Interaction of magnetic field in flow of Maxwell nanofluid with convective effect

Author’s Accepted Manuscript Interaction of magnetic field in flow of Maxwell nanofluid with convective effect T. Hayat, Taseer Muhammad, S.A. Shehzad, G.Q. Chen, Ibrahim A. Abbas www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(15)30030-5 http://dx.doi.org/10.1016/j.jmmm.2015.04.019 MAGMA60078

To appear in: Journal of Magnetism and Magnetic Materials Received date: 27 February 2015 Revised date: 21 March 2015 Accepted date: 5 April 2015 Cite this article as: T. Hayat, Taseer Muhammad, S.A. Shehzad, G.Q. Chen and Ibrahim A. Abbas, Interaction of magnetic field in flow of Maxwell nanofluid with convective effect, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.04.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Interaction of magnetic field in flow of Maxwell nanofluid with convective effect T. Hayat a, b , Taseer Muhammad a ,* , S.A. Shehzad c , G.Q. Chen b, d and Ibrahim A. Abbas b a

Department of Mathematics, Quaid-I-Azam University 45320 , Islamabad 44000 , Pakistan

b

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80203 , Jeddah 21589 , Saudi Arabia

c

Department of Mathematics, Comsats Institute of Information Technology, Sahiwal 57000 , Pakistan

d

Laboratory of Systems Ecology, College of Engineering, Peking University, Beijing 100871, China Corresponding author E-mail: [email protected] (Taseer Muhammad)

Abstract: Magnetohydrodynamic (MHD) three-dimensional flow of Maxwell nanofluid subject to the convective boundary condition is investigated. The flow is generated by a bidirectional stretching surface. Thermophoresis and Brownian motion effects are present. Fluid is electrically conducted in the presence of a constant applied magnetic field. Unlike the previous cases even in the absence of nanoparticles, the correct formulation for the flow of Maxwell fluid in the presence of a magnetic field is established. Newly proposed boundary condition with the zero nanoparticles mass flux at the boundary is employed. The governing nonlinear boundary layer equations through appropriate transformations are reduced in the nonlinear ordinary differential system. The resulting nonlinear system has been solved for the velocities, temperature and nanoparticles concentration distributions. Convergence of the constructed solutions is verified. Effects of emerging parameters on the temperature and nanoparticles concentration are plotted and discussed. Numerical values of local Nusselt number are computed and analyzed. It is observed that the effects of magnetic parameter and the Biot number on the temperature and nanoparticles concentration are quite similar. Both the temperature and nanoparticles concentration are enhanced for the increasing value of magnetic parameter and Biot number. Keywords: Three-dimensional flow; Maxwell fluid; MHD; Nanoparticles; Convective boundary condition.

1

1. Introduction The investigations on non-Newtonian fluids are remarkably enhanced during the past few decades because of their practical implications in technology and industrial processes. Many of the materials in our daily life include apple sauce, sugar solution, muds, chyme, soaps, emulsion, shampoos, blood at low shear rate etc. exhibits the characteristics of non-Newtonian fluids. In the literature, there is no single relation that characterizes all the properties of non-Newtonian fluids which characterize all the properties of such materials. Many models of non-Newtonian fluids are developed by the researchers in the past. Among these models, Maxwell fluid is a simplest subclass of rate type non-Newtonian fluids. This model is widely used to explore the effects of stress relaxation. The involvement of stress relaxation in the stress tensor of Maxwell fluid makes it highly nonlinear and complicated in comparison to Newtonian fluid. Maxwell fluid model reduced into the simple Navier-Stokes relation when extra stress time is zero. The boundary layer flows of viscoelastic non-Newtonian fluids have been widely used in engineering technology and industrial applications. Such flows commonly involved in power engineering and food engineering, petroleum production, polymer solutions and in polymer melt, the cooling of a metallic plate in a cooling bath, drawing on plastic films and many others. Abundant studies on this topic exist in the literature, but few interesting and recent studies can be seen in the refs. [1  8].

Nowadays, the cooling of electronic devices is the major industrial requirements due to the fast technology, but the low thermal conductivity rate of ordinary base fluids includes water, ethylene glycol and oil is the basic limitation. To overcome on such limitation, the nanoscale solid particles are submerged into host fluids which change the thermophysical characteristics of these fluids and enhanced the heat transfer rate dramatically. Choi [9] was the first who identified this colloidal suspension. The recent developments in nanofluids and their mathematical modelling, play vital role in industrial and nanotechnology. The nanofluids are used in the applications such as cooling of electronics, heat exchanger, nuclear reactor safety, hyperthermia, biomedicine, engine cooling, vehicle thermal management and many others. Further the magneto nanofluids are useful in the manufacturing processes of industries and biomedicine applications. Examples include in gastric medications, biomaterials for wound treatment, sterilized devices, etc. The magneto nanoparticles can be employed in the elimination of tumors with hyperthermia, targeted drug release and for

2

magnetic resonance imaging. A bulk of research articles on nanofluids is available in the literature in which few can be seen in the refs. [10  20]. This paper emphasizes on the three-dimensional boundary layer flow of Maxwell fluid induced by a bidirectional stretching surface. Thermophoresis and Brownian motion effects are encountered in the energy and mass species expressions. We considered the thermal convective [21, 22] and zero nanoparticles mass flux conditions at the boundaries. The zero nanoparticles mass flux condition was first introduced by Kuznetsov and Nield [15] for two-dimensional boundary layer flows. Here we used this condition for the three-dimensional boundary layer flow of Maxwell nanofluid. The governing nonlinear ordinary differential equations are solved via homotopy analysis method [23  30]. The obtained results are sketched and discussed in detail. The values of local Nusselt number are tabulated and examined.

2. Mathematical modeling Consider the steady three-dimensional flow of an incompressible Maxwell nanofluid over a bidirectional stretching surface. Fluid is considered electrically conducting in the presence of constant magnetic field B0 applied in the z  direction. The Hall and electric field effects are ignored. The induced magnetic field is not considered for a small magnetic Reynolds number. Thermophoresis and Brownian motion effects are taken into account. The temperature at the surface is controlled by a convective heating process which is characterized by the heat transfer coefficient h f and temperature of the hot fluid T f below the surface. The boundary layer expressions governing the conservations of mass, momentum, energy and nanoparticles concentration are

u v w    0, x y z

  u 2 xu2  v 2 yu2  w 2 z u2 u u u  2 u B02  u v  w  1     2   2uv  2u  2vw  2u  2uw  2u  x y z f  z xy yz xz   2

2

2

(1)

u    u  1 w , z  

(2)

  u 2 x v2  v 2 y v2  w 2 z 2v v v v  2 v B02  v    u  v  w  1  2   v  1 w ,  2v  2v  2v   x y z f  z  z   2uv xy  2vw yz  2uw xz 

(3)

2

2

2

T T T  2T (  c) p u v w  2   c  f x y z z 3

  T C  DT  T  2   DB  ,    z z  T  z    

(4)

u

  2C  D C C C v w  DB  2   T x y z  z  T

  2T   2 .  z 

(5)

The boundary conditions in the present problem are

u  ax, v  by, w  0,  k

T C DT T  h f T f  T , DB   0 at z  0, z z T z

(6)

u  0, v  0, T  T , C  C as z  ,

(7)

In above expressions u , v and w are the velocity components in the x, directions respectively, 1 the relaxation time,    /  f



y  and z 

the kinematic viscosity,  the

dynamic viscosity,  f the density of base fluid,  the electrical conductivity,

T the

temperature,   k /( c) f the thermal diffusivity of the fluid, k the thermal conductivity, ( c) f the heat capacity of the fluid, ( c) p the effective heat capacity of nanoparticles, DB the

Brownian diffusion coefficient, C the nanoparticles concentration, DT the thermophoretic diffusion coefficient, T the temperature far away from the surface and C  the nanoparticles concentration far away from the surface. Using the following transformations

u  axf ( ), v  ayg ( ), w  a   f ( )  g ( ) ,   T T C a 1/ 2  ( )  T f T ,  ( )  C ,     z.   1/ 2

(8)

Eq. 1 is automatically satisfied and Eqs. 2  7 have the following forms

 g   M

   1( f  g ) g   g

    2 f  g  g g    f  g  g   M

f   M 2   1 ( f  g ) f   f    2 f  g  f f    f  g  f   M 2 f   0,

(9)

g   0,

(10)

2



2

2

2

2

2



   Pr ( f  g )   Nb    Nt   0, 2

   Le Pr( f  g )  

Nt    0, Nb

(11) (12)

f  0, g  0, f   1, g   c,     1   0, Nb   Nt   0 at   0,

(13)

f   0, g   0,   0,   0 as   .

(14)

4

where  is the Deborah number, M is the magnetic parameter, c is the ratio of stretching rates, Pr is the Prandtl number, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter,  is the Biot number, Le is the Lewis number and prime stands for differentiation with respect to  . These parameters can be expressed by the following definitions:  c  p DB C  c  f 

  1a, M 2  Ba , c  ba , Pr   , Nb  2 0

f

Nt 

 c  p DT T f T  ,  c  f T

 

hf k

 a

, Le 

 DB

.

,    

(15)

The local Nusselt number Nu x is defined as

Nu x  

x T Tw  T  z

 Re x   (0). 1/ 2

(16)

z 0

It is noted that the dimensionless mass flux represented by a Sherwood number Shx is now identically zero and Re x  ux / is the local Reynolds number.

3. Series solutions The initial guesses and linear operators for homotopic solutions are

f 0 ( )  1  e  , g 0 ( )  c(1  e  ),  0 ( ) 

 1 

e  , 0 ( )  



Nt  e , 1   Nb

L f  f   f , L g  g   g , L      , L     .

(17) (18)

The above operators have the following properties



 

 



L f C1  C2 e  C3e   0, L g C4  C5 e  C6 e   0,       L C7 e  C8 e  0, L C9 e  C10e  0,  





(19)

in which C i (i  1  10) denote the arbitrary constants. The zeroth-order deformation problems can be stated as follows:





(1  p)L f fˆ ( , p)  f 0 ( )  p f N f [ fˆ ( , p), gˆ ( , p)],

(20)

(1  p)L g gˆ ( , p)  g 0 ( )  p g N g [ fˆ ( , p), gˆ ( , p)],

(21)

  (1  p)L ˆ( , p)   ( )  p N [ fˆ ( , p), gˆ ( , p),ˆ( , p),ˆ( , p)],

(22)

(1  p)L ˆ( , p)   0 ( )  p  N [ fˆ ( , p), gˆ ( , p),ˆ( , p),ˆ( , p)], 

0





5

(23)

fˆ (0, p)  0, fˆ  (0, p)  1, fˆ  (, p)  0, gˆ (0, p)  0,   gˆ  (0, p)  c, gˆ  (, p)  0, ˆ  (0, p)   1  ˆ(0, p) ,  ˆ(, p)  0, Nbˆ  (0, p)  Ntˆ  (0, p)  0, ˆ(, p)  0,



Nf











(24)

2 ˆ ˆ  3 fˆ 2 ˆ  gˆ  f   f  fˆ ( ; p), gˆ ( ; p)   M   1 f  3  2    2  2 fˆ  gˆ fˆ  f2ˆ  ˆ     2 f   , 2  3 fˆ   M    fˆ  gˆ  3   









N g gˆ ( ; p), fˆ ( ; p) 











2  3 gˆ 2 ˆ  gˆ  gˆ   gˆ   M   1 f  3  2    2  2 fˆ  gˆ gˆ  gˆ2  ˆ    2 g   , 2  3 gˆ   M     fˆ  gˆ   3  













2

(25)

2





(26)



ˆ ˆ  2ˆ ˆ  gˆ   Pr f ˆ N fˆ ( ; p), gˆ ( ; p),ˆ( , p), ˆ( , p)   Pr f      2   2

 ˆ  ˆ ˆ  ,  Nb Pr  Nt Pr      









2 ˆ ˆ  2ˆ ˆ  gˆ   Nt   . N  fˆ ( ; p), gˆ ( ; p),ˆ( , p), ˆ( , p)   Le Pr f  Nb  2  2

Here p denotes the embedding parameter,  f ,

(27)

(28)

 g ,   and   the non-zero auxiliary

parameters and N f , N g , N  and N  the nonlinear operators. Putting p  0 and p  1 we have

fˆ (; 0)  f 0 ( ), fˆ (;1)  f ( ),

(29)

gˆ (; 0)  g 0 ( ), gˆ (;1)  g ( ),

(30)

ˆ( , 0)   0 ( ), ˆ( ,1)   ( ),

(31)

ˆ( , 0)  0 ( ), ˆ( ,1)   ( ).

(32)

When p varies from 0 to 1 then fˆ ( ; p), gˆ (; p), ˆ( , p) and ˆ( , p) vary from the initial guesses f 0 ( ), g 0 ( ),  0 ( ) and 0 ( ) to the final solutions f ( ), g ( ),  ( ) and

 ( ), respectively. The Taylor series expansion gives the following expressions: 6

 1  m fˆ ( , p) fˆ ( ; p)  f 0 ( )   f m ( ) p m , f m ( )  m! p m m 1 

gˆ ( ; p)  g 0 ( )   g m ( ) p m , g m ( )  m1



ˆ( , p)   0 ( )   m ( ) p m ,  m ( )  m1 

ˆ( , p)  0 ( )  m ( ) p m , m ( )  m1

1  m gˆ ( , p) m! p m

1  mˆ( , p) m! p m 1  mˆ( , p) m! p m

,

(33)

,

(34)

p 0

p 0

,

(35)

,

(36)

p 0

p 0

The convergence of Eqs. 33  36 strongly depends upon the suitable choices of  f ,  g ,   and   . Considering that  f ,  g ,   and   are chosen in such a manner that Eqs. (33)  (36) converge at p  1 then 

fˆ ( ; p)  f 0 ( )   f m ( ), m 1

(37)



gˆ ( ; p)  g 0 ( )   g m ( ), m1

(38)



ˆ( , p)   0 ( )   m ( ), m1

(39)



ˆ( , p)  0 ( )  m ( ).

(40)

m1

The general expressions of solutions with the special solutions f m  , g m  ,  m   and m   are f m ( )  f m ( )  C1  C2 e  C3e  ,

(41)

g m ( )  g m ( )  C4  C5e  C6 e  ,

(42)

 m ( )   m ( )  C7 e  C8e  ,

(43)

m ( )  m ( )  C9e  C10e  .

(44)

7

4. Convergence analysis The series solutions (37)  (40) contain the auxiliary parameters  f ,  g ,   and   . These parameters are useful in adjusting and controlling the convergence of the obtained series solutions. The proper values of these parameters are quite essential to construct the convergent solutions through the homotopy analysis method (HAM). To choose the suitable values of  f ,  g ,   and   , the   curves are drawn at 15th order of approximations. Figs. 1 and 2 show that the convergence region lies within the domain

 1.80   f  0.35,

 1.90   g  0.20,

 1.65     0.15 and  1.70     0.10. Further the presented solutions are convergent in

the whole domain when  f   g  1.0       . Table 1 shows that the 8th order of approximations are sufficient for the convergent series solutions.

( )

8

( )

( )

( )

Table 1: Convergence of HAM solutions for different order of approximations when







 ( )

( )

9

( )

( )

5. Results and discussion The effects of interesting physical parameters, namely Deborah number  , magnetic parameter M , ratio parameter c, Biot number  , Prandtl number Pr and thermophoresis parameter Nt

in the dimensionless temperature profile    are sketched in the Figs. 3  8. Fig. 3 presents the influence of Deborah number  on the temperature profile   . Here the temperature and thermal boundary layer thickness are enhanced when we increase the value of Deborah number. Deborah number is directly proportional to the relaxation time. Relaxation time is higher for larger values of Deborah number. Hence, higher relaxation time causes to enhance the temperature and thermal boundary layer thickness. Fig. 4 presents the variations in temperature profile for different values of magnetic parameter M . Here M  0 corresponds to hydromagnetic flow and

M  0 is for the hydrodynamic flow situation. We observed that the temperature and thermal boundary layer thickness are higher for hydromagnetic flow in comparison to the hydrodynamic flow case. Fig. 5 describes that an increase in the ratio parameter c creates a reduction in the temperature profile and its related thermal boundary layer thickness. Here c  0 corresponds to two-dimensional flow case. We observed that thermal boundary layer thickness is more in the two-dimensional case when compared with the three-dimensional flow. Fig. 6 is displayed to see the influence of the Biot number  on the temperature profile   . An increment in  causes a stronger convection which yields higher temperature and thermal boundary layer thickness. Fig.

7 depicts the behavior of Prandtl number Pr on the temperature profile   . We observed that an increase in the value of Prandtl number results in a reduction in the temperature profile and thermal boundary layer thickness. An enhancement in the Prandtl number corresponds to weaker thermal diffusivity. Physically, larger Prandtl fluids possess weaker thermal diffusivity and smaller Prandtl fluids have stronger thermal diffusivity. This change in thermal diffusivity creates a reduction in the temperature    and thermal boundary layer thickness. Fig. 8 presents that the larger values of thermophoresis parameter Nt causes an enhancement in the temperature profile   . An increase in Nt producing an enhancement in the thermophoresis force which tends to move nanoparticles from hot to cold areas and consequently it enhances the temperature and thermal boundary layer thickness.

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Figs. 9  16 are plotted to examine the effects of Deborah number  , magnetic parameter M , ratio parameter c, Biot number  , Lewis number Le, Prandtl number Pr, thermophoresis parameter Nt and Brownian motion parameter Nb on the dimensionless nanoparticles concentration profile   . Fig. 9 presents that the larger values of Deborah number  causes an enhancement in the nanoparticles concentration profile and its related boundary layer thickness. From Fig. 10 it is observed that the nanoparticles concentration profile is higher for hydromagnetic flow M  0 and lower for hydrodynamic case M  0. It is also observed that the nanoparticles concentration is enhanced and going away from the wall of the sheet for hydromagnetic flow. Impact of ratio parameter c in the nanoparticles concentration profile    is sketched in Fig. 11. Nanoparticles concentration    and its related boundary layer thickness are reduced when we increase the values of ratio parameter. Fig. 12 shows the effects of the Biot number  on the nanoparticles concentration profile   . Here we noticed that an increase in the value of the Biot number creates an enhancement in the nanoparticles concentration profile and its associated boundary layer thickness. Fig. 13 depicts that the larger values of Lewis number

Le cause a reduction in the nanoparticles concentration profile   . Lewis number depends on the Brownian diffusion coefficient. Higher Lewis number leads to the lower Brownian diffusion coefficient, which shows a weaker nanoparticles concentration profile and its related boundary layer thickness. Fig. 14 presents the variations in the nanoparticles concentration profile    for different values of Prandtl number Pr . We noticed that an increase in the value of Prandtl number shows a reduction in the nanoparticles concentration    and its associated boundary layer thickness. Fig. 15 shows that an increase in thermophoresis parameter Nt causes an enhancement in the nanoparticles concentration    and its related boundary layer thickness. Fig. 16 presents that the larger values of Brownian motion parameter Nb creates a reduction in the nanoparticles concentration profile   . In nanofluid system, due to the presence of nanoparticles, the Brownian motion takes place and with the increase in Nb the Brownian motion is affected and consequently the nanoparticles concentration boundary layer thickness reduces.

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The convergent values of  f (0),

 g (0),

  (0) and  (0) at different order of

approximations are examined in Table 1 when   0.1, c  0.2   , M  0.3  Nt , Nb  0.5, Le  1.0, Pr  1.2 and  f   g  1.0       . Clearly the values of  f (0),  g (0),

  (0) and  (0) starts to repeat from 8th order of homotopic deformations. Therefore, we

conclude that 8 th order of homotopic deformations gives us the convergent series solutions of velocities, temperature and nanoparticles concentration. Table 2 shows the comparison for different values of

with homotopy perturbation method (HPM) and exact solutions. Table 2

presents an excellent agreement of HAM solutions with the existing homotopy perturbation method (HPM) and exact solutions in a limiting sense. Table 3 is computed to investigate the heat transfer rate at the wall (local Nusselt number) for different values of  , c, M ,  , Le, Pr, Nt and Nb. The heat transfer rate at the wall increases by increasing the values of Biot number  . Effects of Lewis number Le and Brownian motion parameter Nb on heat transfer rate are quite similar.

( )

12

( )

( )

13

( )

( )

14

( )

( )

15

( )

( )

16

( )

( )

17

( )

( )

18

( )

Table 2: Comparative values of HAM

( ) and

( ) HPM [31]

( ) for various values of

Exact [31]

19

HAM

when

( ) HPM [31]

Exact [31]

Table 3: Numerical values of local Nusselt number (

( )) for various values of ( )

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6. Concluding remarks Magnetohydrodynamic (MHD) three-dimensional boundary layer flow of Maxwell nanofluid over a bidirectional stretching surface with the convective boundary condition is addressed. Main points of the present analysis are listed below. 

Temperature and nanoparticles concentration profiles are enhanced when we increase the values of Deborah number  .



The effects of the Biot number  on the temperature and nanoparticles concentration are quite similar.



An increase in Lewis and Prandtl numbers show a decrease in nanoparticles concentration profile.



Nanoparticles concentration profile is reduced for an increase in Nb while it is increased with larger Nt.



The heat transfer rate at the wall is constant for Lewis number Le and Brownian motion parameter Nb.

Acknowledgments: We are grateful to a reviewer for the useful suggestions. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU) under grant No. (21-130-36-HiCi). The authors, therefore, acknowledge technical and financial support of KAU.

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Highlights 

Three-dimensional flow of Maxwell fluid.



Consideration of nanoparticles effect.



Formulation through convective condition.



Analysis in magnetohydrodynamic regime.



Utilization of new condition associated with mass flux.

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