Slip MHD liquid flow and heat transfer over non-linear permeable stretching surface with chemical reaction

International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

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Slip MHD liquid flow and heat transfer over non-linear permeable stretching surface with chemical reaction M.H. Yazdi a,c,⇑, S. Abdullah a, I. Hashim b, K. Sopian c a

Department of Mechanical & Materials Engineering, Faculty of Engineering & Built Environment, Universiti Kebangsaan Malaysia, Malaysia School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Malaysia c Solar Energy Research Institute (SERI), Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia b

a r t i c l e

i n f o

Article history: Received 3 December 2010 Received in revised form 17 March 2011 Available online 5 May 2011 Keywords: MHD flow Slip boundary condition Local similarity solution Navier–Stokes equations

a b s t r a c t In the present study the magnetohydrodynamic (MHD) liquid flow and heat transfer over non-linear permeable stretching surface has been presented in the presence of chemical reactions and partial slip. By means of proper similarity variables, the fundamental equations of the boundary layer are transformed to ordinary differential equations which for the fixed values of the x-coordinate along the plate local similarity solution would be valid appropriately. The ordinary differential equations are solved numerically using an explicit Runge–Kutta (4, 5) formula, the Dormand–Prince pair and shooting method. As a result, the velocity profiles, the concentration profiles, temperature profiles, the wall shear stress, the local Sherwood number and the local Nusselt number for the various values of the involved parameters of the problem are presented and discussed in details. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Microfluidics as a young research field plays a great role to develop control accuracy of the small devices. Microchannel based systems have emerged as a critical design trend in development of precise control [1] and maneuvering of small devices. A wide variety of microscale devices like sensors, actuators, and valves are now extensively used in our everyday life [2]. Some MEMS devices have also been designed in the field of fluid applications called microfluidics, examples include micro MHD pumps, microvalves and microturbines [3]. Slip-flow happens if the characteristic size of the flow system is small or the flow pressure is very low. In no-slip-flow, as a requirement of continuum physics, the fluid velocity is zero at a solid–fluid interface. But it is non-zero in the existence of slip-flow [4,5]. It is well known that flow past a permeable surface has practical applications. Examples of permeable surfaces which exist in nature are beach sand, limestone, the human lung and in small blood vessels. The magnetohydrodynamic (MHD) flow of a fluid in the open/closed microchannel is of interest in connection with certain problems of the movement of conductive physiological fluids, e.g., the blood plasma, blood plasma pump machines and with the need for theoretical research on the problem of the slip MHD liquid flow along permeable surfaces. As we know, biological fluids are typically non-Newtonian. Fortunately blood plasma is Newtonian and is very similar in ⇑ Corresponding author at: Solar Energy Research Institute (SERI), Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia. E-mail address: [email protected] (M.H. Yazdi). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.04.009

physical properties to water [6]. On the other hand most of the MHD applications in microfluidics are in the liquid fields. Thus considering MHD liquid slip flow has promising potential in numerous practical applications such as MHD micro pumps which are a non-mechanical pump. Consequently, the micro magnetohydrodynamic effects are recognized as a tool for controlling the microstructure of materials. Another specific microfluidic application of the problem is micromixing technology which is vital in biological processes specially on rapid mixing of a biological liquid fluid in a microchannel. Chemical reaction is also another significant process that should be taken into account in micromixing of the biological systems such as cell activation and protein folding particularly when the mixing of reactants for initiation is necessary. Lorentz forces which are produced by magnetic field are able to move liquids in a mixing process. Hence one of the active methods of micromixing of biological samples is using magnetic field [7]. Suction can be applied to biological chemical processes to remove reactants while blowing is used to add reactants in the system and to cool the surfaces by its ability of decreasing heat transfer. The problem of flow and heat transfer over a stretching surface has been investigated and discussed by many researchers [8–17]. Recently, Javed et al. [18] studied heat transfer of viscous fluid over a non-linear shrinking sheet in the presence of magnetic field where they have obtained dual solutions for the exact and numerical solutions in the shrinking sheet problem. Numerous investigations have been done analytically regarding to the slip flow regimes over surfaces. Martin and Boyd [19] analyzed Blasius boundary layer problem in the presence of slip boundary condition. Their results demonstrated that the boundary

3215

M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225 Table 1 Comparison of the velocity gradient at the wall |f0 0 (0)| between the present code results and that obtained before. n

fw

K

M

Cortell [32] (2007)

Javed et al. [18] (2011)

Fourth-order Runge– Kutta method

Finite difference scheme (Keller-box method)

0.6275 0.7667 0.8895 0.9538

0.6275 0.7668 0.8895 0.9539

Van Gorder et al. [28] (2010) Homotopy analysis and BVP method

Fang [30] (2009)

Presented results

Close exact cubic solution

The Dormand–Prince pair and shooting method

0 0.2 0.5 0.75

0

0

0

0.6275 0.7667 0.8896 0.9540

1

0

0 0.5 1 2

0

1 0.5912 0.4302 0.2840

1

1 2 5

0

0

1.6103 2.4142 5.1926

1.6180 2.4142 5.1926

1

1 2 5

0.5

0

0.8284 1.0509 1.4331

0.8284 1.0509 1.4331

1

0

0 0.5 1

0.5

1

1 0.5912 0.4302 0.2840

1.1180 0.6495 0.4691

layer equations can be used to study flow at the MEMS scale and provide useful information to study the effects of rarefaction on the shear stress and structure of the flow. In another task [20], they have analyzed slip flow and heat transfer at constant wall temperature. Based on their boundary layer theory, non-equilibrium effects will cause a drop in drag on airfoils. According to their studies of liquids flow over flat plate at constant wall temperature boundary conditions, there is no temperature jump in liquid fluids. In the analysis of macro flow and heat transfer no distinction is made between liquids and gases. Solutions to gas and liquid flows for similar geometries are identical as long as the governing parameters such as Reynolds number, Prandtl number and the boundary conditions are the same for both. This is not the case under micro scale conditions. Because the mean free paths of liquids are much smaller than those of gases, the continuum assumption may hold for liquids but fail for gases [21]. Using liquid fluids such as water, blood plasma tend to avoid having temperature jump on the boundary conditions. Recently, Matthews and Hill [22] studied the effect of replacing the standard no-slip boundary condition

1.1180 0.6495 0.4691

with a non-linear Navier boundary condition for the boundary layer equations. In another task they have investigated Newtonian flow with non-linear Navier boundary condition for three simple pressure-driven flows through a pipe, a channel and an annulus [23]. The axisymmetric fluid flow of a Newtonian fluid regarding to a stretching sheet in the presence of the partial slip was investigated by Ariel [24]. Yazdi et al. [25] investigated friction and heat transfer in the slip flow boundary layer at constant heat flux boundary conditions. Wang [26] has studied the hydrodynamic flow in the presence of partial slip over a stretching sheet with suction. Recently Yazdi et al. [27] analyzed convective heat transfer of the slip liquid flow past horizontal surface within the porous media at constant heat flux boundary conditions. Their results suggest that slip liquid flow can successfully reduce wall friction through slip-flow boundary conditions in convective heat transfer problems and increase heat transfer rate. It has been found that suction makes a significant effect on the velocity adjacent to the wall in the presence of slip. Hydrodynamic nano boundary layer flow over permeable stretching surface using Homotopy Analysis Method

Table 2 Comparison of the temperature and concentration gradient at the wall |h0 (0)|, |/0 (0)| respectively between the present code results and that obtained before. n

k0

Pr

c

m

Sc

Ali [34] (1994)

Ishak et al. [14] (2009) Finite difference method

Hayat et al. [36] (2010) Homotopy Analysis Method

|h0 (0)| 1

0

0.72 1 3

1

1

0.72 1 3

1

Takhar et al. [35] (2000) Finite difference method

Cortell [33] (2007) Fourth-order Runge–Kutta method

|/0 (0)|

|h0 (0)|

0.4617 0.5801 1.1599

1

10 1 0.1 10 1 0.1

1

|/0 (0)|

0.4631 0.5818 1.1647 0.8086 1 1.9237

10 1 0.1

Presented results The Dormand– Prince pair and shooting method

0.8086 1 1.9236

0.8086 1 1.9238 3.2326 1.1776 0.6704

3.2312 1.1765 0.6689

3.2326 1.1766 0.6689

2

2.6496 1.001 0.6372

2.6485 1 0.6354

2.6482 1 0.6353

3

2.3041 0.9076 0.6231

2.3030 0.9065 0.6213

2.3028 0.9066 0.6213

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M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

a

1

b

1.5

K=0

0.8

K=0

K=1

K=2

|f ( η )|

f' ( η )

K=1

1

K=2

0.6

K=4

"

K=4 0.4

0.5 0.2

0

0

1

2

3

4

0

5

η 1

d

fw=0

0.8

fw=0.4

0.6

f' ( η )

fw=0.8 fw=-0.4

η

3

4

5

0.8

fw=0

0.6

fw=0.2

0.5

fw=0.4 fw=0.8

0.4

fw=-0.2 fw=-0.4

0.3

fw=-0.8

0.2

2

"

fw=-0.2

0.4

1

0.7

fw=0.2

|f (η )|

c

0

fw=-0.8

0.2 0.1

0

0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

4

η 1

f

f (η)

0.6

7

8

9

10

0.8

M=0.4

0.6

M=0.5

0.5

M=0 M=0.4 M=0.5

0.4

M=0.6

"

M=0.6

'

6

0.7

M=0

0.8

|f (η )|

e

5

η

0.4

0.3 0.2

0.2

0.1

0

0

1

2

η

3

4

5

0

0

1

2

3

η

4

5

6

Fig. 1. (a) Velocity distribution and (b) velocity gradient magnitude as function of g for various values of K at fw = 0.2, M = 0.1 and n = 0.5. (c) Velocity distribution and (d) velocity gradient magnitude as function of g for various values of fw at K = 0.5, M = 0.1 and n = 0.5. (e) Velocity distribution and (f) velocity gradient magnitude as function of g for various values of M at fw = 0.2, n = 0.5 and K = 0.5.

(HAM) and Boundary Value Problem solver (BVP) was studied by Van Gorder et al. [28] On the topic of MHD flow modeling, the boundary-layer equation of flow over a nonlinearly stretching sheet in the presence of a chemical reaction and a magnetic field was investigated by Kechil and Hashim [29]. Fang et al. [30] studied analytically hydrodynamic boundary layer of slip MHD viscous flow over a stretching sheet. Their investigation shows the velocity and shear stress profiles are influenced by the slip, magnetic and suction/injection parameters. They have illustrated that wall drag force increases with the increase of magnetic parameter. There have been many theoretical models developed to describe slip flow along the surface. However, to the best of our

knowledge, no investigation has been made yet to analyze the slip MHD flow and heat transfer over non-linear permeable stretching surface at prescribed surface temperature (PST) in the presence of the chemical reaction.

2. Mathematical formulation For modeling of magnetohydrodynamics (MHD) fluid transport in slip boundary layer, the assumptions made for the derivation of the full Navier–Stokes equations have been examined. Examples of MHD fluids are plasmas, liquid metals and salt water. Biological

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M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225 1.1

a

1.6

b K=0 K=0.5 K=1 K=2

0.8 0.7

1.2 1

|f "(0)|

0.9

f '(0)

K=0 K=0.5 K=1 K=2

1.4

1

0.8

0.6

0.6

0.5

0.4

0.4

0.2

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0

0.4

-0.4

-0.3

-0.2

-0.1

0

fw

c

0.2

0.3

0.4

3.5

4

1.4

d

0.9

M=0

0.8

M=0.2

0.7

M=0.4

1.2

M=0 M=0.2

1

M=0.4 |f "(0)|

f '(0)

1

M=0.5

0.6

0.1

fw

0.8

M=0.5

0.6

0.5

0.4 0.4

0.2

0.3 0.2

0

0.5

1

1.5

2

2.5

3

3.5

0

4

0

0.5

1

1.5

2

2.5

3

K

K

Fig. 2. Variation of the (a) f0 (0) and (b) |f0 0 (0)| as function of fw for different slip coefficient K, at n = 0.5 and M = 0.4. Variation of the (c) f0 (0) and (d) |f0 0 (0)| as function of K for different magnetic parameter M, at n = 0.5 and fw = 0.2.

fluids are normally non-Newtonian liquids. Blood plasma is Newtonian having physical properties very much like water. Accordingly considering such fluids can make our problem applicable in the microfluidic devices. The assumptions are the fluid is a continuum and Newtonian. In addition, the fluid can be assumed to be incompressible. We will study the 2-D, steady, laminar flow in the presence of a transverse magnetic field with strength B(x) which is applied in the vertical direction, given by the special form:

BðxÞ ¼ B0 x

n1 2

;

B0 – 0

ð1Þ

where x is the coordinate along the plate measured from the leading edge and n is a power index described as the stretching non-linear parameter. The magnetic Reynolds number is assumed small so that the induced magnetic field is neglected. The viscous dissipation term is important when the velocity gradient is high. It will be considered in the energy equation to be investigated in the slip flow regime. The positive y-coordinate is measured normal to the x-axis in the outward direction towards the fluid. The corresponding velocity components in the x and y directions are u and v, respectively. The surface velocity is given by

uw ðxÞ ¼ u0 xn

ð2Þ

where u0 is a constant rate parameter of the stretching surface velocity and n is non-linear stretching parameter. The wall is stretched keeping the origin fixed at prescribed surface temperature (PST), Tw as follows:

y ¼ 0;

0

T ¼ T w ð¼ T 1 þ axk Þ

ð3Þ

where k0 is the surface temperature parameter in the prescribed surface temperature (PST) boundary condition. Special case of the constant surface temperature (CST) is produced by introducing k0 equal to zero. The concentration neighboring to the surface would be Cw and the solubility of A in B and the concentration of A far away from the plate would be C1. We assume that the reaction of a species A with B be the higher-order homogeneous chemical reaction with rate constant, j. It is desired to analyze the system by a boundary layer method. It is assumed that the concentration of dissolved A is small enough and the related physical property D is constant in the fluid. The steady twodimensional boundary layer equations for this problem, in the usual notation [31] are

@u @ v þ ¼0 @x @y @u @u @ 2 u rB2 u þv ¼ m1 2  @x @y @y q  2 @T @T @ 2 T m1 @u u þv ¼a 2þ @x @y @y cp @y

u

u

@C @C @2C þv ¼ D 2  jC m @x @y @y

ð4Þ

ð5Þ

ð6Þ

ð7Þ

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M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

The associated boundary conditions are

v ¼ v w ;

y ¼ 0 ) u ¼ uw þ us ;

a

C ¼ Cw;

0

ð8Þ

0.675

where q is the fluid density, j is the reaction rate constant, m is order of chemical reaction, r is the electrical conductivity of the fluid, vw is the suction/injection and us is the velocity slip which is assumed to be proportional to the local wall shear stress as follows [4]:

0.67

T ¼ T w ð¼ T 1 þ axk Þ C ¼ C1;

T ¼ T1

 @u us ¼ l  @y w

0.665

T  T1 ; Tw  T1

/¼

0.3

The fundamental partial differential equations (5)–(7) are transformed to ordinary differential equations by substituting similarity variables (10) into Eqs. (5)–(7) as follows:



2M2 0 2n f  ðf 0 Þ2 ¼ 0 nþ1 nþ1  0  0 2k h00 þ Prh0 f  Prf 0 h þ Ec x2nk Prðf 00 Þ2 ¼ 0 nþ1 00

f 000 þ ff 

/00 þ Sc f /0  Sc cð/Þm ¼ 0

ð13Þ ð14Þ ð15Þ

For these equations, the associated boundary conditions are

8 00 > f 0 ð0Þ ¼ 1 þ Kf ð0Þ > > > > < f ð0Þ ¼ fw g¼0 ) > /ð0Þ ¼ 1 > > > > : hð0Þ ¼ 1 8 0 f ð1Þ ¼ 0 > < g ! 1 ) /ð1Þ ¼ 0 > : hð1Þ ¼ 0

0.7

0.75

0.9 0.8 0.7 0.6

ð12Þ



0.6

M=0 Kp=0.0 M=0.2 M=0.3 M=0.4 M=0 Kp=0.5 M=0.2 M=0.3 M=0.4

0.5 0.3

0.4

0.5

ð11Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uw ðn þ 1Þ 2xm1

!

n

1

ð10Þ

where K is the slip coefficient defined for liquids by

K¼l

0.5

1.1

It is helpful to introduce a slip coefficient using similarity variables:

f ð0Þ ¼ 1 þ Kf ð0Þ

0.4

b

C  C1 Cw  C1

00

M=0.3

0.66

|f "(0)|

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u u0 ðn þ 1Þ n1 x2 f 0 ðgÞ ¼ ¼ ; g ¼ y uw u0 xn 2m1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u m ðn þ 1Þ n1 n1 0 x2 fþ v ¼ 0 1 gf 2 nþ1

0

M=0 M=0.2

ð9Þ

where l is slip length as a proportional constant of the velocity slip. To obtain solutions in the slip-flow domain, fluid velocity and thermal conditions must be specified at the boundaries. In liquids, however, the molecules are densely packed and a mean free path is not a meaningful quantity. For liquids, therefore, l is defined as the interaction length. Assuming liquid fluid tends to temperature jump being neglected. By using similarity transformation, the fundamental equations of the boundary layer are transformed to ordinary differential ones that are locally valid. Thus, the mathematical analysis of the problem can be simplified by introducing the following dimensionless coordinates:

hðgÞ ¼

f '(0)

y ! 1 ) u ¼ 0;

ð16Þ

where fw, Sc, c, Pr, Ec, and M show the suction/injection parameter, the Schmidt number, the reaction rate parameter, the Prandtl number, the Eckert number and the magnetic parameter, respectively:

n

0.6

0.7

0.75

Fig. 3. (a) The effects of non-linear stretching parameter, n and magnetic parameter, M on f0 (0) when fwp = 0.2 and Kp = 0.5. (b) The effects of non-linear stretching parameter, n, slip coefficient, Kp and magnetic parameter, M on |f00 (0)| when |fwp = 0.2.

v w fw ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; uw m1 ðnþ1Þ 2x

u2 Ec ¼ 0 ; acp

M2 ¼

rB20 qu0

2jx c¼ ; uw ðn þ 1Þ

ð17Þ Sc ¼

m1 D

fw is negative for mass injection and positive in the presence of the suction along the surface. The reaction rate parameter is positive for destructive (c > 0) and negative due to generative (c < 0) chemical reactions. Considering above Eqs. (12), (17) lead us to mention that n (non-linear stretching parameter) and x (coordinate along the surface) which appear in the fw, c and K tend to break down the similarity solution. Concentrating on the above Eqs. (12) and (17) for fw, c and K can be recognized that n and x are producing in all of them in a special form which we introduce it as the non-linear term, Pnx:

Pnx ¼

xn1 ðn þ 1Þ 2

ð18Þ

This parameter obliges our equations to be solved locally. It is clear that Pnx is equal to 1 in the linear stretching surface problem (n = 1). Rewriting fw, c and K based on the non-linear term Pnx yields an independent fwp, cp and Kp from x and n as follows:

v w v w fwp fw ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi uw m1 ðnþ1Þ u0 m1 P nx Pnx 2x

ð19Þ

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M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

a

1

b 1.2

0.8

K=0

0.6

|φ'(η)|

K=2

φ(η)

K=0

1

K=1 K=4 0.4

K=1 K=2

0.8

K=4

0.6 0.4

0.2

0.2 0

0

1

2

3

0

4

0

1

2

c

3

4

5

η

η 1

d

fw=0

1.5

fw=0

fw=0.2

0.8

fw=0.2

fw=0.4

fw=0.8

|φ'(η)|

φ(η)

fw=-0.2 fw=-0.4

0.4

fw=0.4

1

fw=0.8

0.6

fw=-0.8

fw=-0.2 fw=-0.4

0.5

fw=-0.8

0.2

0

0

1

2

3

4

5

0

0

1

2

3

4

5

6

η

η

Fig. 4. (a) Concentration profiles and (b) concentration gradient as function of g for various values of slip coefficient K, at m = 1, Sc = 1, c = 1, fw = 0.2, M = 0.1 and n = 0.5. (c) Concentration profiles and (d) concentration gradient as function of g for various values of suction/injection parameter at m = 1, Sc = 1, c = 1, K = 0.5, M = 0.1 and n = 0.5.

c¼

cp 2jDC m1 x jDC m1 ¼ ¼ uw ðn þ 1Þ u0 Pnx Pnx

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uw ðn þ 1Þ u0 Pnx ¼l ¼ K p Pnx K¼l 2xm1 m1

ð20Þ

ð21Þ

where fwp, cp and Kp are suction/injection, reaction rate parameter and slip coefficient based on Pnx which are totally independent from x and n. Consequently there is an appropriate possibility by defining these parameters (fwp, cp and Kp) keeping away from difficulties of the dependency of fw, c and K on non-linear stretching parameter n and x-coordinate. Therefore the local similarity solution of the problem for the fixed values of the x-coordinate, varying n would be obtained properly for the various values of involved parameters of the problem for momentum, energy and concentration Eqs. (13)– (15). Momentum transfer results (f-profiles) affect temperature and concentration distributions but not vice-versa. This trend leads us to solve our equations as a one-way coupled problem. These nonlinear differential equations (13)–(15) are solved numerically by using the explicit Runge–Kutta (4, 5) formula, the Dormand–Prince pair and shooting method [37] subject to the slip boundary conditions (16) which is valid locally. After solving this slip flow problem numerically, the wall shear stress, the local Nusselt number and the local Sherwood number exhibits a dependence on the slip coefficient, the magnetic parameter, the suction/injection parameter,

the Prandtl number, the Eckert number, the surface temperature parameter, the Schmitt number, order of chemical reaction and the reaction rate parameter as follows:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ðn þ 1Þ 3n1 00 x 2 jf ð0Þj sw ¼ lu0 2m1 y¼0  s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   x @T @y  u0 ðn þ 1Þ nþ1 y¼0 0 x2 ¼ h ð0Þ Nux ¼ 2m1 Tw  T1 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   x @C u0 ðn þ 1Þ nþ1 x2 ¼ /0 ð0Þ Shx ¼ C w  C 1 @y y¼0 2m1   @u ¼ l  @y

ð22Þ

ð23Þ ð24Þ

3. Results and discussion Table 1 shows comparison of the magnitude of the velocity gradient at the wall |f0 0 (0)| between the present code results using the Dormand–Prince pair and shooting method and that obtained previously by other researchers. This table indicates that our results are compatible with the previous works of Cortell [32] and recent work of Javed et al. [18] regarding to non-linear stretching surface problem and that of Fang [30] due to MHD slip flow analysis over stretching surface. It is apparent that accelerating velocity tends to increase the skin friction at the wall. On the other hand, the slip coefficient tends to decrease wall shear stress. The numerical

3220

a

M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

1.6

M=0.0 γ=1 M=0.3 M=0.4 M=0.0 γ=1.5 M=0.3 M=0.4

1.5

|φ'(0)|

1.4

1.3

1.2

1.1

-0.2

b

-0.15

-0.1

-0.05

0

fw

0.05

0.1

0.15

0.2

3

3.5

4

2.5

|φ'(0)|

2

Sc=1 Sc=2 Sc=3

1.5

1

0

0.5

1

1.5

2

2.5

K Fig. 5. (a) Concentration gradient at the wall as function of fw for various values of magnetic parameter and reaction rate parameter at m = 1, n = 0.5, K = 0.5 and Sc = 1. (b) Concentration gradient at the wall as function of K for various values of Sc at m = 1, M = 0.1, n = 0.5 and fw = 0.2.

results using Boundary Value Problem solver (BVP) obtained by Van Gorder [28] which agree well with their analytical solutions using the Homotopy Analysis Method (HAM) are also wellmatched with the presented work. Table 2 represents comparison of the temperature and the concentration gradient magnitude adjacent to the wall |h0 (0)|, |/0 (0)| respectively between the present code results by Runge–Kutta (4, 5) formula, the Dormand–Prince pair and shooting method and that obtained previously in the linear stretching surface case (n = 1). Fig. 1 (a) and (b) illustrates the variation of the dimensionless velocity component f0 (g) and velocity gradient magnitude |f00 (g)|, respectively, versus the similarity independent variable g for various values of slip coefficient K, at fw = 0.2 M = 0.1 and n = 0.5. Since our problem code has been solved at a specific location on the wall (x-coordinate), a large value of g is relevant to locations far from the surface (large y) at a specified xcoordinate. As slip coefficient increases, the slip at the surface wall increases, as a result reaches to a smaller amount of penetration due to the stretching surface into the fluid. In the no-slip condition, K approaches zero so the velocity slip at the wall is equal to zero, us = 0, accordingly the fluid velocity adjacent to the wall is equal to the velocity of the stretching surface (uw) then f0 (0) = 1. It is clear from Fig. 1(a) that dimensionless velocity component at the wall reduces with an increase in the slip coefficient (K). The velocity

profiles damped out a bit slower for the high amount of the slip coefficients because of an interception which exists among them. According to Fig. 1(b) an increase in the slip coefficient (K) leads to decrease wall shear stress (in absolute sense) over non-linear stretching surface. Fig. 1(c) and (d) shows variations of the dimensionless velocity component f0 (g) and the velocity gradient magnitude |f00 (g)| as function of g respectively for various values of suction/injection parameter at K = 0.5, M = 0.1 and n = 0.5. As an output of Fig. 1(c), it is understandable that velocity of the fluid on the wall decreases by increasing suction parameter whereas increases with injection at specific slip coefficient. An interception point exists among the shear stress profiles. Therefore suction leads the shear stress profiles to damp faster through fluid field. Fig. 1(d) indicates the wall shear stress would be increased with increase in suction whereas injection tends to decrease wall shear stress. The shear stress profiles act in the different way due to the suction/injection parameter before and after interception point. By increasing g to leave behind interception point into fluid field, the shear stress reduces by suction. Fig. 1(e) and (f) shows variations of the velocity profiles and velocity gradient magnitude, respectively, as function of g for various values of magnetic parameter M, at fw = 0.2, n = 0.5 and K = 0.5. Fig. 1(e) indicates that the Lorentz force changes the velocity profile such that the velocity distribution decreases with increasing M. It is clear that there is no interception point among the velocity profiles. An interesting result that can be recognized is velocity of the fluid adjacent to the wall would be changed by different magnetic parameters which this occurrence happens just only in the slip flow regime. Consequently magnetic parameter leads to decrease f0 (0) at the wall. In the presence of the Lorenz force, the skin friction coefficient increases with a raise in the magnetic parameter shown in Fig. 1(f). Since the interception point is obvious in the middle of the velocity gradient profiles, shear stress reduces by increasing magnetic parameter into the fluid regime. Variations of the dimensionless velocity f0 (0) and velocity gradient |f00 (0)| adjacent to the wall as function of fw in different slip coefficient (K) are shown in Fig. 2(a) and (b) respectively at n = 0.5 and M = 0.4. Due to the no-slip condition, K approaches zero so the velocity of the fluid adjacent to the wall is equal to the surface velocity, f0 (0) = 1 shown in Fig. 2(a). It is concluded from this figure that the dimensionless velocity at the wall, f0 (0), decreases with the increase in the slip coefficient (K) at a specify permeability parameter, fw. Injection increases f0 (0) but suction tends to decrease it at the wall. One of the interesting aspects of the slip flow boundary layer which has been shown in this figure is in the slip problems; permeability parameter is liable to modify velocity of the fluid adjacent to the wall, f0 (0), which has no effect on it in the no-slip boundary condition case. It is clear that f0 (0) decreases by increasing suction and increases by injection which is a constant value and equal to 1 in the no-slip flow case. Fig. 2(b) illustrates in the no slip condition problem the highest wall shear stress occurs. On the other hand increasing slip coefficient lead to decrease wall shear stress by decreasing |f00 (0)|. Another interesting feature in this figure is that gradient of the |f00 (0)| versus fw is much lower in the high slip coefficients K, compared with the low slip coefficients. This fact tends to make permeability parameter negligible at high slip coefficients which lead to make no difference between suction and injection at the wall. The reason of this occurrence is that the boundary layer would be much more affected by slip coefficients than suction/ injection parameters. Fig. 2(c) and (d) illustrates the dimensionless velocity f0 (0) and velocity gradient magnitude |f00 (0)| adjacent to the wall versus slip coefficient (K) in the various values of the magnetic parameter (M) at n = 0.5 and fw = 0.2. It is obvious from Fig. 2(c) that the velocity of the fluid at the wall f0 (0) decreases monotonically with the increase in the slip coefficient (K). As slip coefficient (K) increases, the slip velocity (us) at the surface wall

M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

1.8

Sc=1 m=1 Sc=2 Sc=3 Sc=1 m=2 Sc=2 Sc=3 Sc=1 m=3 Sc=2 Sc=3

1.6

1.4

|φ'(0)|

3221

1.2

1

0.8

0.6 0.3

0.35

0.4

0.45

n

0.5

0.55

0.6

Fig. 6. The effects of Sc, m and n, on |/0 (0)| when fwp = 0.2, Kp = 0.2, cp = 0.3.

0.8

γ =0.1 p

γ =0.2 p

0.75

γ =0.3 p

γ =-0.1 P

0.7

γ =-0.2

|φ(0)|

p

-

γ =-0.3 p

0.65

0.6

0.55

0.5 0.3

0.35

0.4

0.45

n

0.5

0.55

0.6

Fig. 7. The effects of cp and n, on |/0 (0)| when fwp = 0.2, Kp = 0.2, m = 2, Sc = 1, M = 0.1.

increases, and the penetration of the stretching wall effects into the fluid decreases. By adding to magnetic parameter, f0 (0) decreases. This behavior is due to the growing effect of the Lorenz force in the flow regime. Fig. 2(d) indicates that influences of the magnetic parameter on wall shear stress are much more significant at low slip coefficients. It is concluded that by increasing K, the Lorenz force has no affect on the wall shear stress. Variations of the dimensionless velocity f0 (0) adjacent to the wall as function of stretching non-linear parameter in different magnetic parameters are shown in Fig. 3(a) when fwp = 0.2 and Kp = 0.5. This figure indicates that the minimum which exists in the profiles moves downward through the left side by increasing magnetic parameter. f0 (0) is connected to f0 0 (0) due to the slip boundary conditions along the surface. On the other hand magnetic field tends to enhance the

Lorenz force in the fluid field which increases velocity gradient at the wall, f00 (0). As a result magnetic parameter tends to decrease f0 (0). Velocity of the fluid adjacent to the wall can be both decreased and increased depending on the stretching non-linear parameter because of the minimum which occurs within the profiles. In Fig. 3(b), we have plotted the velocity gradient magnitude |f00 (0)| adjacent to the wall showing the effects of the non-linear stretching parameter on both magnetic parameter M and slip coefficient, Kp. It can be seen that the combined effect of increasing the values of n and M is to increase wall shear stress. On the other hand slip absolutely tends to decrease wall shear stress. Influences of the slip coefficient K, on concentration profile and concentration gradient at Sc = 1, c = 1, fw = 0.2, M = 0.1, m = 1 and n = 0.5 have been shown in Fig. 4(a) and (b) respectively. Increasing

3222

M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

0.95

m=1 Kp=0.2 m=2 m=3 m=1 Kp=0.5

0.9

m=2 m=3

|φ(0)|

0.85

-

0.8

0.75

0.7 0.3

0.4

0.5

0.6

n Fig. 8. The effects of Kp, m and n, on |/0 (0)| when fwp = 0.2, Sc = 1, M = 0.1, cp = 0.5.

a

a

1

2.6 2.4

θ(η)

0.6

K=0

2.2

K=1

2

K=2

1.8

M=0 M=0.3

|θ(0)|

0.8

K=4

`

0.4

M=0.4

1.6 1.4 1.2

0.2

1 0

b

0

0.5

1

1.5

2

η

2.5

0.8 -0.2

3

-0.15

-0.1

-0.05

0

0.05

b

3

K=0

|θ(0)|

|θ(η)|

K=4

`

3.5

4

2.5

K=2 1.5

0.2

Pr=1 Pr=3 Pr=5 Pr=7

3

K=1

2

0.15

4 3.5

2.5

0.1

fw

2

`

1.5

1

1

0.5

0.5 0

0

0

0.4

0.8

η

1.2

1.6

2

Fig. 9. Temperature profiles (a) and temperature gradient (b) as function of g for various values of slip coefficient K, at Ec = 0, k0 = 0, Pr = 7, n = 0.5, fw = 0.2 and M = 0.1 (CST).

0

0.5

1

1.5

2

2.5

3

K Fig. 10. (a) Temperature gradient on the wall as function of fw for various values of magnetic parameter at Ec = 0, k0 = 0, Pr = 7, n = 0.5 and K = 0.5 (CST). (b) Temperature gradient on the wall as function of K for various values of Prandtl number at Ec = 0, k0 = 0, n = 0.5, fw = 0.2 and M = 0.1 (CST).

M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225 3.2 3 2.8

Ec=0.1

|θ'(0)|

2.6

Ec=0

2.4 2.2 2 1.8 1.6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Kp Fig. 11. The effects of Kp on |h0 (0)| and viscous dissipation term when fwp = 0.0, M = 0.0, k0 = 2n, Pr = 5 (PST).

slip coefficient adds to the velocity slip on the wall. As a result, it increases the concentration and reduces the concentration gradient at the wall (g = 0). Increasing slip coefficient increases boundary layer thicknesses lead to a low concentration gradient magnitude which means a low local Sherwood number. There is an interception in the midst of concentration gradient profiles which tends slip coefficient acts on the contrary through fluid flow. Fig. 4(c) and (d) illustrates variation of the concentration profiles and the concentration gradient magnitude respectively as function of g for various values of permeability parameter fw at Sc = 1, c = 1, K = 0.5, M = 0.1, m = 1 and n = 0.5. Suction/injection makes a significant effect on concentration profiles shown in Fig. 4(c). There is no interception point between concentration distributions. Increasing injection on the wall guides the concentration distribution to be improved into the fluid whereas suction leads to decrease them on the contrary. The concentration boundary layer thickness decreases by suction while injection makes it thicker at the surface. Regarding the magnitude of the concentration gradient profiles

which have been shown in Fig. 4(d), the local Sherwood number increases due to suction whereas it decreases in the injection case. Fig. 5(a) shows variation of the concentration gradient on the wall versus suction/injection parameter for different values of the magnetic parameter and the reaction rate parameter in destructive first order chemical reaction at n = 0.5, K = 0.5, m = 1 and Sc = 1. Combination of different applications which appears in this figure guides us to understand carefully how to reach a high concentration gradient on the wall. Increasing suction tends to increase the concentration gradient in different reaction rate parameters. As the reaction rate parameter increases, the concentration gradient increases whereas magnetic parameter moves it down to reach a lower local Sherwood number. The Schmidt number has a significant effect on mass transfer due to an increased concentration gradient on the wall. Fig. 5(b) illustrates variation of the concentration gradient on the wall versus slip coefficient for different values of the Schmidt number at m = 1, M = 0.1, n = 0.5 and fw = 0.2. The Schmidt number tends to increase the Sherwood number by increasing concentration gradient on the wall. The reason for this trend is that the concentration boundary layer becomes thin for large Sc number. On the other hand slip coefficient tends to decrease it on the contrary. Fig 6 illustrates variation of the concentration gradient on the wall for higher order-chemical reactions and Schmidt numbers. It shows that the concentration gradient magnitude at the wall increases with an increase in the Schmidt number in destructive chemical reaction (cp > 0). The non-linear stretching n leads to slight enhancement in the concentration gradient magnitude in the slip flow domain. As shown in Fig. 6 the concentration gradient magnitude at the wall decreases when the reaction-order parameter increases. Fig. 7 illustrates variation of the concentration gradient magnitude at the wall versus nonlinearity stretching parameter for both destructive (cp > 0) and generative (cp < 0) chemical reactions. This figure shows that non-linear stretching parameter tends to increase surface mass transfer slightly for destructive chemical reactions whereas the concentration gradient decreases at the wall in the generative chemical reactions case. Increasing the reaction rate parameter in absolute sense decreases the concentration gradient at the wall for the generative chemical reaction but the opposite effect is observed for the case of destructive chemical reaction. Fig. 8 presents the combined effect

3.8 Kp=0.0 Ec=0.0 Kp=0.5

3.6

Kp=1 3.4

Kp=2 Kp=0.0 Ec=0.1

|θ'(0)|

3.2

Kp=0.5 Kp=1

3

Kp=2 2.8 2.6 2.4 2.2 0.5

0.55

0.6

0.65

3223

0.7

0.75

n Fig. 12. The effects of Ec, Kp and n, on |h0 (0)| when fwp = 0.2, M = 0.1, k0 = 2n, Pr = 5.

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M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

of reaction-order parameter and slip coefficient on the concentration gradient magnitude at the wall. The surface mass transfer decreases for higher-order reaction in slip flow domain. Slip coefficient has significant effect on the surface mass transfer by decreasing the concentration gradient at the wall. Increasing non-linear stretching parameter n enhances surface mass transfer for destructive chemical reaction. Fig. 9(a) and (b) shows variation of the temperature and temperature gradient profiles versus g for different values of the slip coefficient respectively at constant surface temperature (CST) when Pr = 7, k0 = 0, Ec = 0, n = 0.5, fw = 0.2 and M = 0.1. As we explained before there is no temperature jump in the liquid fluids [20]. On the other hand due to the essential applications of the MHD liquids in microfluidic devices especially in MHD micro pumps, we consider liquid fluids being very similar in physical properties to water (Pr = 7) such as blood plasma [6]. Fig. 9(a) indicates that an increase with slip coefficient tends to increase temperature into the fluid field. By escalating K, thermal boundary layer thickness enhances. There is no interception point among the temperature profiles but exists in the middle of the temperature gradient profiles shown in Fig. 9(b). Higher slip along the surface lower wall temperature gradient (in absolute sense) reaching to low heat transfer rate. Variation of the heat transfer rate versus suction/injection parameter for various values of magnetic parameter at constant surface temperature (CST) when k0 = 0, Ec = 0, Pr = 7, n = 0.5 and K = 0.5 has been shown in Fig. 10(a). It is indicated that increasing suction (fw > 0) at the wall tends to have high heat transfer rate but injection (fw < 0) decreases heat transfer rate (in the absolute sense). Magnetic parameter due to applied the Lorenz force on the liquid field makes the wall temperature gradient to decrease. As a result Nusselt number decreases by magnetic parameter. Fig. 10(b) illustrates variation of the heat transfer rate versus slip coefficient K for various values of Prandtl number at (CST) when k0 = 0, Ec = 0, n = 0.5, fw = 0.2 and M = 0.1. It is illustrated that wall temperature gradient decreases with slip coefficient while increases with the Prandtl number. On the other hand variation of the K is much more considerable for high Prandtl numbers. To reach a high heat transfer rate, less slip on the wall by a liquid with a high Prandtl number is needed. Variation of the rate of heat transfer versus Kp with/without viscous dissipation term has been obtained in Fig. 11. Consideration of the viscous dissipation term is being essential when high velocity gradient happens on the fluid field. This figure illustrates that the viscous dissipation

1.85

|θ'(0)|

1.8

M=0.1 Ec=0.1 M=0.3 M=0.4 M=0.1 Ec=0.0 M=0.3 M=0.4

1.75

1.7 0.5

0.55

0.6

n

0.65

0.7

0.75

Fig. 13. The effects of Ec, M and n, on |h0 (0)| when fwp = 0.2, Kp = 0.2, k0 = 0.2, Pr = 5.

term would be negligible at high slip coefficients in slip flow boundary layer. It is interesting to note that slip coefficient by means of decreasing velocity gradient at the wall leads to viscous dissipation being insignificant. Hence both profiles of the with/ without viscous dissipation would be compatible at high slip coefficients. Variation of the heat transfer rate versus non-linear stretching parameter for various values of Kp and Ec at Pr = 5, k0 = 2n, fwp = 0.2 and M = 0.1 has been shown in Fig. 12. This figure indicates that the viscous dissipation term would be negligible at high slip coefficients. It illustrates that the temperature gradient profiles of Ec = 0.1 and Ec = 0.0 are being close to each other when Kp = 2 which means viscous dissipation is going to be negligible. In both no-slip flow and liquid flow with low slip coefficients can be concluded that the Eckert number is significant which tends to decrease heat transfer rate. Non-linear stretching parameter leads to increase heat transfer rate when k0 > 0. Fig. 13 presents the combined effect of the magnetic parameter M and the Eckert number on the temperature gradient on the wall when k0 < 0. This figure indicates that the rate of heat transfer decreases by both the Eckert number and the magnetic parameters whereas non-linear stretching parameter leads to increase the heat transfer rate in the presence of slip. By considering Figs 12 and 13 it can be seen that non-linear stretching parameter n is able to enhance heat transfer rate in both negative and positive k0 .

4. Conclusion Flow field and heat transfer as well as mass transfer due to higher order chemical reaction have been investigated for an MHD slip flow regime over non-linear permeable stretching surface at prescribed surface temperature (PST) using a local similarity solution subject to the x-coordinate along the surface. A nonlinear term called by Pnx has been introduced where by rewriting fw, c and K it guides us reaching to an independent fwp, cp and Kp from x and n. The results suggest that  The skin friction coefficient increases with the increase in the magnetic parameter M, and non-linear stretching parameter n while skin friction coefficient decreases with slip coefficient (K, Kp).  Influences of the suction/injection parameter (fw, fwp) on the wall shear stress are negligible at high-slip coefficients (K, Kp) which implies that suction/injection parameters (fw, fwp) have no effects at the wall.  High concentration gradients are possible at low-slip coefficients, low-order chemical reaction m and at high reaction rate parameter (c, cp) in destructive chemical reactions, high nonlinear stretching parameters in destructive chemical reactions, and high Schmitt numbers.  Both high reaction rate parameter (c, cp) and n tend to increase mass transfer on the wall in the destructive chemical reactions, but an opposite behavior can be seen in the generative chemical reactions.  High heat transfer rate has been found at liquids with high Prandtl with a low-slip coefficient and a low magnetic parameter.  In both cases of no-slip and low-slip coefficients, the Eckert number is significant to decrease heat transfer rate. In addition, the viscous dissipation term can be negligible for high-slip coefficients.  Non-linear stretching parameter n is able to enhance heat transfer rate in both negative and positive surface temperature parameters k0 . The analytical solution presented here is potentially influential in controlling wall shear stress as well as the local Nusselt and

M.H. Yazdi et al. / International Journal of Heat and Mass Transfer 54 (2011) 3214–3225

Sherwood numbers. These results are able to guide investigation on a variety of systems such as micromixing of biological samples using magnetic field in open microchannels and MHD micro pump.

[17]

Acknowledgments

[18]

The authors would like to acknowledge the Ministry of Higher Education, Malaysia for sponsoring this work. They would like also to thank the reviewers for their time and interests as well as constructive suggestions and comments for improving the paper.

[19]

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