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Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate Swati Mukhopadhyay*, Iswar Chandra Mandal a

Department of Mathematics, The University of Burdwan, Burdwan-713104, W.B., India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 August 2014 Received in revised form 27 September 2014 Accepted 7 October 2014 Available online xxx

The effects of velocity slip and thermal slip on MHD boundary layer mixed convection ﬂow and heat transfer of an incompressible ﬂuid past a plate in presence of suction/blowing are presented. Using similarity transformations the governing partial differential equations are reduced to ordinary differential equations and the nonlinear equations are then solved numerically with the help of shooting method. The present analysis reveals that by reducing the boundary layer thickness, the increasing velocity slip parameter makes the ﬂuid velocity to increase whereas non-dimensional temperature decreases for increasing values of velocity slip parameter. The rate of heat transfer decreases with the increasing values of thermal slip. With increasing values of suction (blowing) parameter the surface temperature decreases (increases). Copyright © 2014, The Authors Published by Elsevier B.V.on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Keywords: Mixed convection MHD Velocity slip Thermal slip Suction/blowing

1. Introduction During the last decades, ﬂow of an incompressible viscous ﬂuid and heat transfer phenomena over a plate have received great attention owing to the abundance of practical applications in chemical and manufacturing processes. The interest in mixed convection boundary layer ﬂows of viscous incompressible ﬂuids is increasing substantially due to its large number of practical applications in industry and manufacturing processes [1]. When the ﬂow arises naturally simply owing to the effect of a density difference, resulting from a temperature or concentration difference in a body force ﬁeld, such as the gravitational ﬁeld, the process is termed free convection. The density difference gives rise to a buoyancy effect that generates the ﬂow. The cooling of the heated body in ambient air generates such a ﬂow in the region surrounding it. The buoyancy forces arising from the simultaneous effects of temperature differences play a signiﬁcant role in mixed convective thermal diffusion when the ﬂow velocity is relatively small and the temperature difference is relatively large [20]. The MHD boundary layer theory has a signiﬁcant place in the development of the magneto hydrodynamics. In recent years, the study of MHD ﬂow and the heat transfer in porous surface has * Corresponding author. E-mail addresses: [email protected] (S. Mukhopadhyay), [email protected] (I. Chandra Mandal). Peer review under responsibility of Karabuk University.

iswar.

attracted much interest of researchers due to the effect of magnetic ﬁeld on the boundary layer ﬂow control [7,15,20]. Convective boundary layer ﬂow of an electrically conducting ﬂuid in the presence of a magnetic ﬁeld has been the subject of a great number of investigations due to its fundamental importance in industrial and technological applications including surface coating of metals, crystal growth and reactor cooling [7,9,13]. The Lorentz force is active and interacts with the buoyancy force in governing the ﬂow and temperature ﬁelds. The effect of the Lorentz force is known to suppress the convection and an external magnetic ﬁeld is applied as a control mechanism in material manufacturing industry [13]. The MHD parameter is one of the important parameters by which the cooling rate can be controlled and the product of the desired quality can be achieved [9]. In all the studies mentioned above, the no-slip boundary condition was assumed. In the recent years, in case of micro-electromechanical systems microscale ﬂuid dynamics received much attention [21e23]. Due to the microscale dimensions, the ﬂuid ﬂow behaviour belongs to the slip ﬂow regime and greatly differs from the traditional ﬂow [21]. The non-adherence of the ﬂuid to a solid boundary, also known as velocity slip, is a phenomenon that has been observed under certain circumstances [14,22]. When the ﬂuid has particulate matter such as emulsions, suspensions, foams and polymer solutions, the no-slip condition is inadequate [23]. Slip occurs also in ﬂuoroplastic coating (e.g. Teﬂon) resists adhesion. The ﬂuids that exhibit slip effect have many applications, for instance, the polishing of artiﬁcial heart valves and internal cavities

http://dx.doi.org/10.1016/j.jestch.2014.10.001 2215-0986/Copyright © 2014, The Authors Published by Elsevier B.V.on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/3.0/).

Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001

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Nomenclature f f0 M Pr S T Tw T∞ u∞

non-dimensional stream function streamwise velocity magnetic parameter Prandtl number suction/blowing parameter temperature of the ﬂuid temperature of the wall of the surface free-stream temperature free stream velocity

Greek symbols b thermal slip parameter d velocity slip parameter h similarity variable k thermal diffusivity l mixed convection parameter m dynamic viscosity n kinematic viscosity j stream function r density of the ﬂuid q non-dimensional temperature

[21]. Martin and Boyd [12] investigated the effects of slip on momentum and heat transfer in a laminar boundary layer ﬂow over a ﬂat plate. Cao and Baker [6] studied the mixed convective ﬂow and heat transfer from a vertical plate by considering velocity slip and temperature jump boundary conditions and they obtained local non-similar solutions. Pal and Shivakumara [17] discussed the effects of sparsely packed porous medium on mixed convection ﬂow and heat transfer from a vertical plate. Datta et al. [8] discussed the surface mass transfer effects on mixed convection ﬂow over a nonisothermal plate. Recently, the combined effects of slip and uniform heat ﬂux condition at the surface on boundary layer ﬂow over a ﬂat plate was studied by Aziz [2] and in his paper the local similarity also appeared in the slip boundary condition. Recently Pal and Talukdar [18], presented an analytical solution of unsteady MHD convective heat and mass transfer past a vertical permeable plate with thermal radiation and chemical reaction in the presence of slip at the boundary. Of late Bhattacharyya et al. [3], investigated the combined effect of magnetic ﬁeld and slip on ﬂow over a plate. The effects of suction/blowing and thermal radiation on forced convection ﬂow over a plate in a Darcy-Forchheimer porous medium was investigated by Mukhopadhyay et al. [16]. Bhattacharyya et al. [5] investigated the mixed convection ﬂow past a vertical plate in presence of slip. They did not consider the buoyancy opposed ﬂow. Recently Patil et al. [19], analyzed the unsteady mixed convection ﬂow from a moving plate in a parallel free stream. They reported the combined effects of thermal radiation and Newtonian heating on mixed convection ﬂow and heat transfer. The aim of the present paper is to extend the study of Bhattacharyya et al. [5] by considering the combined effects of magnetic ﬁeld and suction/blowing. Moreover, the effects of velocity slip and thermal slip on steady mixed convection ﬂow and heat transfer past a vertical plate are analyzed. Both the cases of buoyancy aided and buoyancy opposed ﬂows are considered. Using similarity variables, a third order and a second order differential equations corresponding to the momentum and thermal boundary layer equations are obtained. These equations are solved numerically using shooting method. The effects of the magnetic parameter, mixed convection parameter, velocity slip parameter, thermal slip parameter, suction/blowing

parameter on velocity and temperature ﬁelds are investigated and analyzed with the help of their graphical representations. 2. Formulation of the problem We consider a mixed convection two-dimensional steady laminar boundary-layer ﬂow of an incompressible, viscous liquid over a vertical plate of very small thickness and much larger breadth in presence of a magnetic ﬁeld. The governing equations for MHD boundary layer ﬂows and heat transfer are written as

vu vy þ ¼ 0; vx vy

(1)

u

vu vu v2 u sB2 þy ¼n 2 u u∞ þ gb* T T∞ ; vx vy r vy

(2)

u

vT vT v2 T þy ¼ k 2: vx vy vy

(3)

Since the velocity of the ﬂuid is low (laminar ﬂow), the viscous dissipative heat is assumed to be negligible here. The x co-ordinate is measured from the leading edge of the plate, y co-ordinate is measured along the normal to the plate [see Fig. 1(a)]. Here u and y are the components of velocity respectively in the x and y directions, m is the coefﬁcient of ﬂuid viscosity, r is the ﬂuid density, n ¼ m=r is the kinematic viscosity, s is the electric conductivity, k is the thermal diffusivity of the ﬂuid, u∞ is the free stream velocity, pﬃﬃﬃ B ¼ B0 = x is the non uniform magnetic ﬁeld applied along the yaxis where B0 is a constant, b* is the volumetric coefﬁcient of thermal expansion, T is the temperature, T∞ is the free stream temperature assumed constant. The appropriate boundary conditions for the problem are given by [3,5]

u ¼ L1

vu vT ; y ¼ yw ; T ¼ Tw þ D1 at y ¼ 0; vy vy

u ¼ u∞ ; T ¼ T∞ as y/∞:

(4) (5)

Here the plate temperature Tw is variable with Tw > T∞ and is given by Tw ¼ T∞ þ T0 =x, T0 being a constant. L1 ¼ LðRex Þ1=2 is the velocity slip factor and D1 ¼ DðRex Þ1=2 is the thermal slip factor with L and D being the initial values of velocity and thermal slip factors having the same dimension of length, Rex ¼ u∞ x=n is the local Reynolds number. 2.1. Similarity analysis and solution procedure We also introduce the following dimensionless variables [5]

q¼

T T∞ ; Tw T∞

(6)

rﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u∞ ; u∞ nxf h ; h ¼ y nx

(7)

and

j¼

where the stream function j is given by the following relations as

u¼

vj vj ;y ¼ : vy vx

(8)

Using the relations (6)e(8) in the boundary layer equation (2) and in the energy equation (3) we get the following equations.

Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001

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The boundary conditions (4) and (5) then become

vj v2 j vj u∞ 0 ¼ L1 2 ; ¼ yw ; q ¼ 1 þ D q at y ¼ 0; vy n vy vx

(11)

vj ¼ u∞ ; q ¼ 0 as y/∞: vy

(12)

Using equation (8), the equations (9) and (10) ﬁnally can be put in the following form

000 1 00 f þ ff M f 0 1 þ lq ¼ 0; 2

(13)

1 00 1 0 q þ f q þ f 0 q ¼ 0; Pr 2

(14)

where l ¼ gb* T0 =u2∞ is the mixed convection parameter, M ¼ sB20 =ru∞ is the magnetic parameter and Pr ¼ n=k is the Prandtl number. If l > 0, buoyancy forces act in the direction of the mainstream and ﬂuid is accelerated (favourable pressure gradient) (assisting ﬂow) whereas for l < 0, buoyancy forces oppose the motion, retarding the ﬂuid in the boundary layer (adverse pressure gradient) (opposing ﬂow). Here, l ¼ 0 gives a purely forced convection situation. In view of these, the boundary conditions ﬁnally become 00

f 0 ¼ df ; f ¼ S; q ¼ 1 þ bq0 at h ¼ 0

(15)

and

f 0 ¼ 1; q ¼ 0 at h/∞;

(16)

where d ¼ Lu∞ =n is the velocity slip parameter and b ¼ Du∞ =n is the thermal slip parameter. In the absence of magnetic ﬁeld and buoyancy force, i.e. when M ¼ 0 ¼ l the equation (13) reduces to the equation of boundary layer ﬂow on a ﬂat plate at zero incidence (i.e. in case of a horizontal plate). Equations (13) and (14) along with the boundary conditions are solved by converting them to an initial value problem. In order to integrate these equations as an initial value problem we require a 00 value for f ð0Þ and q0 ð0Þ but no such values are given at the 00 boundary. The suitable guess values for f ð0Þ and q0 ð0Þ are chosen and then integration is carried out. We compare the calculated values for f 0 and q at h ¼ 6 with the given boundary conditions 00 f 0 ð6Þ ¼ 1 and qð6Þ ¼ 0 and adjust the estimated values f ð0Þ and 00 q0 ð0Þ. The series of values for f ð0Þ and q0 ð0Þ are taken and the fourth order classical Runge-Kutta method with step-size h ¼ 0.01 is applied. We repeat the procedure until we get the converged results within a tolerance limit of 105. 00

Fig. 1. (a) Sketch of the physical ﬂow problem. (b)Velocity f 0 (h) and shear stress f (h) proﬁles for M ¼ 0 ¼ S and l ¼ 0 in the absence of slip.

2

2

3

sB2

vj v j vj v j v j ¼n 3 vy vxvy vx vy2 r vy

vj u∞ vy

! þ gb* ðTw T∞ Þq;

(9)

and

vj vq q vj vq v2 q ¼ k 2: vy vx x vx vy vy

(10)

3. Results and discussions In order to get a clear insight of the physical problem, numerical computations have been carried out using the method described in the previous section for various values of different parameters such as the magnetic parameter (M), mixed convection parameter (l), velocity slip parameter (d), thermal slip parameter (b), suction/ blowing parameter (S) and Prandtl number (Pr) encountered in this problem. For illustrations of the results, numerical values are plotted in the ﬁgures Figs. 2(a)eFig. 8(d). For the veriﬁcation of the accuracy of the applied numerical method our results are compared corresponding to the velocity and

Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001

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Fig. 2. (a) Velocity proﬁles for several values of the magnetic parameter M for buoyancy aided ﬂow in presence of suction/blowing. (b) Velocity proﬁles for several values of the magnetic parameter M for buoyancy opposed ﬂow in presence of suction/blowing. (c) Temperature proﬁles for several values of the magnetic parameter M for buoyancy aided ﬂow in presence of suction/blowing. (d) Temperature proﬁles for several values of the magnetic parameter M for buoyancy opposed ﬂow in presence of suction/blowing.

Fig. 3. (a) Velocity proﬁles for several values of velocity slip parameter d for buoyancy aided ﬂow in presence of suction/blowing. (b) Velocity proﬁles for several values of velocity slip parameter d for buoyancy opposed ﬂow in presence of suction/blowing. (c) Temperature proﬁles for variable values of velocity slip parameter d for buoyancy aided ﬂow in presence of suction/blowing. (d) Temperature proﬁles for variable values of velocity slip parameter dfor buoyancy opposed ﬂow in presence of suction/blowing.

Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001

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Fig. 4. (a) Velocity proﬁles for several values of mixed convection parameter l in presence of suction/blowing. (b) Temperature proﬁles for several values of mixed convection parameter l in presence of suction/blowing.

shear stress proﬁles in case of forced convective ﬂow in the absence of magnetic ﬁeld and suction/blowing and also with no-slip boundary conditions with the available published results of [10] in Fig. 1(b) and are found in excellent agreement. Moreover, the computed skin-friction coefﬁcient for forced convection ﬂow past a non-porous plate for no-slip boundary condition and also in the absence of magnetic ﬁeld is compared with the available result of [10] and [4] and found in excellent agreement [see Table 1]. Fig. 2(a)e(d) illustrates the effects of the magnetic parameter (M) on velocity and temperature ﬁled for buoyancy aided/opposed ﬂows. These ﬁgures show the variations of velocity and temperature proﬁles for increasing values of M when the other parameters are kept constant. Fig. 2(a) clearly indicates that the thickness of the velocity boundary layer decreases with increasing values of M for buoyancy aided ﬂow. In this case, velocity is found to increase with the increasing values of M. Same nature of the velocity ﬁeld is observed by [11]. With a rise in the strength of magnetic parameter, the Lorentz force associated with the magnetic ﬁeld makes the boundary layer thinner. Magnetic lines of force move past the plate at the free stream velocity. The ﬂuid which is decelerated by the viscous action, receives a push from the magnetic ﬁeld which counteracts the viscous effects. The temperature decreases with the increase of M within the boundary layer for buoyancy aided ﬂow [Fig. 2(c)]. Thickness of velocity boundary layer also decreases with increasing values of M for buoyancy opposed ﬂow [Fig. 2(b)]. This effect is more pronounced for buoyancy opposed ﬂow [Fig. 2(b)] than that of buoyancy aided ﬂow [Fig. 2(a)]. Here M ¼ 0 represents the case when there is no applied magnetic ﬁeld. The decrease in temperature is more prominent in case of buoyancy opposed ﬂow [Fig. 2(d)] than that of buoyancy aided ﬂow [Fig. 2(c)]. Increase in the magnetic parameter M causes a decrease in the thermal

boundary layer thickness. The rate of heat transfer (the thermal boundary layer thickness becomes thinner) is enhanced [Fig. 2(c), (d)] when the velocity boundary layer thickness decreases [Fig. 2(a), (b)]. Fluid velocity is higher for suction case than that of blowing case [Fig. 2(a), (b)]. But the temperature is higher for blowing case than that in suction case [Fig. 2(c), (d)]. Fig. 3(a)e(d) presents the nature of velocity and temperature proﬁles respectively for the variation of velocity slip parameter d for both buoyancy aided and opposed ﬂows. From Fig. 3(a), (b) it is very clear that due to slip, velocity increases in both cases of buoyancy aided and opposed ﬂows but the velocity increase is higher for buoyancy aided ﬂow [Fig. 3(a)] than that of buoyancy opposed ﬂow [Fig. 3(b)]. Due to the slip condition at the plate the velocity of ﬂuid adjacent to the plate acquires some positive value and accordingly the thickness of boundary layer decreases. With the increases in slip (in magnitude) more ﬂuid get permits to slip past the plate and as a result, the ﬂow accelerates near the plate and away from the plate this effect diminishes whereas shear stress decreases with increasing values of velocity slip parameter d [Fig. 3(a), (b)]. Velocity overshoot is noted only for buoyancy aided ﬂow for higher values of velocity slip parameter for both cases of suction and blowing [Fig. 3(a)]. This is due to the combined effects of buoyancy force and velocity slip. Temperature proﬁles for buoyancy aided and opposed ﬂows are exhibited in Fig. 3(c), (d) respectively. Temperature decreases with the increase in velocity slip parameter d for both buoyancy aided [Fig. 3(c)] and opposed ﬂows [Fig. 3(d)]. The enhanced velocity due to slip near the plate is the cause of increasing heat transfer. Effects of mixed convection parameter l on velocity and temperature are exhibited in Fig. 4(a), (b). Fluid velocity is found to increase [Fig. 4(a)] with increasing values of l for both cases of

Fig. 5. (a) Velocity proﬁles for variable values of suction/blowing parameter S for buoyancy aided/opposed ﬂow. (b) Temperature proﬁles for variable values of suction/blowing parameter S for buoyancy aided/opposed ﬂow.

Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001

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Fig. 6. (a) Velocity proﬁles for variable values of thermal slip parameter b for buoyancy aided ﬂow in presence of suction/blowing. (b) Velocity proﬁles for variable values of thermal slip parameter b for buoyancy opposed ﬂow in presence of suction/blowing. (c) Temperature proﬁles for variable values of thermal slip parameter b for buoyancy aided ﬂow in presence of suction/blowing. (d) Temperature proﬁles for variable values of thermal slip parameter b for buoyancy opposed ﬂow in presence of suction/blowing.

Fig. 7. (a) Velocity proﬁles for variable values of Prandtl number Pr for buoyancy aided ﬂow in presence of suction/blowing. (b) Velocity proﬁles for variable values of Prandtl number Pr for buoyancy opposed ﬂow in presence of suction/blowing. (c) Temperature proﬁles for variable values of Prandtl number Pr for buoyancy aided ﬂow in presence of suction/blowing. (d) Temperature proﬁles for variable values of Prandtl number Pr for buoyancy opposed ﬂow in presence of suction/blowing.

Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001

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Fig. 8. (a) Variation of skin friction coefﬁcient with magnetic parameter M for three values of mixed convection parameter l in presence of suction/blowing. (b) Variation of heat transfer coefﬁcient with magnetic parameter M for three values of mixed convection parameter l in presence of suction/blowing. (c) Variation of skin friction coefﬁcient with thermal slip parameter b for three values of velocity slip parameter d in presence of suction/blowing. (d) Variation of heat transfer coefﬁcient with thermal slip parameter b for three values of velocity slip parameter d in presence of suction/blowing.

suction and blowing but the temperature decreases [Fig. 4(b)] with increasing values of l. Due to favourable buoyancy effects ﬂuid velocity increases within the boundary layer for buoyancy aided ﬂow (l > 0) whereas for buoyancy opposed ﬂow (l < 0) opposite nature is noted [Fig. 4(a)]. Velocity overshoot is noted for l ¼ 0.3 only for blowing [Fig. 4(a)]. This fact is also reported by [5]. With the increase in l, temperature ﬁeld is suppressed and consequently thermal boundary layer thickness becomes thinner and as a result rate of heat transfer from the plate increases [Fig. 4(b)]. Actually l > 0 means heating of the ﬂuid or cooling of the surface (assisting ﬂow). Increase in l can lead to increase the effects of temperature ﬁeld in the velocity distribution which causes the enhancement of the velocity due to enhanced convection currents. Physically, in the process of cooling, the free convection currents are carried away from the plate to the free stream and since the free stream is in the upward direction and thus the free currents induce the velocity to enhance. As a result, boundary layer thickness increases. For opposing ﬂow l < 0, opposite effects are noted. No overshoot in temperature ﬁeld as reported by [5] is noted. This is due to the presence of magnetic ﬁeld and suction/blowing. Fig. 5(a), (b) exhibits the effects of suction/blowing parameter S on velocity and temperature proﬁles respectively. With the increasing suction (S > 0), ﬂuid velocity is found to increase

Table 1 Skin friction coefﬁcient for forced convective ﬂow past a non-porous plate in the absence of magnetic ﬁeld and slip at the boundary. 00

f ð0Þ

[10] 0.33206

[4] 0.332058

Present study 0.332058

[Fig. 5(a)] while due to injection (S < 0), ﬂuid velocity decreases. With the increase of the blowing velocity, boundary layer becomes thicker. Since the effect of suction is to suck away the ﬂuid near the wall, the momentum boundary layer is reduced due to suction (S > 0). Consequently the velocity increases. It is found that in case of suction (S > 0), the velocity proﬁles have no point of inﬂexion whereas in case of injection (S < 0), they exhibit a point of inﬂexion only for buoyancy opposed ﬂow. It is noticed that ﬂow separation occurs when S ¼ 1. This type of ﬂow separation is occurred due to decreasing ﬂuid velocity (for applied injection) as shown in Fig. 5(a). Therefore, with the help of suction through a porous wall the velocity proﬁle is made more stable. Fig. 5(b) demonstrates that the temperature decreases with the increasing suction parameter S. The thermal boundary layer thickness decreases with the suction parameter S which causes an increase in the rate of heat transfer. As the ﬂuid is brought closer to the surface, the thermal boundary layer thickness reduces. But the temperature increases due to injection. The thermal boundary layer thickness increases with injection which causes a decrease in the rate of heat transfer. Therefore, it is clear that suction enhances the heat transfer coefﬁcient much better than blowing. Thus suction can be used for cooling the surface much faster than blowing for both cases of buoyancy aided and opposed ﬂows. Fig. 6(a)e(d) displays respectively the nature of velocity and temperature proﬁles for the variation of thermal slip parameter b for buoyancy aided/opposed ﬂow. As b increases ﬂuid velocity decreases in case of buoyancy aided ﬂow (l > 0) [Fig. 6(a)] but for buoyancy opposed ﬂow (l < 0) opposite nature is noted [Fig. 6(b)]. As the thermal slip increases, less heat is transferred from the plate to the ﬂuid and hence the temperature decreases for both buoyancy

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S. Mukhopadhyay, I. Chandra Mandal / Engineering Science and Technology, an International Journal xxx (2014) 1e8

aided [Fig. 6(c)] and opposed ﬂows [Fig. 6(d)]. As the distance from the origin increases, the temperature of the plate decreases which is very clear from the expression of the plate temperature and as a result heat transfer decreases. With increasing Prandtl number, ﬂuid velocity is found to decrease in case of assisting ﬂow (l > 0) [Fig. 7(a)] but velocity increases for opposing ﬂow (l < 0) [Fig. 7(b)] for both cases of suction and blowing. Fluid velocity is higher for suction case. Velocity overshoot is noted for assisting ﬂow only [Fig. 7(a)]. Due to increasing Prandtl number Pr, temperature decreases in both the cases of assisting (l > 0) [Fig. 7(c)] and opposing (l < 0) [Fig. 7(d)] ﬂows. Consequently, thermal boundary layer thickness rapidly decreases with increasing Pr. An increase in Prandtl number means an increase of ﬂuid viscosity. This causes a decrease in the ﬂow velocity and the temperature decreases. 00 Fig. 8(a) displays the behaviour of skin friction coefﬁcient f ð0Þ with the magnetic parameter for three values of mixed convection 00 parameter l. f ð0Þ increases with increasing M and also with the increasing values of l. On the other hand, temperature gradient at the wall q0 ð0Þ decreases with increasing M and also with increasing l [Fig. 8(b)]. Skin friction coefﬁcient decreases with the increase in velocity slip as well as thermal slip parameters [Fig. 8(c)]. The skinfriction coefﬁcient is maximum at the no-slip condition which is similar to the observations of [6]. Temperature gradient at the wall decreases with velocity slip d but increases with thermal slip b [Fig. 8(d)] i.e. the rate of heat transfer increases with the increase of velocity slip d as well as magnetic ﬁeld but it decreases with thermal slip parameter b. The negative value of q0 ð0Þ physically explains that there is heat ﬂow from the plate. 4. Conclusions The numerical solutions for steady MHD mixed convection boundary layer ﬂow and heat transfer over a porous plate in presence of velocity slip and thermal slip are presented. The effect of magnetic ﬁeld on a viscous incompressible ﬂuid is to increase the ﬂuid velocity by reducing the drag on the ﬂow which in turn causes a decrease in the temperature ﬁeld. Due to slip, ﬂuid velocity increases but temperature is found to decrease with increasing values of velocity slip as well as thermal slip. It is noted that with increasing values of suction (blowing) parameter the surface temperature decreases (increases). The rate of heat transfer increases with the increasing values of velocity slip, magnetic parameter and also with increasing Prandtl number. Acknowledgement Thanks are indeed due to the reviewers for their constructive suggestions which helped a lot in improving the quality of the paper. One of the authors (S.M.) acknowledges the ﬁnancial support received from UGC, New Delhi, India through Special Assistance Programme DSA Phase 1.

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Please cite this article in press as: S. Mukhopadhyay, I. Chandra Mandal, Magnetohydrodynamic (MHD) mixed convection slip ﬂow and heat transfer over a vertical porous plate, Engineering Science and Technology, an International Journal (2014), http://dx.doi.org/10.1016/ j.jestch.2014.10.001