanofluid free convection heat transfer in a chamber. They showed that the uncertainties of heat transfer change with the variation of the nanofluid volume fraction. Qing et al. [5] examined the entropy production of a Casson nanofluid over a

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porous stretching plate in the presence of Lorentz forces. They utilized the Chebyshev spectral collocation method. Bhatti et al. [6] investigated the fluid flow over a porous

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stretching plate using SLM. Slip motion impact on EMHD nanofluid flow over a sheet

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has been examined by Ayub et al. [7]. They presented impacts of active parameters on the Sherwood number. Bhatti and Rashidi [8] presented the impact of thermo-diffusion on

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a Williamson nanofluid over a sheet. Parvin et al. [9] studied Al2O3-water nanofluid free convection in a curved cavity. Selimefendigil and Oztop [10] examined nanofluid

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conjugate conduction-convection mechanism in a titled cavity. They proved that the temperature gradient augmented with the increase of the Grashof number. Sheikholeslami et al. [11] examined the non-uniform viscosity impact on natural

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convection of magnetic nanofluid. They illustrated that Nusselt number reduces with rise

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of viscosity parameter. Nanofluid mixed convection on a cone has been reported by Chamkha et al. [12]. They presented radiation impact in their article. Sheikholeslami et al.

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[13] simulated about the influence of Lorentz forces on nanofluid radiative heat transfer. They depicted that temperature gradient decrease with augment of magnetic field. Influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [14]. He concluded that improvement in heat transfer reduces with the rise of the Kelvin forces. Reddy et al. [15] utilized a two-phase model simulation of mixed convection over a plate. The problem of a wavy duct in existence of Brownian forces has been examined by Shehzad et al. [16]. They selected the Nelder-Mead method to find the solution. Nanofluid behaviors in various applications have been investigated by several researchers [17-29]. The aim of this article is to study the influence of magnetic field on a nanofluid with the existence of Marangoni convection. The Runge-Kutta integration scheme is chosen to simulate this problem. The influences of the active parameters on the hydrothermal treatment are examined.

3

ACCEPTED MANUSCRIPT 2. Problem statement Magnetohydrodynamic Marangoni boundary layer flow of a nanofluid is studied

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using a two-phase model. The influence of the induced magnetic field and the Hall effects are neglected [30]. The interface temperature is considered as a function of x. Fig. 1

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depicts the interface condition and the velocity components. The governing equations for

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this investigation are as follows:

u u  2u B 2 u   2  0 el u y x y f

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v

NU

v u  0 y x

(2)

(3)

C C  2C  DT  d 2C  v  DB    x y y 2  T 0   dy 2 

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u

TE

D

2  dC dT  T T  2T  c  p   dT    D B  v u  2  .   (DT / T 0 )    y x y  c f   dy dy   dy    

(1)

(4)

 is the surface tension and it is a function of temperature and concentration [1]:

   0 1   C C  C    T T T   C  

1   0 C

, T  

T

1   0 T

(5) (6)

, C

The corresponding boundary conditions are as follows [1]:

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ACCEPTED MANUSCRIPT C  x ,    C  , u  x ,    0, T  x ,    T  ,

(7)

C  x , 0  C   C 0X , X  x / L 2

T  x , 0   T  T 0 X 2 , u y

 y 0

y 0

 C   0  C  x 

 T y 0

T x

  y 0 

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v  x , 0   0,

 x

PT



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In order to transform the above equations in self-similar form, the following equations are defined.

(8)

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C  x , y   C 0 X 2    C  T  x , y   T 0 X 2    T 

  x , y    X f   ,   y / L

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u   / y ,v   / x

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By using Eq. (8), the final dimensionless equations and conditions are given by: (9)

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f   ff   f 2  Mf   0

   Pr  f    2f    Nb    Nt  2  0

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(10)

   Sc  f    2f   

(11)

Nt    0 Nb

  0   1,   0   1, f  0   0, f   0   2 1  r  ,

(12)

f      0,      0,      0

r can be defined as: r

C 0 C T 0 T

(13)

 C p  p D BC 0 X  C p  p DT X   and Pr  , Nb  , Sc  , Nt   D  C p    C p   2

f

2

and

f

Ha  B 0 L el /  ,M=Ha 2 are the Prandtl number, Brownian motion parameter, Schmidt

5

ACCEPTED MANUSCRIPT number, thermophoretic parameter and the Hartmann number (magnetic number), respectively. L is a reference length and is defined as in [31]:

r

can

(14)

be

introduced

r  MaC / MaT

as

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  0T 0 T

where MaC   0 C C 0 L /  

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L 

3. Numerical Runge-Kutta method

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and MaT   0T T 0 L /   .

In the Runge-Kutta method, at first, the following definitions are applied:

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x 2  f , x 1   , x 3  f  , x 4  f , x 5   , x 6   , x 7   , x 8    .

are:

The final system and initial conditions

(15)

0   x1       0  x 2    x   u 1   3   x 4   2 1  r    x    1   5    x 6   u2     1  x7    x   u 3  8  

(16)

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TE

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x   1   1     x  x3   2     x x   4   3   2    x x  Mx  x 3 2 3 4 x    4   x6  x     5    Pr  x x  2x x   Nt x 2  Nbx x  2 6 3 5 6 6 8 x      6   x8   x    Nt 7   Sc  x 2x 8  2x 3x 7   x 6   x    Nb   8 

6

ACCEPTED MANUSCRIPT Eqs. (15) and (16) are solved utilizing the 4th order Runge-Kutta method. According to f      0,      0,      0 , unknown initial conditions can be obtained by the

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Newton’s method.

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4. Results and discussion

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Magnetohydrodynamic nanofluid hydrothermal behavior is examined considering Marangoni convection. Two-phase model is taken into account. The Runge-Kutta

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integration scheme is utilized to solve this problem. The MAPLE code has been validated by a comparison with the results of a previously published paper [1]. Fig. 2(a) indicates

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good accuracy of the present code. Fig. 2(b) shows the good agreement between the exact and numerical solutions. The concentration, temperature and the velocity profiles for

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several values of the Marangoni ratio, thermophoretic parameter, Brownian motion

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parameter, magnetic number and the Schmidt number are presented. Fig. 3 depicts the impact of the Lorentz forces on the flow, mass and heat transfer.

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As the Lorentz forces augment, a back flow appears and in turn, both components of velocity reduce. The hydraulic boundary layer thickness reduces in existence of the magnetic field. The influence of the Lorentz forces on the normal velocity is more pronounced than on the tangential velocity. Also, the temperature gradient near the hot surface reduces with augmentation of the Hartmann number. The concentration augments with the augmentation of the Lorentz forces. The influence of the Marangoni ratio on the temperature, concentration and the velocity distributions is depicted in Fig. 4. Both the tangential and the normal velocity enhance with the augmentation of the Marangoni ratio. The temperature reduces with the increase of this parameter. The influence of the Marangoni ratio on the concentration is similar to that obtained for the temperature. Fig. 5 illustrates the impact of the Schmidt number on the concentration and the temperature distributions. The temperature reduces with the rise of the Schmidt number but the opposite trend is seen for the concentration. The influences of the thermophoretic and the Brownian motion parameters on the concentration and the temperature 7

ACCEPTED MANUSCRIPT distributions are illustrated in Figs. 6 and 7. As these parameters rise, the temperature enhances. The concentration enhances with the augmentation of the thermophoretic

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parameter but it reduces with increase of the Brownian motion. The Nusselt number for various values of the active parameters is depicted in Figs.

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8 and 9 and Tables 1 and 2. According to these data, a correlation is presented for the

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Nusselt number as follows:

Nu  6.22372  0.31371 M  1.4172 r  0.039731Sc  0.858Nb  4.33Nt 3

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0.0129M r  2.66 10 M Sc  0.046096M Nb  0.23M Nt  0.079 r Sc

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0.0183Sc 2  0.0758 Nb 2  1.0896 Nt 2

NU

0.0298 r Nt  0.20721Sc Nb  1.794 Nt Nb  6.21 10 3 M 2  0.078 r 2

It is predicted that the Nusselt number enhances with the augmentation of the Marangoni ratio. Enhancing the Lorentz forces causes the Nusselt number to augment due to the

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increment in the boundary layer thickness. Impacts of the thermophoretic parameter,

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Hartmann number.

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Schmidt number and the Brownian motion parameter on Nu are similar to that of the

5. Conclusion

MHD nanofluid hydrothermal flow in the presence of Marangoni convection is studied using a two-phase model. The Runge-Kutta integration method is selected to solve this problem. The impacts of the Marangoni ratio, thermophoretic parameter, Schmidt number, Brownian motion parameter and the magnetic number on hydrothermal characteristics are examined. The results indicate that the temperature gradient augments with the rise of the Marangoni ratio but it reduces with the increase of the other active parameters. As the Lorentz force increases, the hydraulic boundary layer thickness reduces but the opposite tendency is shown for the concentration and the temperature boundary layer thicknesses.

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ACCEPTED MANUSCRIPT References [1] E. Magyari, A.J. Chamkha, International Journal of Thermal Sciences 47 (2008) 848–

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857 [2] A.I. Alsabery, A.J. Chamkha, H. Saleh, I. Hashim, International Journal of Heat and

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Mass Transfer, Volume 100, September 2016, Pages 835-850

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[3] M. Sheikholeslami, D.D. Ganji, Journal of the Taiwan Institute of Chemical Engineers, Volume 65, August 2016, Pages 43-77

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[4] R. Nasrin, M.A. Alim, A.J. Chamkha, Prandtl number and aspect ratio, International

7355-7365

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Journal of Heat and Mass Transfer, Volume 55, Issues 25–26, December 2012, Pages

[5] J. Qing, M.M. Bhatti, M. Ali Abbas, Mohammad Mehdi Rashidi and Mohamed El-

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Sayed Ali, Entropy 18 (4), 123

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[6] M. M. Bhatti, T. Abbas, M. M. Rashidi, Iranian Journal of Science and Technology,

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Transactions A: Science, 1-7

[7] M. Ayub, T. Abbas, M. M. Bhatti, The European Physical Journal Plus 131 (6), 1-9. [8] M.M. Bhatti, M.M. Rashidi, Journal of Molecular Liquids, 221 (2016) 567-573. [9] S. Parvin, R. Nasrin, M.A. Alim, N.F. Hossain, A.J. Chamkha, International Journal of Heat and Mass Transfer 55 (2012) 5268-5274 [10] F. Selimefendigil, H. F. Öztop, Journal of Molecular Liquids 216 (2016) 67-77. [11] M. Sheikholeslami, M.M. Rashidi, T. Hayat, D.D. Ganji, Journal of Molecular Liquids 218 (2016) 393–399. [12] A. J. Chamkha, S. Abbasbandy, A.M. Rashad and K. Vajravelu, Meccanica 48 (2013) 275-285. [13] M. Sheikholeslami, T. Hayat, A. Alsaedi, International Journal of Heat and Mass Transfer 96 (2016) 513–524 9

ACCEPTED MANUSCRIPT [14] M. Sheikholeslami Kandelousi, The European Physical Journal Plus (2014) 129248.

Journal of Heat and Mass Transfer 64 (2013) 384-392

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[15] Ch. RamReddy, P.V.S.N. Murthy, A. J. Chamkha, A.M. Rashad, International

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[16] N. Shehzad, A. Zeeshan, R. Ellahi, K. Vafai, Journal of Molecular Liquids 222

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(2016) 446-455.

[17] M. Sheikholeslami, K. Vajravelu, Mohammad M. Rashidi, International Journal of

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Heat and Mass Transfer 92 (2016) 339–348

[18] M. Sheikholeslami, S. Abelman, IEEE Transactions on Nanotechnology 14 (3)

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(2015) 561-569

[19] M. Sheikholeslami Kandelousi, Physics Letters A 378(45) (2014) 3331-3339.

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[20] M. M. Bhatti, T. Abbas, M. M. Rashidi, M. El-Sayed Ali, Entropy 18 (6), 200

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[21] M. M. Bhatti, T. Abbas, M. M. Rashidi, M. El-Sayed Ali, Z. Yang, Entropy 18 (6),

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224

[22] M. Sheikholeslami, T. Hayat, A. Alsaedi, International Journal of Heat and Mass Transfer, In Press, 2016, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.077 [23] M. Sheikholeslami, D.D. Ganji, Journal of Molecular Liquids 224 (2016) 526–537 [24] M. M. Bhatti, M. M. Rashidi, International Journal of Applied and Computational Mathematics, 1-15 [25] A. J. Chamkha, S. Abbasbandy, A.M. Rashad and K. Vajravelu, Transport in Porous Media, 91 (2012) 261-279. [26] Mohsen Sheikholeslami, A. J. Chamkha, Part A, 69 (2016) 1186–1200, DOI:10.1080/10407782.2015.1125709 [27] M. Sheikholeslami, A. J. Chamkha, NUMERICAL HEAT TRANSFER, PART A, 2016, VOL. 69, NO. 7, 781–793, http://dx.doi.org/10.1080/10407782.2015.1090819 10

ACCEPTED MANUSCRIPT [28] M. Sheikholeslami, H.R. Ashorynejad, P. Rana, Journal of Molecular Liquids 214 (2016) 86–95

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[29] M. Sheikholeslami, International Journal of Hydrogen Energy, (2016), DOI

[30] K.R. Cramer, S.-I. Pai, Washington, DC, 1973.

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information: 10.1016/j.ijhydene.2016.09.185

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[31] C. Golia, A. Viviani, L’Aerotechnica Missili e Spazio 64 (1985) 29–35.

11

MA

NU

SC

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PT

ACCEPTED MANUSCRIPT

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TE

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Fig. 1. Physical model of the problem

12

ACCEPTED MANUSCRIPT

6 Present work

M =3

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Magyari and Chamkha Present work Magyari and Chamkha

M =6

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4.5

SC

Nu

2

3

4

5

MA

1.5

NU

3

6

7

8

Pr

D

(a)

1.6

TE

1.4

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1.2 1

f

0.8 0.6 0.4

NM Exact solution

0.2 0

0

1

2



3

4

5

(b) Fig. 2. (a) Comparison between the present results with that of obtained by Magyari and Chamkha [1]; (b) Comparison between the numerical and exact solution results when M=0.

13

2

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ACCEPTED MANUSCRIPT

2.4

M= 1 M= 5 1.6

2

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M = 10 M = 20

M= 1 M= 5 M = 10 M = 20

1.6

1.2

SC

f ' 1.2

f 0.8

0.4

0

1

2



3

4

1.2

M= 5 M = 10 M = 20

0

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0.8

0

1

2

0



3

1

2



3

4

5

(b) 1

M= 1

TE

M= 1

0.4

0

5

D

(a)



0.4

MA

0

NU

0.8

M= 5 0.8

M = 10 M = 20

0.6

 0.4

0.2

4

5

0

0

1

(c)

2



3

4

5

(d)

Fig. 3. Effect of magnetic number on velocity, temperature and concentration profiles when r  1, Sc  1, Nb  1, Nt  0.001,Pr  10 .

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ACCEPTED MANUSCRIPT 2

5

r= 1 r= 2 4

1.6

r= 3 r= 4

3

1.2

f

r= 1 r= 2

0.8

2

r= 3

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r= 4

1

1

2



3

4

5

NU

0

SC

0.4

0

PT

f

'

(a)

0

1

2



3

4

1

r= 1

r= 1

r= 2

r= 2

0.8

0.8

r= 3

r= 3 r= 4

r= 4

0.6

D

0.6



TE

 0.4

0 0

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0.2

0.2

0.4

0.6



0.8

5

(b)

MA

1

0

0.4

0.2

1

1.2

0

0

1

(c)

2



3

(d)

Fig. 4. Effect of r on velocity, temperature and concentration profiles when M  1, Sc  1, Nb  1, Nt  0.001,Pr  10 .

15

4

5

ACCEPTED MANUSCRIPT 1

1

0.8

0.8

0.6

0.6



Sc = 1 0.4

Sc = 2

PT



0.4

Sc = 3 0.2

0.2

SC

0

Sc = 2 Sc = 3 Sc = 4

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Sc = 4

Sc = 1

0

0.4

0.6



0.8

1

0

1.2

1

NU

0.2

(a)

2



3

4

(b)

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Fig. 5. Effect of Schmidt number on temperature and concentration profiles

TE

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when M  1, r  1, Nb  1, Nt  0.001,Pr  10 .

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0

16

5

ACCEPTED MANUSCRIPT 1

1

0.8

0.8

0.6

0.6



Nb = 0.01 0.4

Nb = 0.1

PT



0.4

Nb = 0.5 Nb = 1 0.2

0.2

0.4

0.6



0.8

1

0

1.2

NU

0

SC

0

(a)

Nb = 0.1 Nb = 0.5 Nb = 1

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0.2

Nb = 0.01

0

1

2



3

4

(b)

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Fig. 6. Effect of Brownian motion parameter on temperature and concentration

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TE

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profiles when M  1, r  1, Sc  1, Nt  0.0001,Pr  10 .

17

5

ACCEPTED MANUSCRIPT 1

Nt = 0.0001 Nt = 0.001 Nt = 0.01

0.8

Nt = 0.1 0.6

PT



SC

0.2

RI

0.4

0

1

NU

0

(a)

2



3

4

5

(b)

MA

Fig. 7. Effect of thermophoretic parameter on temperature and concentration profiles

AC CE P

TE

D

when M  1, r  1, Sc  1, Nb  1,Pr  10 .

18

ACCEPTED MANUSCRIPT 7

9

r= 1 r= 2 r= 4

7

Sc = 1 Sc = 2 Sc = 4

6

Nu

Nu

5

PT

8

6 4

RI

5 3

1

5

10

15

2

20

Sc  1, Nb  1, Nt  0.001,Pr  10

NU

M

1

5

MA

D TE 19

10

15

M

r  1, Nb  1, Nt  0.001,Pr  10

Fig. 8. Effect of M , r , Sc on Nusselt number.

AC CE P

3

SC

4

20

SC

RI

PT

ACCEPTED MANUSCRIPT

r  2.5, Nb  0.5005, Nt  0.05005

AC CE P

TE

D

MA

NU

Sc  2.5, Nb  0.5005, Nt  0.05005

r  2.5, Sc  2.5, Nt  0.05005

r  2.5, Sc  2.5, Nb  0.5005

M  10.5, Nb  0.5005, Nt  0.05005

M  10.5, Sc  2.5, Nt  0.05005

20

SC

RI

PT

ACCEPTED MANUSCRIPT

M  10.5, r  2.5, Nt  0.05005

AC CE P

TE

D

MA

NU

M  10.5,, Sc  2.5, Nb  0.5005

M  10.5, r  2.5, Nb  0.5005

M  10.5, r  2.5, Sc  2.5

Fig. 9. Contour plots for effect on active parameters on Nusselt number when Pr  10 .

21

ACCEPTED MANUSCRIPT Table1. Effect of M , Nt on Nusselt number when r  1, Sc  1, Nt  0.001,Pr  10

0.1

1

7.070521

6.990635

5

5.644733

5.590999

5.079263

10

4.636935

4.599673

4.242494

20

3.485774

3.461264

3.225136

RI

AC CE P

TE

D

MA

NU

M

PT

0.01

SC

Nb

22

1

6.235772

ACCEPTED MANUSCRIPT Table2. Effect of M , Nb on Nusselt number when r  1, Sc  1, Nb  1,Pr  10 Nt 0.001

1

6.236324

6.235772

5

5.079616

5.079263

10

4.242724

4.242494

4.21738

20

3.225285

3.225136

3.208817

SC

RI

PT

0.0001

AC CE P

TE

D

MA

NU

M

23

0.1

6.175476 5.040762

ACCEPTED MANUSCRIPT Highlights:

TE

D

MA

NU

SC

RI

PT

Nanofluid heat transfer is studied considering Marangoni convection. Runge-Kutta method used to solve the governing equations. Effect of Lorentz forces on flow and heat transfer is studied. Nusselt number increases with increase of Marangoni ratio. Nusselt number decreases with increase of M , Nt , Nb , Sc .

AC CE P

    

24