Study of hydromagnetic heat and mass transfer flow over an inclined heated surface with variable viscosity and electric conductivity

Commun Nonlinear Sci Numer Simulat 15 (2010) 2073–2085

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Study of hydromagnetic heat and mass transfer flow over an inclined heated surface with variable viscosity and electric conductivity Mohammad M. Rahman a,*, K.M. Salahuddin b a b

Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Muscat, Oman Department of Management Information Systems (MIS), University of Dhaka, Dhaka 1000, Bangladesh

a r t i c l e

i n f o

Article history: Received 8 March 2009 Received in revised form 14 August 2009 Accepted 23 August 2009 Available online 27 August 2009 Keywords: Convective flow Variable Prandtl number Variable electric conductivity Radiation Internal heat generation Local similarity solution

a b s t r a c t The effects of variable electric conductivity and temperature dependent viscosity on hydromagnetic heat and mass transfer flow along a radiate isothermal inclined permeable surface in a stationary fluid in the presence of internal heat generation (or absorption) are analyzed numerically presenting local similarity solutions for various values of the physical parameters. The research shows that the difference in the results between variable Prandtl number and constant Prandtl number are significant when fluid viscosity strongly dependents on the temperature. The results also show that skin friction coefficient, Nusselt number and Sherwood number are lower for the fluids of constant electric conductivity than those of the variable electric conductivity. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The study of thermal boundary layer flows of variable viscosity on isothermal heated surfaces not only possesses theoretical interest but also models many fluid transport mechanisms encountered in industries and engineering systems. Amongst others, we can name hot rolling, wire drawing, glass fiber production, paper production, gluing of labels on hot bodies, drawing of plastic films, etc. When a cooler fluid flows around a hot body, the temperature of the fluid will rise in a thin layer around the body and in a wake behind it. This thin layer is known as the thermal boundary layer. In this layer, flow and thermal phenomena interact nonlinearly and governed by the so-called thermal boundary layer equations. In classical treatment of thermal boundary layers, the kinematic viscosity is assumed to be constant; however, experiments indicate that this assumption only makes sense if temperature does not change rapidly for the application of interest. Indeed, for liquids, experimental data shows that viscosity decreases with temperature. Viscosity changes with temperature, for example the absolute viscosity of water decreases by 240% when the temperature increases from 10  C to 50  C which has been shown by Herwig and Wickern [1]. Film of fluids with constant viscosity along an inclined heated plate was investigated by Saouli and Saouli [2]. Meanwhile, several authors have investigated the effects of temperature dependent viscosity on the flow of non-Newtonian fluids in a channel under various conditions (e.g. Makinde [3], Szeri and Rajagopal [4], Yurusoy and Pakdemirli [5]). Ali [6] has studied the effect of

* Corresponding author. Fax: +968 2414 1490. E-mail address: [email protected] (M.M. Rahman). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.08.012

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Nomenclature B0 C Cf Cp Cw C1 Dm f fW g Gr Gc Nux Pn Pr Pr1 qw Q Q0 R Rex Sc Shx T Tw T1 u U0 U1 v

v0 x y

applied magnetic field (Wb m2) species concentration (kg m2) skin friction coefficient specific heat at constant pressure (J kg1 K1) concentration at the porous plate (kg m2) species concentration at infinity (kg m2) molecular diffusivity (m2 s1) dimensionless stream function dimensionless suction parameter acceleration due to gravity (m s2) Grashof number modified Grashof number local Nusselt number radiative Prandtl number variable Prandtl number ambient Prandtl number surface heat flux (W m2) local heat generation parameter heat generation parameter (W) radiation parameter local Reynolds number Schmidt number local Sherwood number temperature within boundary layer (K) temperature at the plate (K) temperature of the ambient fluid (K) velocity along x-axis (m s1) characteristic velocity (m s1) velocity outside the boundary layer (m s1) velocity along y-axis (m s1) suction velocity (m s1) coordinate along the surface (m) coordinate normal to the surface (m)

Greek symbols angle of inclination (rad) b coefficient of volume expansion (K1) coefficient of volume expansion with concentration (K1) b q1 mass density of the ambient fluid (kg m3) l coefficient of dynamic viscosity (Pa s) t apparent kinematic viscosity (m2 s1) r electric conductivity (mX m1) r0 magnetic permeability (N A2) j thermal conductivity (W m1K1) g similarity parameter w stream function (m2 s1) h dimensionless temperature H viscosity parameter / dimensionless concentration

a

temperature dependent viscosity on laminar mixed convection boundary layer flow and heat transfer on a continuously moving vertical surface. Laminar falling liquid film with variable viscosity along an inclined heated plate has been studied by Makinde [7]. A steady two-dimensional flow of an electrically conducting incompressible fluid over a heated stretching sheet with variable viscosity has been investigated by Mukhopadhyay et al. [8]. Pop et al. [9], and Elbashbeshy and Bazid [10] have studied the effect of variable viscosity using the similarity solution with no buoyancy force. Although viscosity varies with temperature, all of the afore-mentioned works considered constant Prandtl number (a parameter directly proportional to fluid viscosity, see Section 3) within the boundary layer. So one of the objectives of this study is to investigate the roll of variable Prandtl number on the heat and mass transfer flows.

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The study of Magnetohydrodynamic (MHD) flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal-working processes. There has been a great interest in the study of Magnetohydrodynamic flow and heat transfer in any medium due to the effect of magnetic field on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. This type of flow has attracted the interest of many researchers (see for examples Sparrow and Cess [11], Romig [12], Hossain [13], Takhar and Ram [14], Chaim [15], Vajravelu and Hadjinicolaou [16], Pop and Na [17], Jha [18]) due to its applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. Hydromagnetic flow and heat transfer problems have become more important industrially. In many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through an electrically conducting fluid subject to a magnetic field, the rate of cooling can be controlled and final product of desired characteristics can be achieved. On the other hand, all the above investigations are restricted to MHD flow and heat transfer problems. However, of late the radiation effect on MHD flow and heat transfer problems has become more important industrially. At high operating temperatures, radiation effects can be quite significant. Many processes in engineering areas occur at high temperatures and knowledge of radiative heat transfer becomes very important for the design of pertinent equipment. Due to the direct engineering applications of radiation many authors have studied and reported results on radiative heat transfer flow of which the names of Soundalgekar et al. [19], Hossain and Takhar [20], Hossain et al. [21], Seddek [22], Duwairi and Damseh [23], Rahman and Sultana [24], Alam et al. [25], Alam et al. [26], Rahman [27] and Prasad and Vajravelu [28] are worth mentioning. All of the afore-mentioned works related with constant electric conductivity of the fluid. The study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Possible heat generation effects may change the temperature distribution; consequently, the particle deposition rate in nuclear reactors, electronic chips and semi conductor wafers. Due to the practical application of the heat generation or absorption effect several authors have addressed this issue on different geometry considering various flow conditions (see for examples [29–34]). To the best of our knowledge, the heat and mass transfer flow of a viscous incompressible fluid of variable electric conductivity flowing over a radiate isothermal inclined permeable surface taking into account the temperature-dependence of viscosity (hence variable Prandtl number inside the boundary layer) and internal heat generation (or absorption) has remained unexplored. Therefore, the objective of the present communication is to carryout a numerical simulation of this problem exploring the effects of the pertinent parameters on the shear stress, rate of heat transfer and rate of mass transfer. 2. Physical model 2.1. Governing equations of the flow Let us consider a steady two-dimensional MHD convective flow of viscous incompressible electrically conducting fluid past a semi-infinite inclined flat plate with an acute angle a to the vertical. With x-axis measured along the plate, a magnetic field of uniform strength B0 is applied in the y-direction which is normal to the flow direction. Fluid suction is imposed at the surface of the plate and the suction hole size is taken to be constant. The fluid of density ðq1 Þ is quiescent ðU 1 ¼ 0Þ and the convective motion is induced by the buoyancy forces. The temperature of the surface is held uniform at T W which is higher than the ambient temperature T 1 . The flow configurations and coordinate system are shown in Fig. 1.

x

Vertical direction

v0(x)

e

at

ed

pl

Tw

in

B0

l nc

r

I

ye

y

α

la

u

U=0

v

a nd

T

ou

B

∞

T∞

g0

y Fig. 1. Flow configurations and coordinate system.

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The external electric field is assumed to be zero and the magnetic Reynolds number is assumed to be small. Hence the induced magnetic field is small compared with the external magnetic field. Moreover, the electrical conductivity r is assumed to vary with the velocity of the fluid and have the form (Helmy [35]):

r ¼ r0 u; r0 is a constant:

ð1Þ

For the flow under study, it is relevant to assume that the applied magnetic field strength; BðxÞ has the form:

B0 BðxÞ ¼ pffiffiffi ; x

B0 is a constant ðsee Helmy½35; Aissa and Mohammadein ½36Þ

ð2Þ

Under the usual boundary layer and Boussinesq approximations, the flow, heat and mass transfer are governed by the following equations (see [6,9,25]):

@u @ v þ ¼ 0; @x @y

ð3Þ 

 @u r0 B20 u2 þ gbðT  T 1 Þ cos a þ gb ðC  C 1 Þ cos a  ; @y q1 x

u

@u @u 1 @ þv ¼ @x @y q1 @y

u

@T @T j @2T 1 @qr Q0  ðT  T 1 Þ; þv ¼ þ @x @y q1 cp @y2 q1 cp @y q1 cp

ð5Þ

u

@C @C @2C þv ¼ Dm 2 ; @x @y @y

ð6Þ

l

ð4Þ

where u; v are the velocity components along x; y co-ordinates respectively, l is the coefficient of dynamic viscosity, q1 is the mass density of the fluid, B0 is the uniform magnetic field acting normal to the plate, g is the acceleration due to gravity, r0 is the magnetic permeability, cp is the specific heat at a constant pressure, Dm is the molecular diffusivity, j is the thermal conductivity, T and C are respectively the temperature and concentration of the fluid, T w is the temperature of the fluid within the boundary layer, T 1 is the temperature of the fluid outside the boundary layer, b is the volumetric coefficient of thermal expansion, b is the volume expansion with concentration, Q 0 is the heat generation parameter, C is the concentration of the fluid within the boundary layer and C 1 is the concentration of the fluid outside the boundary layer. The fluid is considered to be gray; absorbing-emitting radiation but non-scattering medium and Rosseland approximation is used to describe the radiative heat flux in the energy equation. By using Rosseland approximation qr takes the form,

qr ¼ 

4r1 @T 4 ; 3j1 @y

ð7Þ

where r1 , the Stefan-Boltzmann constant and j1 , the mean absorption coefficient. It is assumed that the temperature differences within the flow are sufficiently small such that T 4 may be expressed as a linear function of temperature. This is accomplished by expanding T 4 in a Taylor series about T 1 and neglecting higher-order terms, thus

T 4  4T 31 T  3T 41 :

ð8Þ

Using (7) and (8) Eq. (5) becomes

u

@T @T j @ 2 T 16r1 T 31 @ 2 T Q 0 þ þ ðT  T 1 Þ: þv ¼ @x @y q1 cp @y2 3j1 q1 cp @y2 q1 cp

ð9Þ

2.2. Boundary conditions of the model The appropriate boundary conditions of the model are: i. On the plate surface ðy ¼ 0Þ:

u ¼ 0;

v ¼ v 0 ðxÞ ðno-slip and permeable surface conditionsÞ;

T ¼ T w ; C ¼ C w ðuniform surface temperature and concentrationÞ:

ð10aÞ ð10bÞ

ii. Marching with the quiescent free stream ðy ! 1Þ:

u ¼ 0;

T ¼ T1;

C ¼ C1;

where the subscripts w and 1 refer to the surface and boundary layer edge, respectively.

ð10cÞ

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3. Non-dimensionalisation To proceed we introduce the following non-dimensional variables:

sffiffiffiffiffiffiffiffiffiffiffiffi U0 ; g¼y 2t1 x

w¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t1 U 0 xf ðgÞ;

/¼

C  C1 ; Cw  C1

h¼

T  T1 ; Tw  T1

ð11Þ

where w is the stream function and U 0 is some characteristic velocity. and v ¼  @w we have from Eq. (11) that, Since u ¼ @w @y @x

u ¼ U0 f 0

and

rffiffiffiffiffiffiffiffiffiffiffiffi t1 U 0 ðf  gf 0 Þ: 2x

v ¼

ð12Þ

Here f is non-dimensional stream function and prime denotes differentiation with respect to g. The increase of temperature leads to the increase in the transport phenomena by reducing the viscosity across the momentum boundary layer as a consequence heat transfer rate at the wall is also affected. Therefore, to predict the flow and heat transfer rates it is necessary to take into account the temperature dependent viscosity of the fluid. For a viscous fluid, Ling and Dybbs [37], and Lai and Kulacki [38] (later used by several authors [39,40]) suggested a viscosity dependence on the temperature of the form

l ¼ l1 =½1 þ cðT  T 1 Þ;

ð13Þ

so that viscosity is an inverse linear function of temperature T. Eq. (13) can be rewritten as 1=l ¼ AðT  T 1 Þ; where

A ¼ c=l1

and T r ¼ T 1  1=c:

ð14Þ

In the above relation (14), both A and T r are constants and their values depend on the reference state and c, a thermal property of the fluid. In general, for liquids A > 0 and for gases A < 0. Typical values of c and A for air are c ¼ 0:026240 and A ¼ 123:2 (see West [41]). The dimensionless temperature h can also be written as

h¼

T  Tr þ H; Tw  T1

ð15Þ

1 where H ¼ TTwr T ¼  cðT w1T 1 Þ ¼ const is known as fluid viscosity parameter, and its value is determined by the viscosity/temT 1 perature characteristics of the fluid under consideration and the temperature difference

DT ¼ T w  T 1 : Using ð15Þ Eq: ð14Þ becomes

l ¼ l1





H : Hh

ð16Þ

Now substituting Eqs. (11)–(16) into Eqs. (4), (9) and (6) we obtain

f 000 þ

H  h 00 h0 Hh Hh H  h 02 ff þ h þ Gc Cosa /M f ¼ 0; f 00 þ Gr Cosa H Hh H H H

ð17Þ

h00 þ

3R Pr1 ½f h0 þ Q h ¼ 0; 3R þ 4

ð18Þ

/00 þ Scf /0 ¼ 0:

ð19Þ

In the above equations we have used the following non-dimensional parameters:  Gr ¼ 22gbðTUw2T 1 Þx is the local Grashof number where as Gc ¼ 2gb ðCUw2C 1 Þx is known as the local modified Grashof number, 0 0 2r0 b0 q1 t1 cp is the Prandtl number related to the constant viscosity, t1 ¼ lq1 is M ¼ q is the magnetic field parameter, Pr1 ¼ k 1 1 kj1 0x is the local heat the kinematic coefficient of viscosity of the ambient fluid, R ¼ 4r T 3 is the radiation parameter, Q ¼ U2Q 0 q1 cp 1 1 generation parameter, and Sc ¼ Dt1m is the Schmidt number. The corresponding boundary conditions (10) become,

/ ¼ 1 at g ¼ 0;

f ¼ fw ;

f 0 ¼ 0;

h ¼ 1;

f 0 ¼ 0;

h ¼ 0;

/ ¼ 0 as g ! 1;



where fw ¼ v 0 ð2x=t1 U 0 Þ1=2 is the suction velocity at the plate for

ð20Þ

v 0 < 0.

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3.1. Parameters of engineering interest The parameters of engineering interest for the present problem are the local skin friction coefficient ðCf Þ, local Nusselt number ðNux Þ and the local Sherwood number ðShx Þ which indicate physically wall shear stress, rate of heat transfer and rate of mass transfer respectively. These physical quantities can be calculated from the following relations:

 12   Rex H1 Cf ¼ f 00 ð0Þ; 2 H

Nux

ð21Þ

  12  Rex 3R ¼ h0 ð0Þ; 3R þ 4 2

ð22Þ

 12 Rex ¼ /0 ð0Þ; 2

ð23Þ

and Shx

0x where Rex ¼ Ut1 is the local Reynolds number. Thus from Eqs. (21)–(23) we see that skin friction coefficient Cf, Nusselt number Nux and Sherwood number Shx are proportional to the numerical values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ, respectively.

3.2. Variable Prandtl number The Prandtl number is a function of viscosity and as viscosity varies across the boundary layer, the Prandtl number varies, too. The assumption of constant Prandtl number inside the boundary layer may produce unrealistic results (see Pantokratoras [42–44]). Therefore, Prandtl number related to the variable viscosity is defined by

lc p Pr ¼ ¼ j



H Hh







l1 c p H Pr1 : ¼ j Hh

ð24Þ

At the surface (g ¼ 0Þ of the plate this can be written as

Prw ¼



H H1

 Pr1 :

ð25Þ

In light of the above discussions, using (24) the non-dimensional temperature Eq. (18) can be rewritten as

  h h00 þ Pn 1  f h0 þ Qh ¼ 0;

H

ð26Þ

where the radiative Prandtl number corresponding to the variable viscosity can be defined as

Pn ¼

3RPr : 3R þ 4

ð27Þ

From Eq. (24) it can be seen that for large H, i.e. H ! 1, variable Prandtl number Pr equals to the ambient Prandtl number Pr1 , in that case Eq. (26) reduces to Eq. (18). For g ! 1; hðgÞ becomes zero therefore Pr equals Pr1 regardless of the values of H. Eq. (26) is the corrected non-dimensional form of the energy equation in which Prandtl number is treated as variable. To the best of our knowledge nobody ever yet mentioned this correction into the non-dimensional energy Eq. (26) for modeling thermal boundary layer flows with temperature dependent viscosity. 4. Numerical simulations In this paper, the effects of variable electric conductivity and temperature dependent viscosity on a steady two-dimensional convective flow of a viscous incompressible fluid past a radiate isothermal inclined porous surface has been investigated numerically by using Nachtsheim–Swigert [45] shooting iteration technique. It can be seen that the solutions are affected by the nine parameters, namely suction parameter fw , magnetic field parameter M, heat generation parameter Q, viscosity parameter H, radiative Prandtl number Pn, Schimdt number Sc, Grashof number Gr, modified Grashof number Gc, and angle of inclination a. Because the parameters Gr; Gc; Q and fw depend on the coordinate x, the solutions are locally similar. Such local similarity analyses have been performed by many authors (see for example Alam et al. [25,26,32,33], Rahman [27], Rahman et al. [34], Raptis [46], El-Arabawy [47], Rahman and Sattar [48], Cortell [49,50], Hayat et al. [51], and Aziz [52]). We treat Eqs. (17), (19), (26) are ordinary differential equations and solve them to derive locally similar solutions for a range of values of the physical parameters characterizing the flow. Since experimental data of the physical parameters are not available therefore in the numerical simulations the values of the pertinent parameters are chosen arbitrarily. The default values of the afore-mentioned parameters which we considered are fw ¼ 0:5; M ¼ 0:5; Q ¼ 0:5; H ¼ 2:0; Pn ¼ 0:7; Sc ¼ 0:62; Gr ¼ 5:0; Gc ¼ 2:0, and a ¼ 450 unless otherwise specified.

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Table 1 Comparison of f 00 ð0Þ and h0 ð0Þ to previously published data at Pnð¼ Pr1 Þ ¼ 0:7 and fw ¼ 0; M ¼ 0; Q ¼ 0; Sc ¼ 0; Gr ¼ 0; Gc ¼ 0 for different values of H using the same expression of g ¼ ðy=xÞRex1=2 defined by Pop et al. [9] with rescaled Eqs. (17) and (26) and boundary conditions (20).

H

Values of

Pop et al. [9]

Ali [6]

Present result

8.0

f 00 ð0Þ h0 ð0Þ

0.4773578 0.3493189

0.4763230 0.3432339

0.4763424 0.3436768

0.1

f 00 ð0Þ h0 ð0Þ

1.5061732 0.2191391

1.4965150 0.1652394

1.4964382 0.1663341

0.01

f 00 ð0Þ h0 ð0Þ

4.4856641 0.1544918

4.4683560 0.0561845

4.4670284 0.0604795

8.0

f 00 ð0Þ h0 ð0Þ

0.4089153 0.3605226

0.4083475 0.3555822

0.4083569 0.3560127

4.1. Code verification To assess the accuracy of our code, we reproduced the values of f 00 ð0Þ and h0 ð0Þ which are proportional to the local skin friction coefficient and rate of heat transfer considering the model of Pop et al. [9] and Ali [6]. Table 1 shows the comparison of the data produced by our code and those of Pop et al. [9] and Ali [6]. In these calculations the value of the Prandtl number has been considered as constant throughout the boundary layer. In fact the results show a close agreement, hence justify the use of the present code for the current model. 4.2. Effect of step size To see the effects of the integration step size Dg, we ran the code for our model with three different step sizes namely Dg ¼ 0:01; Dg ¼ 0:005; and Dg ¼ 0:001. In each case, we found excellent agreement among the results. It was also found that Dg ¼ 0:001 provided sufficiently accurate results and further refinement of the grid size was therefore not warranted. 5. Results and discussion In the earlier section we have shown that the solutions of the present model depend on the nine physical parameters. Due to brevity here we present solutions discussing from the physical point of view through graphs and tables for selected values of the radiative Prandtl number Pn, viscosity parameter H, heat generation parameter Q, and magnetic filed parameter M keeping all other parameters values fixed. Fig. 2 depicts variable Prandtl number within the boundary layer for various values of the viscosity parameter H at an ambient Prandtl number Pr1 ¼ 0:7. From this figure we see that Pr asymptotically converges to the value of Pr1 as g ! 1. It is also notable that at the surface of the plate Pr approaches to Pr1 for large values of H. For H > 0; Pr decreases with the increase of H while an opposite effect is observed for H < 0.

1.4

1.2

Θ = 2, 10

1

Pr

0.8

Pr∞= 0.7

0.6

0.4

Θ =−0.05, −0.10, −2.0 0.2

0

0

1

2

3

4

η Fig. 2. Variable Prandtl number Pr versus g for different values of H.

5

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Fig. 3 illustrates the radiative Prandtl number Pn for different values of the radiation parameter R. From here we observe that Pn increases rapidly with the increase of the radiation parameter. Furthermore the variation in Pn becomes less pronounced for large R values due to the fact that the coefficient 3R=ð3R þ 4Þ in Eq. (27) approaches to 1 as R approaches its asymptotic value infinity. Combining Figs. 2 and 3 we found that for a fixed value of the radiation parameter R, the radiative Prandtl number Pn decreases with the increase of the viscosity parameter when H is positive whereas an opposite effect is seen when H is negative. Fig. 4(a)–(c), respectively, depict the velocity, temperature and concentration profiles for different values of the heat generation (or absorption) parameter Q. From Fig. 4(a) we observe that when heat is generated the buoyancy force increases which induces the flow rate to increase giving rise to the increase in the velocity profiles. It is also observed that the peak of the velocity profiles moves away from the surface of the plate as Q increases. The maximum values of the velocity are 0.8698, 0.9542, 1.0718, and 1.3630 for Q ¼ 0; 0:5; 1:0, and 2.0 which occurs at 0.962, 0.994, 1.038 and 1.152, respectively. It is seen that the velocity increases by 23.22% when Q increases from 0 to 1 while the corresponding reduction in velocity is 15.24% when Q decreases from 0 to 1. From Fig. 4(b) it can be seen that temperature of the fluid increases significantly with the increase of the heat generation parameter as expected. For large values of Q the temperature profile overshoots the free stream value, consequently the gradient of the temperature profile altered due to the strong buoyancy effect. Fig. 4(c) shows the decreasing effect of Q on the concentration profile. The opposite phenomenon is observed for the case of heat absorption. Fig. 5(a) presents the effect of the viscosity parameter H on the velocity profiles. From here it can be seen that for H > 0 velocity profiles increase within the domain g 6 1:2 with the increase of H. Outside this domain velocity profiles overlap and decrease with the increase of H. For H < 0, velocity profile also increases with the increase of H. It is also clear that the maximum of the velocity profile occurs near the surface of the plate. From Fig. 5(b) we see that the thickness of the thermal boundary layer decreases with the increase of H. The effect of H on the concentration boundary layer is similar to that of thermal boundary layer. Table 2, shows that the values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ which are respectively proportional to the skin friction coefficient, Nusselt number and Sherwood number for different values of the magnetic field parameter M and viscosity parameter H. For both variable electric conductivity (VEC) and constant electric conductivity (CEC) cases the values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ decrease with the increase of M for a fixed value of H regardless of its sign whether it is positive or negative. For the negative values of H these values are larger than those of the corresponding positive values of H. It is also noticeable that for constant electric conductivity case these values are smaller than those of the corresponding variable electric conductivity case. On the other hand values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ increase with the increase of the viscosity parameter H for a fixed value of M for both the cases of VEC and CEC. Here H ! 1, indicates viscosity is independent of temperature, i.e. constant. In this case we also notice that skin friction coefficient, Nusselt number and Sherwood number are higher for VEC than those of CEC. The values within the parenthesis represent the corresponding values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ without correcting the energy Eq. (18). Now this table shows that the absolute error between the corresponding values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ with corrected (Eq. (26)) and without corrected (Eq. (18)) form of energy equation is less than one percent for large values of H while the difference between the two cases are markedly large for small negative values of H. In particular, for H ¼ 0:05 this absolute error for skin friction is 17%, for Nusselt number 83%, for Sherwood number 14% for the case of VEC while the corresponding errors for CEC are 32%, 84% and 21%, respectively. The afore-mentioned results strongly recommend that modeling with temperature dependent viscosity there needs to be a correction in the energy equation as like Eq. (26).

1.6 1.4 1.2

R = 0.5, 1, 10, 100, ∞ 1

Pn

0.8 0.6 0.4 0.2 0

0

1

2

η

3

4

Fig. 3. Radiative Prandtl number Pn versus g for different values of R.

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a 1.4 1.2

Q = -1.0, -0.5, 0, 0.5, 1.0, 2.0

1

f

/

0.8

0.6

0.4

0.2

0

0

1

2

3

4

5

6

4

5

6

4

5

6

η

b 1.2 Q = -1.0, -0.5, 0, 0.5, 1.0, 2.0 1

θ

0.8

0.6

0.4

0.2

0

0

1

2

3

η

c

1

0.8

0.6

φ

Q = -1.0, -0.5, 0, 0.5, 1.0, 2.0 0.4

0.2

0

0

1

2

η

3

Fig. 4. (a) Velocity, (b) temperature, and (c) concentration profiles for various values of the heat generation parameter Q.

Table 3, presents the values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ for different values of the radiative Prandtl number Pn and viscosity parameter H. From this table we see that values of f 00 ð0Þ and /0 ð0Þ decrease whereas values of h0 ð0Þ increase with the increase of Pn (hence also with the increase of the radiation parameter R, see Fig. 3) for both the cases of variable electric

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a

1.4

Θ = −0.05, −0.1, −2

1.2

Θ = 10, 4, 2

1

f

/

0.8

0.6

0.4

0.2

0

0

1

2

3

4

5

3

4

5

3

4

5

η

b

1

0.8

Θ = −2, −0.1, −0.05

θ

0.6

Θ = 2, 4, 10

0.4

0.2

0

0

1

2

η

c

1

0.8

φ

0.6

0.4

Θ = 2, 4, 10

0.2

0

0

1

2

η Fig. 5. (a) Velocity, (b) temperature, and (c) concentration profiles for various values of the viscosity parameter H.

conductivity and constant electric conductivity. Physically increasing the radiation parameter leads to decrease the boundary layer thickness and to enhance the heat transfer rate. For constant electric conductivity case values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ are smaller compared to the corresponding values when the conductivity is considered as variable.

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M.M. Rahman, K.M. Salahuddin / Commun Nonlinear Sci Numer Simulat 15 (2010) 2073–2085 Table 2 Values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ for various values of H and M. h0 ð0Þ

f 00 ð0Þ

Parameters

H

M

VEC

2

0.0 0.5 1.0 2.0

2.1339 2.0176 1.9286 1.7992

10

0.0 0.5 1.0 2.0

50

CEC

/0 ð0Þ

VEC

CEC

VEC

CEC

2.1339 1.9812 1.8566 1.6671

0.4581 0.4318 0.4095 0.3732

0.4581 0.4204 0.3853 0.3218

0.7187 0.7022 0.6889 0.6682

0.7187 0.6966 0.6775 0.6465

3.3073 3.0870 2.9222 2.6886

3.3073 3.0404 2.8237 2.4961

0.5332 0.4986 0.4696 0.4223

0.5332 0.4872 0.4439 0.3647

0.7543 0.7341 0.7181 0.6936

0.7543 0.7288 0.7069 0.6713

0.0 0.5 1.0 2.0

3.5179 3.2773 3.0982 2.8446 (2.8432)

3.5179 3.2301 2.9961 2.6428 (2.6414)

0.5475 0.5114 0.4811 0.4319 (0.4327)

0.5475 0.4999 0.4552 0.3730 (0.3734)

0.7591 0.7384 0.7220 0.6969 (0.6966)

0.7591 0.7332 0.7108 0.6745 (0.6743)

1

0.0 0.5 1.0 2.0

3.5697 3.3240 3.1413 2.8828

3.5697 3.2766 3.0384 2.6788

0.5511 0.5145 0.4840 0.4343

0.5511 0.5031 0.4580 0.3751

0.7603 0.7395 0.7229 0.6976

0.7603 0.7342 0.7117 0.6753

50

0.0 0.5 1.0 2.0

3.6212 3.3704 3.1840 2.9207 (2.9221)

3.6212 3.3228 3.0804 2.7145 (2.7160)

0.5546 0.5177 0.4868 0.4367 (0.4359)

0.5546 0.5063 0.4608 0.3773 (0.3769)

0.7614 0.7404 0.7238 0.6984 (0.6986)

0.7614 0.7352 0.7126 0.6760 (0.6763)

2

0.0 0.5 1.0 2.0

4.7689 4.4000 4.1297 3.7550

4.7689 4.0537 4.0117 3.5028

0.6383 0.5933 0.5558 0.4949

0.6383 0.5819 0.5286 0.4296

0.7813 0.7579 0.7394 0.7114

0.7813 0.7528 0.7282 0.6884

0.05

0.0 0.5 1.0 2.0

30.5537 26.6653 24.0529 20.7486 (24.2461)

30.5537 26.6907 23.6660 19.3770 (25.6362)

4.0527 3.8927 3.7671 3.5772 (0.5918)

4.0527 3.8773 3.7162 3.4263 (0.5598)

0.7955 0.7649 0.7409 0.7051 (0.8077)

0.7955 0.7523 0.7158 0.6594 (0.7991)

Table 3 Values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ for various values of Pn and H. h0 ð0Þ

f 00 ð0Þ

Parameters

/0 ð0Þ

Pn

VEC

CEC

VEC

CEC

VEC

CEC

2.0

0.7 1.0 7.0 10.0

2.0176 1.8819 1.1376 1.0156

1.9812 1.8403 1.0822 0.9593

0.4318 0.5277 2.3241 3.3319

0.4204 0.5125 2.2890 3.2986

0.7022 0.6788 0.5847 0.5781

0.6966 0.6715 0.5663 0.5586

2.0

0.7 1.0 7.0 10.0

4.4000 4.1382 2.7580 2.5469

4.3507 4.0617 2.6019 2.3838

0.5933 0.7508 3.9115 5.6484

0.5819 0.7339 3.8665 5.6068

0.7579 0.7291 0.6153 0.6012

0.7528 0.7215 0.5953 0.5832

H

The values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ for different values of heat generation (or absorption) parameter Q and viscosity parameter H are shown in Table 4. For variable electric conductivity as well as of constant electric conductivity the values of f 00 ð0Þ and /0 ð0Þ increase and the values of h0 ð0Þ decrease with the increasing values of the heat generation parameter ðQ > 0Þ. The opposite effect is observed for the case of heat absorption parameter ðQ < 0Þ. It is also clear that for large Q ðQ P 1:4Þ the gradient of the temperature profile becomes positive which confirms Fig. 4(b) when H is positive. This is due to the fact that large Q induces the buoyancy force which intern increases the temperature of the fluid. On the other hand for negative H the gradient of the temperature profile changes its sign for slightly lower values of Q P 1:2. For constant electric conductivity case values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ are smaller for the corresponding values of the variable electric conductivity case for Q < 1:4 ðQ < 1:2Þ when H is positive (negative). Beyond these values of Q opposite trend is observed.

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Table 4 Values of f 00 ð0Þ; h0 ð0Þ and /0 ð0Þ for various values of Q and H. h0 ð0Þ

f 00 ð0Þ

Parameters

/0 ð0Þ

Q

VEC

CEC

VEC

CEC

VEC

CEC

2.0

1.0 0.5 0.0 0.5 1.0 1.4 1.5 2.0

1.6747 1.7562 1.8649 2.0176 2.2380 2.4722 2.5394 2.9001

1.6166 1.7018 1.8169 1.9812 2.2199 2.4732 2.5417 2.9201

0.9365 0.7963 0.6320 0.4318 0.1803 0.0422 0.1238 0.4478

0.9311 0.7897 0.6235 0.4204 0.1666 0.0599 0.1349 0.4502

0.6550 0.6666 0.6819 0.7022 0.7286 0.7524 0.7583 0.7866

0.6431 0.6562 0.6735 0.6966 0.7263 0.7526 0.7590 0.7893

2.0

1.0 0.5 0.0 0.5 1.0 1.2 1.5 2.0

3.8126 3.9519 4.1367 4.4000 4.8026 5.0235 5.4365 6.3714

3.6924 3.8454 4.0517 4.3507 4.8153 5.0687 5.5393 6.5791

1.3571 1.1527 0.9079 0.5933 0.1399 0.1122 0.6160 2.0067

1.3514 1.1459 0.8992 0.5819 0.1260 0.1240 0.6138 1.8999

0.7022 0.7153 0.7330 0.7579 0.7945 0.8134 0.8464 0.9124

0.6896 0.7044 0.7244 0.7528 0.7944 0.8154 0.8516 0.9212

H

6. Conclusions The problem considered in this paper is to study the effects of variable electric conductivity and temperature dependent viscosity on MHD heat and mass transfer flows past a radiate inclined porous surface in the presence of heat generation. Using similarity transformations the governing equations of the problem have been transformed into nonlinear ordinary differential equations and solved for local similar solutions by using Nachtsheim–Swigert shooting iteration technique. From the present study the following conclusions can be drawn; 1. The rate of friction decreases with the increase of the magnetic field parameter M, radiative Prandtl number Pn (hence also radiation parameter RÞ and viscosity parameter H whereas it increases with the increase of the heat generation parameter Q. 2. The rate of heat transfer increases monotonically with the increase of the viscosity parameter H and the radiative Prandtl number Pn whereas it decreases with the increase of the magnetic field parameter M and the heat generation parameter Q. 3. Sufficiently strong heat generation parameter may alter the temperature gradient. 4. The rate of mass transfer increases monotonically with the increase of the heat generation parameter Q, and viscosity parameter H whereas it decreases with the increase of the magnetic field parameter M and the radiative Prandtl number Pn. 5. Angle of inclination and radiation strongly controls the flow, heat and mass transfer characteristics. 6. Absolute errors in the skin friction coefficient, Nusselt number, and Sherwood number are significant for lower negative values of the viscosity parameter H if the Prandtl number is considered as constant rather than variable inside the boundary layer. 7. The velocity and temperature profiles in the case of constant viscosity are higher than the corresponding case of variable viscosity. 8. For modeling thermal boundary layers with temperature dependent viscosity the Prandtl number must be treated as variable inside the boundary layer. 9. The skin friction coefficient, rate of heat transfer and rate of mass transfer are lower for the constant property case (e.g. constant viscosity, constant electric conductivity) than the variable case (e.g. variable viscosity, variable electric conductivity).

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