shrinking surface with convective boundary conditions

Author’s Accepted Manuscript Flow and Heat Transfer of Magnetohydrodynamic Three-Dimensional Maxwell Nanofluid over a Permeable Stretching/Shrinking Surface with Convective Boundary Conditions Rahimah Jusoh, Roslinda Nazar, Ioan Pop www.elsevier.com/locate/ijmecsci

PII: DOI: Reference:

S0020-7403(16)31010-4 http://dx.doi.org/10.1016/j.ijmecsci.2017.02.022 MS3607

To appear in: International Journal of Mechanical Sciences Received date: 6 December 2016 Revised date: 27 January 2017 Accepted date: 26 February 2017 Cite this article as: Rahimah Jusoh, Roslinda Nazar and Ioan Pop, Flow and Heat Transfer of Magnetohydrodynamic Three-Dimensional Maxwell Nanofluid over a Permeable Stretching/Shrinking Surface with Convective Boundary Conditions, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2017.02.022 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.

Flow and Heat Transfer of Magnetohydrodynamic ThreeDimensional Maxwell Nanofluid over a Permeable Stretching/Shrinking Surface with Convective Boundary Conditions Rahimah Jusoh1*, Roslinda Nazar2, Ioan Pop3 1

Faculty of Industrial Sciences and Technology, Universiti Malaysia Pahang, 26300 Kuantan, Pahang, Malaysia. 2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia. 3 Department of Mathematics, Babeş-Bolyai University, R-400084 Cluj-Napoca, Romania. *

Corresponding author. Tel. : +6019-9253729, Email : [email protected]

Abstract – The flow and heat transfer of magnetohydrodynamic three-dimensional Maxwell nanofluid over a permeable stretching/shrinking surface with convective boundary conditions is numerically investigated. The partial differential equations governing the flow and heat transfer are transformed to a set of ordinary differential equations by using the suitable transformations for the velocity, temperature and concentration components. These equations have been solved numerically by employing the bvp4c function in Matlab. Numerical solutions are obtained for the skin friction coefficient and the local Nusselt number. Dual solutions are discovered and hence the stability analysis has been done to identify which solution is stable and physically realizable and which is not stable and which is not stable. Solutions are obtained for the skin friction coefficients and local Nusselt number for several values of the parameters, namely the suction parameter, Deborah number, Biot number and Prandtl number. The solutions are presented in some graphs and tables and are analyzed and discussed in detail. Keywords – Maxwell Nanofluid, Stretching/Shrinking Surface, Three-Dimensional Flow, Dual Solutions. Nomenclature

a, b

constants

B0

magnetic field ( Nm1 A1 )

C

nanoparticle volume fraction

C fx , C fy

skin friction coefficients along the x- and y- directions, respectively 1

C

ambient nanoparticle volume fraction

DB

Brownian diffusion coefficient (m2s- 1)

DT

thermophoretic diffusion coefficient (m2s- 1)

f ,g

dimensionless stream function along the x- and y- directions, respectively

hf

heat transfer coefficient ( Wm2 K 1 )

k

thermal conductivity ( Wm1K 1 )

Le

Lewis number

M

magnetic parameter

Nb

Brownian motion parameter

Nu x

local Nusselt number

Nt

thermophoresis parameter

Pr

Prandtl number

qw

heat flux ( Wm2 )

Re x , Re y

local Reynolds numbers along the x- and y- directions, respectively

s

suction parameter

T

nanofluid temperature ( K )

Tf

temperature of the hot nanofluid underneath the surface ( K )

T

ambient temperature of the nanofluid ( K )

t

time ( s )

u, v, w

velocity components along the x-, y- and z- directions, respectively ( ms 1 )

uw , vw

velocities of the surface in the x- and y- directions, respectively ( ms 1 )

W

mass flux velocity ( ms 1 )

x, y, z

Cartesian coordinates ( m )

Greek Letters



thermal diffusivity of the fluid ( m2 s 1 )



Deborah number



dimensionless similarity variable



stretching/shrinking parameter 2



dimensionless temperature



dimensionless nanoparticle volume fraction



kinematic viscosity ( m2 s 1 )



dynamic viscosity ( kg m1s 1 )

f

density of the base fluid ( kg m3 )



electrical conductivity ( Sm1 )

 C  f

heat capacity of the fluid

 C  p

heat capacity of nanoparticles



relaxation time (s)



Biot number



eigenvalue parameter

 wx , wy

surface shear stresses along x- and y- axes, respectively

1. Introduction In the recent decades, the boundary layer over a stretching sheet has been one of the major interesting research subjects due to its importance in various applications in engineering processes and industries such as wire drawing, metal spinning, glass fiber production, condensation process of metallic plate, hot rolling and aerodynamic extrusion of plastic sheets, etc. (see Fisher [1]). An analytical solution on boundary layer flow past a stretching sheet was pioneered by Crane [2]. Ever since, numerous investigations had been conducted on stretching sheet problems such as by Gupta and Gupta [3], Andersson [4], Wang [5], Ishak et al. [6], Nazar et al. [7], Mandal and Mukhodpadhay [8], Bakar et al. [9], etc. Turkyilmazoglu [10] studied the magnetohydrodynamic (MHD) flow and heat transfer of viscoelastic fluid past a permeable stretching surface and pioneered a new parameter called porous magneto-convection concentration parameter. He also found the uniqueness solutions of the flow and temperature profiles in the absence of viscoelasticity. On the other hand, not much attention has been paid to shrinking sheet problem. Miklavčič and Wang [11] discovered that shrinking sheet would induce a velocity away from the sheet. Therefore, by forcing a satisfactory mass suction on the boundary will let the flow towards a shrinking sheet potentially to occur. Moreover, Wang [12] considered the stagnation flow towards a shrinking sheet and discovered that for large shrinking rates, non3

uniqueness and non-existence of the similarity solutions may exist. Furthermore, studies on the shrinking sheet flow problems were explored by Mansur et al. [13], Zaimi et al. [14], Rohni et al. [15], Bachok et al. [16], Hayat et al. [17] and Bhattacharyya et al. [18], among others. As Goldstein [19] has pointed out, the new type of shrinking sheet flow is essentially a backward flow and it shows physical phenomena quite distinct from the forward stretching flow. Research of stretching/shrinking surfaces is not only applicable to viscous and nonviscous fluids, but also to nanofluids. Nanofluids are generally an advanced class of fluids which comprises of a base fluid with suspended nano-sized particles as discovered by Choi [20], Buongiorno [21] and Buongiorno et al. [22]. The emergence of nanofluid as an alternative medium for heat transfer is due to its effectiveness in thermal conductivity as discussed by Choi and Eastman [23] and Wong and Leon [24]. As described by Saidur et al. [25], nanofluid is widely applied in various industrial and engineering sectors. For instance, in the cooling of electronics, engine, cameras, microdevices, chillers, solar water heating and diesel combustion. By employing effective medium theory, the earlier investigations of heat conduction for solid-in-liquid suspensions is that of Maxwell. The details of how this result is derived have been described by Das et al. [26]. The Maxwell fluid model is the incomplex subclass of rate type non-Newtonian fluids and portraying the feature of stress relaxation through constant strain as mentioned by Hayat et al. [27] .Eapen et al. [28] discovered that several experiments with well-dispersed nanoparticles have shown modest conductivity enhancements with the Maxwell mean-field theory. Heat transfer analysis of the unsteady flow of Maxwell fluid over a stretching surface has been investigated by Mukhopadhyay [29] followed by Megahed [30] with consideration of slip velocity. Nadeem et al. [31] studied the flow and heat transfer of Maxwell fluid

past a stretching sheet with the effects of

nanoparticles and an applied magnetic field. They observed low thermal conductivity for high Prandtl numbers and concluded that there is a reduction in the conduction and the heat transfer rate increased at the surface of the sheet. Ramesh and Gireesha [32] and Hayat et al. [33] investigated the flow of Maxwell nanofluid with convective boundary conditions. In thermal system, one of the considerable control parameter for heat and fluid flow is the magnetic field. Miroshnichenko et al. [34] discovered a remarkable suppression of flow and heat transfer occurs when the magnetic field is parallel to the gravity force. The study of magnetohydrodynamic (MHD) which describing the significance of magnetic properties of electrically conducting fluid is crucial among engineers and scientists. The concept of MHD can be seen in the thermal power generating system. The efficiency of the system is 4

depending on heat. MHD generators can convert the thermal energy directly to the electrical energy as reported by Krishnan and Jinshah [35]. Application of MHD also can be found in nuclear power plants and hydroelectric power plants. As reported by Hayat et al. [36], in order to attain the maximum quality product in industrial manufacturing, an adapted magnetic field can be taken into account for handling the electrically conducting nanoliquids. Hayat et al. [37] also studied the inclusion of nonlinear thermal radiation in the MHD flow of viscoelastic nanofluid and discovered the enhancement of the thermal boundary layer thickness with the larger values of radiation parameter. Although the above investigations examined the boundary layer flow over stretching surfaces, only few references in the literature systematically described the boundary layer flow of Maxwell nanofluid over shrinking surfaces and to the best of our knowledge, no research has been carried out on the existence of dual solutions and stability analysis involving Maxwell nanofluid. Therefore, this was the motivation of the present study. We extend the problem of the steady three-dimensional MHD boundary layer flow of Maxwell nanofluid as presented by Hayat et al. [33] for the case of stretching sheet to the case of shrinking sheet with suction effect at the surface. The boundary layer equations governed by the partial differential equations are first transformed into a system of nonlinear ordinary differential equations, before being solved numerically using built-in bvp4c function in Matlab. The effect of suction parameter, Deborah number, Biot number and Prandtl number on the flow and heat transfer characteristics are thoroughly examined and discussed. Following Mustafa et al. [38], it is worth pointing out at this end that non-Newtonian fluid mechanics has been an important subject of research during the last several years it has been utilized in food processing, petroleum, chemical and polymer industries. Furthermore, the features of the flow of the common fluid in industry like biological fluids, motor oils and polymeric liquids can be described by using non-Newtonian fluid. Maxwell fluid model is one of the non-Newtonian fluid which exhibiting the viscoelasticity characteristic. There is a phenomenon called the stress relaxation in viscoelastic fluids where the deformation rate steadily diminishes when the shear stress is removed and the relaxation time is the time taken by the fluid for partial elastic recuperation upon the expulsion of stress. Therefore, researchers have given special focus towards the boundary layer flows of Maxwell fluid in the recent past.

2. Basic Equations 5

We consider the three-dimensional magnetohydrodynamic (MHD) flow and heat transfer of Maxwell nanofluid past a permeable shrinking sheet. Considering the mathematical Maxwell-nanofluid model, the basic equations governing the conservations of mass, momentum, energy and nanoparticle concentration can be written in vectorial form as (see Mustafa et al. [38], and Kuznetsov and Nield [39]) .  v  0

 v

(1)



 f   ( v ) v   p    S   B02  v   (v )  t    D  T  ( v )T    2T    DB C T   T  T T  t  T   

D C  ( v ) C  DB  2C   T t  T

 2  T 

(2)

(3)

(4)

where the extra stress tensor S in Eq. (2) for the upper-convected Maxwell fluid obeys the following relation

 D  1    S   f A1 Dt  

(5)

Here  is the fluid relaxation time, A1  v  (v)t is the first Rivlin-Ericksen tensor and D is the upper-convected time derivative. For a second rank tensor S and a vector a , the Dt

following relation exist

DS S   ( v ) S  L S  S Lt Dt t Da a   ( v ) a  L a Dt t

(6)

In these equations t is the time, T is the nanofluid temperature, C is the nanoparticle volume fraction, T is the ambient nanofluid temperature, B0 in the constant applied magnetic field,  f is the density of the nanofluids and the physical meaning of the other quantities are mentioned in the Nomenclature.

3. Solution for the Steady Case

6

A locally orthogonal set of coordinates ( x, y, z ) is chosen with the origin O in the plane of the shrinking sheet as shown in Fig. 1. The x - and y - coordinates are in the plane of the shrinking sheet, while the coordinate z is measured in the perpendicular direction to the shrinking surface. The shrinking sheet is situated at z  0 . It is assumed that the flat surface is stretched in the x - direction with the velocity u( x)  uw ( x) and shrunk in the y -direction with the velocity v( y)  vw ( y) . Further, we assume that the mass flux velocity is w  W , where W  0 is for suction and W  0 is for injection or withdrawal of the fluid. In the z direction, there exist a constant magnetic field of strength B0 . Under these conditions, the basic equations (1) to (4) can be reduced to the following boundary layer equations (see Hayat et al. [33])

u v w   0 x y z

(7)

u

  2u u u u  2u  2u  2u  2u  2u   v  w    u 2 2  v 2 2  w2 2  2uv  2vw  2uw  x y z y z xy yz xz   x



 2u  B02  u   u  w , 2 z f  z 

u

  2v v v v  2v  2v  2v  2v  2v   v  w    u 2 2  v 2 2  w2 2  2uv  2vw  2uw  x y z y z xy yz xz   x

(8)

 2v  B02  v   2 v  w , z f  z 

(9)

T T T  2T   C  p u v w  2  x y z z  C  f

C C C  2C DT u v w  DB 2  x y z z T

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

  2T   2  z 

(10)

(11)

along with the boundary conditions given by (see Kuznetsov and Nield [39], and Aziz [40])

u  uw ( x)  ax, v  vw ( y )  by , w  W   T h f  C DT  T  at z  0  T f  T  , DB  0  z k  z T  z  u  0, v  0, T  T , C  C as z  

7

(12) (13)

where u, v and w are the velocity components along x, y  and z  axes, respectively, T is the nanofluid temperature, C is the nanoparticle volume fraction, T and C are the ambient temperature and nanoparticle volume fraction of the nanofluid, correspondingly. With the presence of constant magnetic field B0 in the z-direction, the fluid is assumed electrically conducting. A convective heating process which is denoted by the heat transfer coefficient h f and temperature of the hot fluid T f underneath the surface influence the temperature at the surface. Other parameters involved in the present problem are  the dynamic viscosity,  

 the kinematic viscosity,  f the density of base fluid,  the f

relaxation time,  the electrical conductivity,   C  f the heat capacity of the fluid, k the thermal conductivity,  

k the thermal diffusivity of the fluid,   C  p the effective  C  f

heat capacity of nanoparticles, DT the thermophoretic diffusion coefficient, DB the Brownian diffusion coefficient, a is a positive constant and b is a constant corresponding to a stretching (b  0) or to a shrinking (b  0) sheet, respectively. The boundary condition DB

 C DT  T   0 at z  0 in (12) representing that the normal flux of nanoparticles is  z T  z

zero at the boundary with consideration of thermophoresis, (Kuznetsov and Nield [39]).

z B0

O

T k  h f T f  T  z

x uw  x   ax

vw  y   by y

b < 0 (shrinking), b > 0 (stretching)

Fig. 1. Physical model and coordinate system

8

In order to solve Eqs. (7) to (11) along with the boundary conditions (12) and (13), we use the following similarity transformations: u  axf    ,

v  ayg    , w    a 

T  T     , T f  T

12

C  C     , C

 f    g   , 12

a    z  

(14)

where primes denote differentiation with respect to η. Substituting the similarity variables (14) into (7)-(11), it is found that the continuity equation (7) is automatically satisfied, and equations (8) to (11) are reduced to the following ordinary (similarity) differential equations:

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

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

2

2

2

2

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

(15) (16) (17)

Nt    0 Nb

(18)

subject to the boundary conditions f  0, g  s, f   1, g    ,     1    0   , Nb   Nt   0 at   0. f   0, g   0,   0,   0 as   

(19) (20)

where  is the constant stretching (  0) or shrinking (  0) parameter, s is the constant mass flux parameter with s  0 for suction, s  0 for injection and s  0 (impermeable plate), respectively, Pr is the Prandtl number, Le is the Lewis number, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter,  is the Deborah number, M is the magnetic parameter, and  is the Biot number which are defined as

   a,

M2 

 B02 b ,  , f a a

  C  p DT T f  T  Nt  ,   C  f T



Pr  hf



k

a

 ,  ,

Nb 

  C  p DBC ,  C  f 



(21)

W Le  , s DB a

The physical quantities of practical interest are the local skin friction coefficients C fx and C f y and the local Nusselt number Nu x which are defined as

Cf x 

 wx wy x qw , Cf y  , Nux  , 2 2  uw  vw  (T f  T )

(22)

where  w x and  w y are the skin frictions along the x and y directions whereas qw is the heat flux from the surface of the sheet, which are given by

9

 u   ,   z  z 0

 w x   1    

 v   T   , qw        z  z 0   z  z 0

 w y   1    

(23)

Substituting (8) into (17) and using (16), we obtain

Re x1 2 C fx  1    f   0  ,

Re y1 2  3 2C fy  1    g   0  ,

Re x 1 2 Nux     0 

(24)

where Re x  uw ( x) x /  and Re y  vw ( y) y /  are the local Reynolds numbers. It should be mentioned that due to zero nanoparticles mass flux condition, the Sherwood number becomes zero (see Kuznetsov and Nield [39]). Therefore, we have excluded the results for the Sherwood number from the present analysis.

4. Stability Analysis Since the numerical results reveal the existence of dual solutions for equations (15) to (18) subject to the boundary conditions (19) to (20), thus a stability analysis is required to identify which of these solutions are physically realizable in practice. As proposed by Merkin [41], the unsteady case for equations (8)-(11) need to be considered, which are replaced by 2  2  2u 2  2u u u u u  2u  2u  2u  2  u  u  v  w   u v w  2uv  2vw  2uw  2 t x y z y 2 z 2 xy yz xz   x



 2u  B02  u   u  w , 2 z f  z 

(25)

  2v v v v v  2v  2v  2v  2v  2v   u  v  w    u 2 2  v 2 2  w2 2  2uv  2vw  2uw  t x y z y z xy yz xz   x



 2v  B02  v   v  w , 2 z f  z 

T T T T  2T   C  p u v w  2  t x y z z  C  f

(26)

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

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

(27)

(28)

where t representing the time. We introduce now the following new dimensionless variables:

10

u  ax

f  ,  g  ,  , v  ay , w   a f  

 f  ,   g  ,  

T  T a   ,   ,  z,   at T f  T  so that equations (8) to (11) can be written as

(29)

2

3  3 f  2 f  f  f  2 f 2  f  2  M   1 f  g    2 f  g  f  g              3  2      2  3   f 2 f M 2  0  

(30)

2

3  3 g  2 g  g  g  2 g 2  g  2  M   1 f  g    2 f  g  f  g              3  2      2  3   g  2 g M 2  0  

(31)

2       2     Pr   f  g   Nb  Nt      2            

(32)

 0  

  2 Nt  2      Le Pr   f  g   0 2 2  Nb     

(33)

subject to the boundary conditions f g  f  0,   0,  0,   1, g  0,   s,  0,    ,  0,    1    0,   ,      Nb  0,   Nt  0,   0   f g  ,   0,  ,   0,   ,   0,   ,   0 as     

(34)

(35)

According to Weidman et al. [42], stability of the steady flow solution f  f0   , g  g0   ,   0   and   0   satisfying the boundary value problem (15) to (20), is tested using the following : f  ,   f 0    e  F   g  ,   g 0    e  G  

  ,    0    e  H  

(36)

  ,   0    e  J  

where  is an unknown eigenvalue parameter, and F   , G   , H   and J   are small relative to f0   , g0   , 0   and 0   . Substituting (36) into (30) to (33), we obtain the following linearized equations

11

 f  g  F    F  G  f    2 f F     f  g  F   M F  2   f  g   f  F   f  F     F  G  f  f     F   0 F    M 2   1 0

0

2

0

0

0

0

0

0

0

0

2

0

(37)

0

 f  g  G    F  G  g    2g G     f  g  G   M G  2   f  g   g G   g G     F  G  g  g     G   0

(38)

 f  g  H    F  G    Nb  J    H    2Nt H    H   0

(39)

G    M 2   1 0

H   Pr

0

0

2

0

0

0

0

0

0

0

Nt  J   H  Le Pr Nb

0

0

0

0

0

0

 f  g  J    F  G     J   0 0

0

2

0

0

0

(40)

with the boundary conditions F  0   0, F   0   0, G  0   0, G   0   0, H   0   H  0  , NbJ   0   NtH   0   0

F     0, G     0, H    0, J    0 as   

(41) (42)

The stability of the corresponding steady flow solutions f0   , g0   , 0   and 0   is determined by the smallest eigenvalue  1 . The range of possible eigenvalues can be

determined by relaxing a boundary condition on F    , G   , H   and J   as suggested by Harris et al. [43]. In this study, we relax the condition that F     0 as

   and for a fixed value of  1 , we solve the system of equations (37) to (40) along with the new boundary conditions (41), (42) and F   0   1 .

5. Results and Discussion The main results of interest are the influence of the suction parameter s, Prandtl number Pr, Biot number  , and Deborah number  on the skin friction coefficients and the local Nusselt number. The system of nonlinear ordinary differential equations (15) to (18) subject to the boundary conditions (19) and (20) has been solved numerically by using the bvp4c function which is a finite-difference code that implements the 3-stage Lobatto IIIa formula in Matlab. Table 1 shows the comparison values of the skin friction coefficients, the local Nusselt number and the local Sherwood number with those of Hayat et al. [33] for   0.2 . The results indicate that the values of the skin friction coefficients for both methods are in

12

excellent agreement whereas the results for the local Nusselt number are slightly dissimilar due to the different approach of analytical and numerical methods. Table 1 Comparison of the skin friction coefficients and local Nusselt number for Pr  1.2,   0.1,   0.2,   0.2, Le  1.0, M  Nt  0.3, and Nb  0.5 when s  0 .

 f   0 

 g   0 

   0 

Hayat et al. [33]

1.11540

0.16474

0.17011

Present

1.11540

0.16474

0.15590

Table 2 shows the comparison values of  f   0  and  g   0  for various values of  specifically for the stretching case. It has been found that they are in good agreement. Therefore, it is concluded that this method works efficiently, hence, the results presented here are accurate.

Table 2 Comparison values of  f   0  and  g   0  for various

λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

values of  when   M  s  0 .  f   0   g   0  Present 1.000008 1.020265 1.039498 1.057957 1.075789 1.093096 1.109947 1.126398 1.142489 1.158254 1.173721

Hayat et al. [33] 1 1.020260 1.039495 1.057955 1.075788 1.093095 1.109947 1.126398 1.142489 1.158255 1.173722

Present 0 0.066849 0.148738 0.243361 0.349209 0.465205 0.590529 0.724532 0.866683 1.016539 1.173721

Hayat et al. [33] [ 0 0.066847 0.148737 0.243359 0.349209 0.465205 0.590529 0.724532 0.866683 1.016540 1.173722

As shown in Figs. 2-9, it is obviously seen that dual solutions exist for the particular suction parameter s  2.0, 2.3, 2.5 and 2.7. These are supported by the results reported by Fang et al. [44] for a permeable shrinking sheet that mentioned the dual solutions occurred only for mass suction s  2 . Miklavčič and Wang [11] also highlighted the same fact that by exerting adequate suction s at the boundary, the flow will generate two solutions when s  2. 13

Two solutions exist up to a critical value c . There are no solution exists for   c and a distinctive solution exists when   c . Table 3 presents some values of c for various values of s, Pr, , Nt and  . It is observed that the increasing values of the suction s and Deborah number  contribute to an upsurge of absolute value c . On the other hand, there is no significant effect of various values of the Prandtl number Pr, Biot number  and thermophoresis parameter Nt on critical value c since the results obtained are the same.

Table 3 The critical values c for various values of s, Pr, , Nt and  .

s 2.0 2.3 2.5 2.7 2.7 2.7 2.7 -

Pr 1.2 1.0 1.5 1.2 1.2 1.2 -

 0.2 0.2 0.6 0.8 1.0 0.2 0.2 -

Nt 0.3 0.3 0.3 0.6 0.7 0.8 0.3 -

 0.1 0.1 0.1 0.1 0.11 0.12

Fig. 2 illustrates the variation of the skin friction coefficient

c - 1.6190 - 2.0360 -2.3805 - 2.8043 - 2.8043 - 2.8043 - 2.8043 - 2.8043 - 2.8043 - 2.8043 - 2.8043 - 2.8043 - 2.8457 - 2.9857

f   0  against the

stretching/shrinking parameter  for various values of the suction parameter s. It is noted from this figure that the value of f   0  decreases when the suction parameter s increases. This is also reflected in Fig. 3 where the skin friction coefficient in y-direction g   0  increases with the enlargement of s when shrinking surface (   0 ) is considered. The flow near the solid surface is enhanced since the application of suction cause the reduction of momentum boundary layer thickness. In Fig. 3, the value of skin friction coefficient g   0  equals to zero for all values of suction parameter s when   0 which means that the surface is at rest/static since we consider suction parameter s in g-direction as in boundary condition

14

(19). However, the skin friction coefficient decreases as s increases when the stretching surface (   0 ) is considered.

Fig. 2. Variations of skin friction f   0  with  for several values of s .

First Solution Second Solution

Fig. 3. Variations of skin friction g   0  with  for several values of s . Application of suction also affects the rate of heat transfer. The existance of suction leads to the movement of the heated fluid towards the wall where the buoyancy forces will slow down

15

the fluid flow due to the strong effect in viscosity. Consequently, this effect tends to lessen the wall shear stress and the local Nusselt number    0  increases as depicted in Fig. 4

Fig. 4. Variations of the local Nusselt number    0  with  for several values of s . Figs. 5 and 6 demonstrate the effects of the Deborah number  on the skin friction coefficients f   0  and g   0  respectively. The skin friction coefficients are lessening as  increasing. Physically, as Deborah number rising, the material behaviour changes to progressively influenced by elasticity which exhibiting solid like act. Consequently, the flow velocity is decelerating and the temperature of the fluid is getting higher. The increment in the temperature of the fluid leads to the decrement in the rate of heat transfer. This is in accordance with the result shown in Fig. 7 where the local Nusselt number    0  decreases with increasing  . Therefore, the usage of coolant having a small Deborah number can improve the cooling of the heated sheet.

16

Fig. 5. Effects of the Deborah number  on the skin friction coefficient f   0  when

  0.5 and s  2.7 .

Fig. 6. Effects of the Deborah number  on the skin friction coefficient g   0  when

  0.5 and s  2.7 .

17

Fig. 7. Effects of the Deborah number  on    0  when   0.5 and s  2.7 . Figs. 8 and 9 illustrate the variation of the local Nusselt number    0  with the Prandtl number Pr and Biot number  . Generally, it can be seen from these figures that the surface heat transfer decreases as  decreases. Further, it is shown that the local Nusselt number is consistently higher for a nanofluid with higher values of Pr and  . Since  is directly proportional to the heat transfer coefficient h f , thus it is inversely proportional to the thermal resistance. Consequently, as  is enhanced, the heat resistance is then reduced and hence increases the heat transfer rate at the surface.

18

Fig. 8. Effects of the Prandtl number Pr on the local Nusselt number    0  when s  2.7

Fig. 9. Effects of the Biot number  on the local Nusselt number    0  when s  2.7 .

The existence of dual solutions has been found for equations (15) to (18) with boundary conditions (19) and (20). A stability analysis has been done using bvp4c function in Matlab to test the stability of the solutions. Furthermore, the eigenvalues parameter for the first and 19

second solutions are found. There is an initial growth of disturbances if the smallest eigenvalue is negative which reflects that the flow is unstable. On the other hand, if the smallest eigenvalue is positive, there is an initial decay and the flow is stable. The smallest eigenvalue  1 for some values of s and  are presented in Table 4. Weidman et al. [42] and Merkin [41] proposed that as   0 , equations (27)-(30) yield the homogeneous problem which defines the critical value c , which is the turning point value that separates the stable and unstable branches. The results in Table 4 show that positive values of  1 are found at the first solution while negative values of  1 are obtained at the second solution which indicates that the first solution is stable whereas the second solution is not. It is worth mentioning that for both first and second solutions,  1 decreases as  approaches c which is consistent with the study by Merkin [41] where   0 at   c . Table 4 Smallest eigenvalue  1 for some values of s when   0.1, M  Nt  0.3, Nb  0.5,   0.2, and Le  1.0 s



 1 (first solution)

 1 (second solution)

2.3

-2.0 -2.03 -2.036 -2.3 -2.38 -2.3805 -2.8 -2.804 -2.8043

1.0308 0.8941 0.2931 1.4996 0.5658 0.3891 1.4993 0.9015 0.5648

-0.9315 -0.5134 -0.1712 -0.9164 -0.7479 -0.4457 -1.329 -0.8018 -0.3255

2.5

2.7

5. Conclusions Fluid flow and heat transfer of three dimensional Maxwell nanofluid over a permeable stretching/shrinking surface with convective boundary conditions is studied. Numerical results are obtained using the bvp4c function in Matlab. Discussions have been carried out for the effects of various parameters on the skin friction coefficient and local Nusselt number. An enlargement in the magnitude of suction and Deborah number reduce the skin friction coefficient. The rate of heat transfer is enhanced with the increment of the Prandtl number, Biot number and suction parameter whereas an upsurge in Deborah number contributes to a decrement in the heat transfer rate. It has been discovered that dual solutions exist for both 20

stretching and shrinking cases. Finally, the stability analysis has been conducted and as a conclusion the first solution is found to be stable while the second solution is unstable. Acknowledgement This work was supported by the research university grant (DIP-2015-010) from the Universiti Kebangsaan Malaysia and the fundamental research grant (FRGSTOPDOWN/2014/ SG04/UKM/01/1) from the Ministry of Higher Education, Malaysia. We also appreciate the valuable comments and suggestions made by the three very competent reviewers.

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Highlights     

Magnetohydrodynamic three-dimensional flow of Maxwell nanofluid is investigated. A permeable stretching/shrinking surface is considered. The rate of heat transfer is enhanced with the increment of the Prandtl number, Biot number, and suction parameter. Dual solutions exist for a certain range of the suction parameter. A stability analysis is performed to determine which solution is stable and physically realizable.

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