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ORIGINAL ARTICLE

Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating S. Das a b c

a,*

, R.N. Jana b, O.D. Makinde

c

Department of Mathematics, University of Gour Banga, Malda 732 103, India Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

Received 28 October 2014; revised 27 January 2015; accepted 8 March 2015

KEYWORDS Magnetohydrodynamic; Mixed convection; Boundary layer; Slip ﬂow and inclined plate

Abstract The combined effects of viscous dissipation and Joule heating on the momentum and thermal transport for the magnetohydrodynamic ﬂow past an inclined plate in both aiding and opposing buoyancy situations have been carried out. The governing non-linear partial differential equations are transformed into a system of coupled non-linear ordinary differential equations using similarity transformations and then solved numerically using the Runge–Kutta fourth order method with shooting technique. Numerical results are obtained for the ﬂuid velocity, temperature as well as the shear stress and the rate of heat transfer at the plate. The results show that there are signiﬁcant effects of pertinent parameters on the ﬂow ﬁelds. ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

1. Introduction Magnetohydrodynamic (MHD) mixed convective ﬂows or combined free and forced convection past a ﬂat plate has been widely studied from both theoretical and experimental standpoints over the past a few decades. MHD mixed convective ﬂows occur in many technological and industrial applications, e.g. solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency * Corresponding author. Tel.: +91 3222 261171. E-mail addresses: [email protected], [email protected] (S. Das). Peer review under responsibility of Faculty of Engineering, Alexandria University.

shutdown, heat exchangers placed in a low-velocity environment, lubrication purposes, drying technologies, ﬂows in the ocean and in the atmosphere [1,2]. Depending on the forced ﬂow direction, the buoyancy forces may aid (aiding or assisting mixed convection) or oppose (opposing mixed convection) the forced ﬂow, causing an increase or decrease in heat transfer rate [3]. The problem of mixed convection resulting from the ﬂow over a heated vertical plate is of considerable theoretical and practical interest. A detailed review of the subject, including exhaustive lists of references, can be found in the books by Bejan [4], Pop and Ingham [5], Jaluria [6] and Chen and Armaly [7]. References [8–17] are some examples of the recent relevant studies existing in the literature. Mukhopadhyay et al. [18] have presented the MHD combined convective ﬂow past a stretching surface. The mixed convection of a viscous

http://dx.doi.org/10.1016/j.aej.2015.03.003 1110-0168 ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

2 dissipating ﬂuid about a vertical ﬂat plate has been studied by Aydin and Kaya [19]. The deviation from interfacial thermodynamic equilibrium will lead to a ﬂow regime where the conventional no-slip wall condition is not valid. According to the value of Knudsen number, the ﬂows can be classiﬁed into three categories: continuum ﬂow (Kn < 0:01), slip ﬂow (0:01 6 Kn 6 0:1) and transitional ﬂow (0:1 < Kn < 10) [20]. As the ﬂow deviates away from the continuum limit, the conventional no-slip wall boundary condition fails to accurately model the surface interaction between the ﬂuid and the wall boundary due to the low collision frequency. Slip models have been proposed to ameliorate the prediction of the non-continuum phenomenon near wall boundaries within the framework of the continuum assumption. For large values of Knð> 10Þ, the Navier–Stokes equations are not applicable and the kinetic theory of gases must be employed. In many practical applications, the particle adjacent to a solid surface no longer takes the velocity of the surface. The particle has a ﬁnite tangential velocity; it slips along the surface. The ﬂow is called slip-ﬂow and this effect cannot be neglected. Cao and Baker [21] have illustrated the slip effects on a mixed convective ﬂow and heat transfer from a vertical plate. Aziz [22] has presented the hydrodynamic and thermal slip boundary layer ﬂow over a ﬂat plate with constant heat ﬂux boundary condition. The combined effects of Joule heating and viscous dissipation on a magnetohydrodynamic free convective ﬂow past a permeable stretching surface with radiative heat transfer have been determined by Chen [23]. Mukhopadhyay [24] has illustrated the slip effects on a unsteady mixed convective ﬂow and heat transfer past a porous stretching surface. Bhattacharyya et al. [25] have investigated an MHD boundary layer slip ﬂow and heat transfer over a ﬂat plate. Rohni et al. [26] have studied an unsteady mixed convective boundary layer slip ﬂow near the stagnation point on a vertical permeable surface embedded in a porous medium. Bhattacharyya et al. [27] have presented a mixed convective boundary layer slip ﬂow over a vertical plate. The unsteady mixed convective ﬂow from a moving vertical plate in a parallel free stream has been studied by Patil et al. [28]. Ellahi et al. [29] have presented a non-Newtonian MHD ﬂuid ﬂow with slip boundary conditions in porous space. A magnetohydrodynamic peristaltic ﬂow of a Jeffrey ﬂuid in eccentric cylinders has been investigated by Nadeem et al. [30]. The effects of temperature dependent viscosity on an MHD ﬂow of non-Newtonian nanoﬂuid in a pipe have been examined by Ellahi [31]. Zeeshan and Ellahi [32] have presented an MHD slip ﬂow of non-Newtonian ﬂuid in a porous space. Sheikholeslami et al. [33] have studied the Cu-water magneto-nanoﬂuid ﬂow and heat transfer. Ellahi et al. [34] have examined the effects of heat transfer and nonlinear slip on the steady Couette ﬂow. Ellahi [35] has presented the magnetohydrodynamic peristaltic ﬂow of Jeffrey ﬂuid in a rectangular duct through a porous medium. Sheikholeslami et al. [36] have reported the CuO-water nanoﬂuid ﬂow and convective heat transfer considering Lorentz forces. Khana et al. [37] have investigated the effects of heat transfer on a peristaltic motion of Oldroyd ﬂuid in the presence of inclined magnetic ﬁeld. Akbar et al. [38] have investigated the inﬂuence of heat generation and heat ﬂux in peristalsis with interaction of nanoparticles. Sheikholeslami et al. [39] have studied the natural convection of a nanoﬂuid in an enclosure with elliptic inner cylinder. A mixed convective boundary layer ﬂow over a vertical slender cylinder has been presented by Ellahi et al. [40].

S. Das et al. The object of this paper was to investigate the combined effects of viscous dissipation and Joule heating on an MHD mixed convective ﬂow past an inclined porous plate. The viscous and Joule dissipation effects are taken into consideration. The governing equations describing the problem are transformed into a non-linear ordinary differential equations by using similarity transformation. The transformed ordinary differential equations were solved numerically using fourth order Runge–Kutta method with the shooting technique. The effects of pertinent parameters on the ﬂuid velocity and temperature have been shown graphically. 2. Mathematical formulation Consider a mixed convective ﬂow of a viscous incompressible electrically conducting ﬂuid past a porous plate which is inclined from the vertical with an acute angle c measured in the clockwise direction and situated in an otherwise quiescent ambient ﬂuid at temperature T1 . Choose a Cartesian coordinates system with x-axis along the plate and the y-axis is measured normal to the sheet in the outward direction toward the ﬂuid (see Fig. 1(a)). A transverse magnetic ﬁeld of strength B is applied normal to the plate. The plate coincides with the plane y ¼ 0 and the ﬂow being conﬁned to y > 0. It is assumed that the variation of ﬂuid properties is taken to be negligible except for the essential density variation appearing in the gravitational body force. Ohm’s law is Cowling [41] ~þ ~ ~ ; J~ ¼ r E ð1Þ qB ~ E; ~ J~and r are respectively the velocity vector, the where ~ q; B; magnetic ﬁeld vector, the electric ﬁeld vector, the current density vector and the electrical conductivity. It is assumed that the magnetic Reynolds number is very small, so that induced magnetic ﬁeld can be neglected [41]. This assumption is justiﬁed since the magnetic Reynolds number is generally very small for metallic liquid or partially ionized ﬂuid. Liquid metals can be used in a range of applications because they

Figure 1(a)

Geometry of the problem.

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

Magnetohydrodynamic mixed convective slip ﬂow

3

are nonﬂammable, nontoxic and environmentally safe. That is why, liquid metals have number of technical applications in source exchangers, electronic pumps, ambient heat exchangers and also used as a heat engine ﬂuid. Moreover, in nuclear power plants sodium, alloys, lead-bismuth and bismuth are extensively utilized in the heat transfer process. Besides that, mercury play its role as a ﬂuid in high-temperature Rankine cycles and also used in reactors in order to reduce the temperature of the system. For power plants which are exerted at extensively high temperature, sodium is treated as heat-engine ﬂuid. Under the above assumptions and following Chen [23], the governing equations of the conservation of mass, momentum, energy in the presence of magnetic ﬁeld are @u @v þ ¼ 0; @x @y

ð2Þ

@u @u 1 @p @2u B u þv ¼ þ m 2 þ gb ðT T1 Þ cos c Jz ; ð3Þ @x @y q @x @y q 1 @p 0¼ ; ð4Þ q @y 2 @T @T @2T @u 1 qcp u ¼k 2 þl þv þ J2z ; ð5Þ @x @y @y @y r where u and v are the velocity components along the x and ydirections, respectively, T the temperature of the ﬂuid, l the dynamic viscosity of the ﬂuid, q the ﬂuid density, g the acceleration due to gravity, b the coefﬁcient of thermal expanpﬃﬃﬃ sion, cp the speciﬁc heat at constant pressure, B ¼ B0 = x is the non-uniform magnetic ﬁeld applied along the y-axis where B0 a constant. The last term in Eq. (3) characterizes the Lorentz force. The last two terms in Eq. (5) indicate the effects of viscous dissipation and Joule heating respectively. The physical boundary conditions are @u @T u¼L ; v ¼ vw ; T ¼ Tw þ K at y ¼ 0; @y @y u ! U1 ; T ! T1 as y ! 1;

ð6Þ

where Tw ¼ T1 þ Tx0 is the variable temperature of the plate, T0 a constant which measures the rate of increase of temperature along the plate, T1 the free stream temperature assumed constant with Tw > T1 and U1 the uniform free stream velocity. When Tw > T1 , the ﬂow is presented as aiding ﬂow since buoyancy effects have a positive component with the free stream velocity. On the other hand, if Tw < T1 , it is presented as opposing ﬂow as buoyancy effects are in the opposite direction with the free stream velocity. L and K are the velocity and thermal slip factors, respectively and when L ¼ K ¼ 0, the noslip condition is recovered. Since the magnetic ﬁeld is uniform at inﬁnity, therefore Jx ! 0; Jy ! 0 and Jz ! 0 as y ! 1. This implies that Ex ¼ 0; Ey ¼ 0 and Ez ¼ B U1 . Hence, Ohm law yields Jx ¼ 0; Jy ¼ 0 and Jz ¼ r Bðu U1 Þ. Eq. (4) suggests that the pressure p is a function of x only. On the use of the above assumptions and inﬁnity condition, equations (3) and (5) become @u @u @2u r B2 þv ¼ m 2 þ gb ðT T1 Þ cos c ðu U1 Þ; @x @y @y q 2 @T @T @2T @u qcp u ¼k 2 þl þv þ rBðu U1 Þ2 : @x @y @y @y u

ð7Þ ð8Þ

Figure 1(b) Velocity proﬁle f 0 ðgÞ and shear stress proﬁle f 00 ðgÞ for M2 ¼ 0, k ¼ 0:2; d ¼ 0:2; b ¼ 0:1; Ec ¼ 0 and c ¼ 0.

The continuity Eq. (1) is satisﬁed by introducing a stream function wðx; yÞ such as u¼

@w @w ; v¼ : @y @x

ð9Þ

The following similarity variables are introduced rﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ U1 T T1 ; w ¼ U1 m x fðgÞ; hðgÞ ¼ g¼y ; mx Tw T1

ð10Þ

where g is the similarity variable, fðgÞ the non-dimensional stream function and hðgÞ the non-dimensional temperature. On the use of (9) and (10) in Eqs. (7) and (8), we obtain the following ordinary differential equations 1 f000 þ ff 00 M2 ðf 0 1Þ þ k h cos c ¼ 0; 2 h i 1 2 00 h þ Prf h0 þ Ec Pr f 00 þ M2 ðf 0 1Þ ¼ 0: 2

ð11Þ ð12Þ

2

where M2 ¼ qrB the magnetic parameter, k ¼ gb UT10 the mixed U1 2

1 convection parameter, Ec ¼ cp ðTUw T the Eckert number and 1Þ

lc

Pr ¼ k p the Prandtl number which measures the ratio of momentum diffusivity to the thermal diffusivity. The prime denotes the differentiation with respect to g. The corresponding boundary conditions are fð0Þ ¼ S; f 0 ð0Þ ¼ d f 00 ð0Þ; hð0Þ ¼ 1 þ b h0 ð0Þ; f 0 ! 1; h ! 0 as g ! 1;

ð13Þ q ﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ where S ¼ 2 U1x m vw is the suction parameter, d ¼ L Um 1x is qﬃﬃﬃﬃﬃﬃ the velocity slip parameter, b ¼ K Um 1x the thermal slip

parameter. It is noted that S is positive for suction, but S is negative for blowing at the plate. The slip parameters d and b are different for gaseous and liquid ﬂows, they bear similar physical meanings for both. First, the values of d and b are 1 proportional to x2 and hence they describe the stream-wise location along the plate. Second, the slip parameters control the slip boundary conditions for both ﬂows. The magnitudes of the slip parameter can be used to describe the degree of non-continuum condition at the plate, or in other words, how much the ﬂow deviates from the no-slip condition. For

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

4

S. Das et al. 3. Numerical solution The non-linearity of the governing Eqs. (11) and (12) leads to use of numerical method. The transformed non-dimensional governing Eqs. (11) and (12) with boundary conditions (13) are converted into simultaneous ﬁrst order ordinary differential equations and then solved numerically by fourth order Rung–Kutta method with shooting technique [42]. The resulting higher order ordinary differential equations are reduced to ﬁrst order differential equations by letting y1 ¼ f; y2 ¼ f 0 ; y3 ¼ f 00 ; y4 ¼ h; y5 ¼ h0 : 2

Figure 2(a) Velocity proﬁle for different M when Pr ¼ 0:72; d ¼ 0:5 and c ¼ p4 for buoyancy aided ﬂow (k > 0).

ð14Þ

Thus, the corresponding higher order non-linear differential equations become y01 ¼ y2 ; y02 ¼ y3 ; 1 y03 ¼ M2 ðy2 1Þ k y4 cos c y1 y3 ; 2 y04 ¼ y5 ; h i 1 y05 ¼ Pry1 y5 EcPr y23 þ M2 ðy2 1Þ2 ; 2

instance, if d ¼ b ¼ 0 indicates that the x-location is far downstream from the leading edge of the plate where the slip effects are negligible. On the other hand, a larger values of the slip parameters indicate that the boundary condition deviates more from the no-slip case. As d and b approach to inﬁnity, the velocity and temperature slips at the plate become inﬁnity large, giving a nearly uniform velocity and temperature distribution with the condition: f00 ð0Þ ¼ 0 and h0 ð0Þ ¼ 0. It is noted that for a very large value of d and b that corresponds to a very small x at the leading edge, the boundary layer assumption is not appropriate, and as a consequence, the boundary-layer equations become inaccurate. Kundsen number (Kn) is a deciding factor, which is a measure of molecular mean free path to characteristic length. When Kundsen number is very small, no slip is observed between the surface and the ﬂuid. However, when Kundsen number lies in the range 103 to 0.1, slip occurs at the surface-ﬂuid interaction. Moreover, if a large value of d and b is due to a Knudsen number greater than 0.1, then the Navier–Stokes equation fails to model the transitional or even free molecule ﬂow regime. For this reason, a discussion regarding large values of d and b could be prone to error in nature. We therefore limit the discussion in this paper to a relatively small range of d and b from 0 to 5 as this will cover the slip ﬂow region.

where b and c are unknown which are to be determined such that the boundary conditions y2 ð1Þ and y4 ð1Þ are satisﬁed. The shooting method is used to guess b and c by iterations until the boundary conditions are satisﬁed. The resulting differential equations can be integrated by Runge–Kutta fourth order integration scheme. The accuracy of the assumed missing initial condition is checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If a difference exists, improved values of the missing initial conditions must be obtained and the process is repeated. The numerical computations are done by MATLAB package. The mesh size is taken as g ¼ 0:01. The numerical computation has been carried out for more reﬁned mesh sizes and the results are found to be independent of the

Figure 2(b) Velocity proﬁle for different M2 when Pr ¼ 0:72; d ¼ 0:5 and c ¼ p4 for buoyancy opposed ﬂow (k < 0).

Figure 3(a) Velocity proﬁle for different d when M2 ¼ 0:5; Pr ¼ 0:72 and c ¼ p4 for buoyancy aided ﬂow (k > 0).

ð15Þ

with the initial conditions: y1 ð0Þ ¼ 0; y2 ð0Þ ¼ d y3 ð0Þ; y3 ð0Þ ¼ b; y4 ð0Þ ¼ c; y4 ð0Þ ¼ 1 þ b y5 ð0Þ;

ð16Þ

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

Magnetohydrodynamic mixed convective slip ﬂow mesh size. The process is repeated until we get the results correct up to the desired degree of accuracy 106 . Our problem reduces to the problem studied by Chen [23] when the thermal radiation and free stream velocity were not considered. It should be mentioned that in the absence of magnetic ﬁeld ðM2 ¼ 0Þ and neglecting the viscous and Joule dissipations and inclination ðc ¼ 0Þ, the relevant results are in agreement with the results reported by Bhattacharyya et al. [27] with slight change of notations. The results obtained in particular cases are compared with the results presented by Bhattacharyya et al. [27]. Fig. 1(b) shows the excellent agreement, which justiﬁes the accuracy of present numerical scheme. 4. Results and discussion In order to gain a clear insight of the physical problem, we have discussed the effects of different values of magnetic parameter M2 , mixed convection parameter k, velocity slip parameter (d), thermal slip parameter b, suction/blowing parameter S, Prandtl number Pr and Eckert number Ec and inclination c on the velocity, temperature, heat transfer rate and shear stress at the plate. M2 ¼ 0 represents the case when there is no applied magnetic ﬁeld. The mixed convection parameter k represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream ﬂow on the heat and ﬂuid ﬂow. Forced convection is the dominant mode of transport when k ! 0, whereas free convection is the dominant mode when k ! 1. For the heated plate case ðTw > T1 Þ, the upward free convection ﬂow caused by the buoyancy is in the same direction with the external forced convection ﬂow. This case is called buoyancy aided mixed convection ﬂow. For the cooling plate case ðTw < T1 Þ, the buoyancy causes a downward free convective ﬂow which is in the opposite direction to that of upward external forced convection ﬂow, which is called the buoyancy opposed mixed convection ﬂow. The values of Pr are chosen 0.72 which represents air at 20 C temperature and 1 atmospheric pressure. Ec ¼ 0 corresponds to no Joule and viscous heating. The values of magnetic parameter and slip parameters are chosen arbitrarily.

Figure 3(b) Velocity proﬁle for different d when M2 ¼ 0:5; Pr ¼ 0:72 and c ¼ p4 for buoyancy opposed ﬂow (k < 0).

5 The ﬂuid velocity fðgÞ0 increases with an increase in magnetic parameter M2 for both cases of buoyancy aided and opposed ﬂows as shown in Fig. 2(a) and 2(b). An increase in the strength of magnetic parameter M2 , the Lorentz force associated with the magnetic ﬁeld makes the boundary layer thinner. The magnetic lines of forces move past the plate at the free stream velocity. The ﬂuid which is decelerated by the viscous force, receives a push from the magnetic ﬁeld which counteracts the viscous effects. Hence the velocity of the ﬂuid increases as the parameter M2 increases. Fig. 3(a) and 3(b) shows that the ﬂuid velocity f 0 ðgÞ increases and as a result the momentum boundary layer thickness decreases with an increase in velocity slip parameter d for buoyancy aided/opposed ﬂows. An increase in slip (in magnitude) more ﬂuid get permits to slip past the plate and ﬂuid experiences less drag and as a result, the ﬂow accelerates near the plate and away from the plate this effect diminishes. There is a deceleration in the ﬂuid velocity f 0 ðgÞ as the inclination c increases for both cases of buoyancy aided and opposed ﬂows as shown in Fig. 4(a) and 4(b). The momentum boundary layer is found to be thickened for increasing values of c. For c ¼ p2, the plate is horizontal and for c ¼ 0 the plate assumes a vertical position. The gravitational effect is minimum for c ¼ p2 and maximum for c ¼ 0. The inclination parameter c arises only

Figure 4(a) Velocity proﬁle for different c when Pr ¼ 0:72; M2 ¼ 0:5 and d ¼ 0:5 for buoyancy aided ﬂow (k > 0).

Figure 4(b) Velocity proﬁle for different c when Pr ¼ 0:72; M2 ¼ 0:5 and d ¼ 0:5 for buoyancy opposed ﬂow(k < 0).

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

6 in the buoyancy term k cos c in the momentum Eq. (11). Thus, the ﬂuid velocity is found to be maximized at the vertical position of the plate (c ¼ 0) and minimized for the horizontal position of the plate (c ¼ p2). Fig. 5(a) and 5(b) displays the effects of Prandtl number on the ﬂuid velocity f 0 ðgÞ. The ﬂuid velocity f 0 ðgÞ decreases for increasing values of Prandtl number Pr for buoyancy aided/opposed ﬂows. Physically speaking, the Prandtl number is an important parameter in heat transfer processes as it characterizes the ratio of thicknesses of the viscous and thermal boundary layers. Increasing the value of Pr causes the ﬂuid temperature and its boundary layer thickness to decrease signiﬁcantly as seen from Fig. 9(a) and 9(b). This decrease in temperature produces a net reduction of the thermal buoyancy effect in the momentum equation which results in less induced ﬂow along the plate and consequently, the ﬂuid velocity decreases. The momentum boundary layer thickness generally increases with increasing values of Pr. Physically, it is true as the Prandtl number describes the ratio between momentum diffusivity and thermal diffusivity and hence controls the relative thickness of the momentum and thermal boundary layers. As Pr increases the viscous forces (momentum diffusivity) dominate the thermal diffusivity and consequently decreases the velocity. Fig. 6 shows that the ﬂuid velocity f 0 ðgÞ increases for increasing values of mixed convection parameter k for

Figure 5(a) Velocity proﬁle for different Pr when M2 ¼ 0:5; d ¼ 0:5 and c ¼ p4 for buoyancy aided ﬂow (k > 0).

Figure 5(b) Velocity proﬁle for different Pr when M2 ¼ 0:5; d ¼ 0:5 and c ¼ p4 for buoyancy opposed ﬂow (k < 0).

S. Das et al. suction/blowing. An increase at k in positive direction results in increasing the ﬂuid velocity due to addition of buoyancy-induced ﬂow onto the external forced convection ﬂow. An increase at k in negative direction results in decreasing velocities due to retarding effect of downward buoyancy-induced ﬂow onto the upward external forced convection ﬂow. For the buoyancy-opposing case (k < 0), the velocity proﬁles are quite similar to the buoyancy-assisted case (k < 0), this is to say, the velocity proﬁles increase as k enlarges. The physical explanation is that in the buoyancy-opposing case, the buoyancy force plays a negative effect on the ﬂuid motion in the boundary layer. Fig. 7 illustrates that the ﬂuid velocity f 0 ðgÞ increases for increasing values of suction parameter S for both cases of buoyancy aided and opposed ﬂows. Since the effect of suction is to suck away the ﬂuid near the plate, the momentum boundary layer is reduced due to suction velocity. Consequently the velocity increases. Fig. 8(a) and 8(b) reveals that the ﬂuid temperature hðgÞ decreases with an increase in magnetic parameter M2 for both cases of buoyancy aided and opposed ﬂows. The thermal boundary layer thickness increases for increasing values of M2 . This can be attributed to the inﬂuence of Ohmic heating due to magnetic ﬁeld in the ﬂow. Fig. 9(a) and 9(b) represents the variation of ﬂuid temperature hðgÞ with respect to Prandtl number Pr. The graph depicts that the ﬂuid temperature decreases when the values of Prandtl number Pr increase for both cases of buoyancy aided and opposed ﬂows. This is due to the fact that a higher Prandtl number ﬂuid has relatively low thermal conductivity, which reduces conduction and thereby the thermal boundary layer thickness; and as a result, temperature decreases. Increasing Pr is to increase the heat transfer rate at the surface because the temperature gradient at the surface increases. Fig. 10(a) and 10(b) shows that the ﬂuid temperature hðgÞ increases for increasing values of Eckert number Ec for buoyancy aided/opposed ﬂows. The thermal boundary layer thickness decreases with increasing values of Ec. The viscous dissipation, as a heat generation inside the ﬂuid, increases the bulk ﬂuid temperature. This can be attributed to the additional heating in the ﬂow system due to viscous dissipation. For the case of k ¼ 0, the Eckert number does not have any inﬂuence on temperature proﬁle since the momentum and energy equations are not coupled. The ﬂuid temperature hðgÞ decreases for increasing values of

Figure 6 Velocity proﬁle for different k when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and c ¼ p4 in the presence of suction/injection.

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

Magnetohydrodynamic mixed convective slip ﬂow

7

Figure 7 Velocity proﬁle for different S when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and c ¼ p4 for buoyancy aided/ opposed ﬂow.

Figure 9(a) Temperature proﬁles for different Pr when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided ﬂow (k > 0).

Figure 8(a) Temperature proﬁles for different M2 when Pr ¼ 0:72; d ¼ 0:5, b ¼ 0:2 and Ec ¼ 0:1 for buoyancy aided ﬂow (k > 0).

Figure 9(b) Temperature proﬁles for different Pr when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed ﬂow (k < 0).

Figure 8(b) Temperature proﬁles for different M2 when Pr ¼ 0:72; d ¼ 0:5, b ¼ 0:2 and Ec ¼ 0:1 for buoyancy opposed ﬂow (k < 0).

Figure 10(a) Temperature proﬁles for different Ec when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided ﬂow (k > 0).

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

8

S. Das et al.

Figure 10(b) Temperature proﬁles for different Ec when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed ﬂow (k < 0).

Figure 12(a) Temperature proﬁles for different b when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided ﬂow (k > 0).

Figure 11(a) Temperature proﬁles for different d when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided ﬂow (k > 0).

Figure 12(b) Temperature proﬁles for different b when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed ﬂow (k < 0).

Figure 11(b) Temperature proﬁles for different d when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed ﬂow (k < 0).

Figure 13 Temperature proﬁles for different k when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 in the presence of suction/injection.

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

Magnetohydrodynamic mixed convective slip ﬂow

9

Figure 14 Temperature proﬁles for different S when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided/ opposed ﬂow.

velocity slip parameter d for suction/blowing as shown in Fig. 11(a) and 11(b). The thermal boundary layer thickness increases with increasing values of d. Fig. 12(a) and 12(b) exhibits that the ﬂuid temperature hðgÞ decreases for increasing values of thermal slip parameter b for both cases of buoyancy aided and opposed ﬂows. As the thermal slip parameter b increases, less heat is transferred from the plate to the ﬂuid and hence the temperature decreases for both buoyancy aided and opposed ﬂows. The thermal boundary layer thickness becomes thicker for increasing values of b.

Table 1 The shear stress f 00 ð0Þ and the rate of heat transfer h0 ð0Þ at the plate g ¼ 0. M2

S

k

Pr

Ec

d

b

c

f 00 ð0Þ

h0 ð0Þ

0.5 1 2

0.5 0.5 0.5

0.2 0.2 0.2

0.72 0.72 0.72

0.1 0.1 0.1

0.5 0.5 0.5

0.2 0.2 0.2

p 4 p 4 p 4

0.82450 0.87451 0.95597

0.90292 0.89996 0.89573

0.5 0.5 0.5

0.5 0 0.5

0.72597 0.77541 0.82450

0.77433 0.83771 0.90292

0.77918 0.80187 0.82450

0.90121 0.90207 0.90292

0.82522 0.82484 0.82450

0.85728 0.88146 0.90292

0.82428 0.82450 0.82473

0.92146 0.90292 0.88438

0.82450 0.59019 0.45904

0.90292 0.91717 0.92393

0.82946 0.81975 0.81519

1.10717 0.70725 0.51961

0.83388 0.82450 0.80185

0.90326 0.90292 0.90207

0.2 0 0.2 0.25 0.5 0.72 0 0.1 0.2 0.5 1 1.5 0 0.5 1 0 p 4 p 2

Fig. 13 reveals that the ﬂuid temperature hðgÞ decreases for increasing values of mixed convection parameter k for suction/blowing. 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 more ﬂuid velocity to enhance. As a result, the thermal boundary layer thickness increases. Fig. 14 shows that the ﬂuid temperature hðgÞ decreases for increasing values of suction parameter S for both cases of buoyancy aided and opposed ﬂows. The explanation for such behavior is that the ﬂuid is brought closer to the plate surface and reduces the thermal boundary layer thickness. As the distance x from the origin increases, the temperature of the plate decreases which is very clear from the expression of the plate temperature Tw ¼ T1 þ Tx0 as shown in temperature ﬁgures. For engineering purposes, one is usually interested in the values of the shear stress (skin friction) and the rate of heat transfer at the plate. The shear stress is an important parameter in the heat transfer studies, since it is directly related to the heat transfer coefﬁcients. The increased shear stress is generally a disadvantage in the technical applications, while the increased heat transfer can be exploited in some applications such as heat exchangers, but should be avoided in other such as gas turbine applications, for instance. The numerical values of the rate of heat transfer h0 ð0Þ and the shear stress f 00 ð0Þ at the plate g ¼ 0 are entered in the Table 1 for several values of M2 ; S; k; Pr; Ec; d; b and c. It is seen from the Table 1 that the rate of heat transfer h0 ð0Þ decreases for increasing values of M2 ; Ec; b; c whereas it decreases for increasing values of S; k; Pr and d. This implies that an increase in Prandtl number is accompanied by an enhancement of the heat transfer rate at the surface of the sheet. The underlying physics behind this can be described as follows. When ﬂuid attains a higher Prandtl number, its thermal conductivity is lowered down and so its heat conduction capacity diminishes. Thereby the thermal boundary layer thickness gets reduced. As a consequence, the heat transfer rate at the surface is increased. The increasing magnetic ﬁeld also increases the thermal boundary layer thickness and as a result the dimensionless heat transfer rate decreases with an increase in the magnetic ﬁeld. Buoyancy forces can enhance the rate of heat transfer at the plate when they assist the forced convection, and vice versa. The thermal boundary layer thickness decreases with the suction parameter S which causes an increase in the rate of heat transfer. The explanation for such behavior is that the ﬂuid is brought closer to the surface and reduces the thermal boundary layer thickness. On the other hand, h0 ð0Þ < 0 means the heat transfer takes place from the surface to ambient ﬂuid. The shear stress f 00 ð0Þ increases for increasing values of M2 ; S; k; Ec whereas it decreases for increasing values of Pr; d; b and c. 5. Conclusion In this paper, we examine the combined effects of Joule heating and viscous dissipation on an MHD mixed convective ﬂow past an inclined porous plate. Using similarity transformation, the governing equations are transformed into self-similar equations. The numerical solutions of the self-similar equations are

Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip ﬂow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003

10 obtained using shooting technique with the help of forth order Runge–Kutta method. The effects of the pertinent parameters are discussed. Numerical results for temperature and velocity are presented graphically for pertinent parameters. Based on the obtained graphical and tabular results, the following conclusions can be summarized as follows: The ﬂuid velocity and temperature enhance when the strength of magnetic ﬁeld increases. The ﬂuid velocity temperature accelerates due to increasing thermal buoyancy force. The ﬂuid velocity reduces for increasing values of velocity slip parameter. An increase in velocity slip at the sheet leads to increase the ﬂuid temperature within the boundary layer, while an increase in thermal slip reduces the temperature distribution. The ﬂuid temperature inside the boundary layer reduces when the values of the Prandtl number increase. The slip parameters always lead to thinning of the thermal boundary layer. Joule heating and viscous dissipation have the effect to increase the ﬂuid velocity, and accordingly decrease the rate of heat transfer and shear stress at the plate.

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