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Dec 12, 2015 - set of linear slip-flow conditions. ... studied the effect of Hall current and variable viscosity on an unsteady MHD ... rotating disk in an electrically conducting fluid with temperature ... the cylindrical coordinate system with r and ϕ as the radial and ... uniform suction or injection through the disk is considered.
Columbia International Publishing American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189 doi:10.7726/ajhmt.2015.1012

Research Article

Transient Hydromagnetic Forced Convective Heat Transfer Slip Flow Due to a Porous Rotating Disk with Variable Fluid Properties M. S. Alam1,2, Nayan K. Poddar2, M. M. Rahman1,*, and K. Vajravelu3 Received 27 May 2015; Published online 12 December 2015 © The author(s) 2015. Published with open access at www.uscip.us

Abstract In this paper we have studied the problem of an unsteady hydromagnetic forced convective heat transfer slip flow over a porous rotating disk taking into account the temperature dependent density, viscosity and thermal conductivity. The governing non-linear partial differential equations of the flow are transformed into a set of non-linear ordinary differential equations using similarity transformations. The resulting nondimensional equations have been solved numerically by applying Nachtsheim-Swigert shooting iteration technique along with sixth-order Runge-Kutta iteration scheme. Comparison with previously published work for the steady case of the problem is performed and the results are found to be in very good agreement. The results of the numerical solution are presented graphically in the form of velocity (i.e. radial, tangential as well as inward axial), temperature profiles and variable Prandtl number for various values of the model parameters. The corresponding skin friction coefficients (i.e. radial and tangential) and the rate of heat transfer coefficient (i.e. Nusselt number) are also calculated and tabulated. The obtained numerical results show that when modeling a thermal boundary layer with temperature dependent fluid properties, consideration of Prandtl number as constant within the boundary layer produces unrealistic results. Therefore, it must be treated as variable throughout the boundary layer. Results also show that the slip factor significantly controls the flow and heat transfer characteristics. Keywords: Heat Transfer; Hydromagnetic; Unsteady Flow; Rotating Disk; Slip Flow; Variable Fluid Properties

1. Introduction Rotating disk flows along with heat transfer is one of the classical problems of fluid mechanics, which has both theoretical and practical values. Rotating disk flows have practical applications in many areas, such as rotating machinery, lubrication, oceanography, computer disk drives, ______________________________________________________________________________________________________________________________ *Corresponding email: [email protected] (M. M. Rahman), Phone: +968 24141423, Fax: +968 24141490 1 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P. O. Box 36, P.C. 123 Al-Khod, Muscat, Sultanate of Oman 2 Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh 3 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA 165

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

viscometry and crystal growth processes etc. Due to the practical applications of rotating disk flows, many researchers (von-Karman, 1921; Cochran, 1934; Benton, 1966; El-Mistikawy and Attia, 1990; El-Mistikawy et al., 1991; Hassan and Attia, 1997; Attia, 1998; Rahman and Sattar, 1999; Maleque and Sattar, 2005; Rahman, 2010; Rahman and Postelnicu, 2010; Zueco and Rubio, 2012) have studied and reported results on rotating disk flow with various flow conditions. Slip velocity is a function of the velocity gradient near the wall. It is known that for gaseous flow there always exist a non–zero velocity near the wall and based on a momentum balance at the wall. In certain situations, the assumption of no slip boundary condition does no longer apply. When fluid flows in micro electro mechanical systems (MEMS), the no-slip condition at the solid fluid interface is no longer applicable. A slip flow model more accurately describes the non- equilibrium near the interface. A partial slip may occur on a stationary and moving boundary when the fluid is particulate such as emulsions, suspensions, foams, and polymer solutions. Sparrow et al. (Sparrow et al., 1971) studied the flow of Newtonian fluid due to the rotation of a porous-surfaced disk with a set of linear slip-flow conditions. A substantial reduction in torque then occurred as a result of surface slip. Miklavcic and Wang (2004) further revisited the problem of Sparrow et al. (Sparrow et al., 1971) and pointed out that the slip flow boundary conditions could also be used for slightly rarefied gases or for flow over grooved surfaces. Arikoglu and Ozkol (2006) studied MHD slip flow over a rotating disk with heat transfer. It is observed that both the slip factor and the magnetic flux decrease the velocity in all directions and thicken the thermal boundary layer. Osalusi et al. (2008) studied thermal-diffusion and diffusion-thermo effects on MHD slip flow due to a rotating disk. Rahman (2010) analyzed the convective hydromagnetic slip flow with variable fluid properties due to a porous rotating disk and obtained numerical results show that the slip factor significantly controls the flow and heat transfer characteristics. Daniel (2015) investigated the effect of heat generation/absorption on boundary layer slip flow of a nanofluid over a porous stretching sheet. He noticed that velocity profile, the flow field at the surface of a porous stretching sheet decreases as the velocity slip parameter increases. In the classical treatment of thermal boundary layers, fluid properties such as density, viscosity and thermal conductivity are assumed to be constant; however experiments indicate that this assumption only makes sense if the temperature does not change rapidly for the application of interest. To predict the flow behavior accurately, it may be necessary to take into account this properties as variables. Zakerullah and Ackroyd (1979) studied laminar natural convection boundary layers on horizontal circular discs. Herwig and Klemp (1988) investigated variable properties effects of fully developed laminar flow in concentric annuli. Maleque and Sattar (2002) studied the effect of Hall current and variable viscosity on an unsteady MHD laminar convective flow due to a rotating disk. The effects of variable fluid properties (density, viscosity and thermal conductivity) and Hall current on the steady MHD laminar convective flow due to a porous rotating disk have been investigated by Maleque and Sattar (2005). Attia (2006) analyzed the problem of unsteady flow and heat transfer of viscous incompressible fluid with temperature dependent viscosity due to a rotating disk in a porous medium. The thickness of the boundary layer relative to the velocity boundary layer depends on the Prandtl number which by its definition varies directly with the fluid viscosity and inversely with the thermal conductivity of the fluid. As the viscosity vary with temperature so does the Prandtl number. Despite this fact, all of the afore-mentioned studies treated the Prandtl number as a 166

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

constant. The use of a constant Prandtl number within the boundary layer when the fluid properties are temperature dependent introduces errors in the computed results. Pantokratoras (2005) investigated some new results on forced and mixed convection boundary layer flow along a flat plate with variable viscosity and variable Prandtl number while Pantokratoras (2007) further studied non-Darcian forced convection heat transfer over a flat plate in a porous medium with variable viscosity and variable Prandtl number. Recently, Rahman and his co-workers (Rahman et al., 2009; Rahman and Salauddin, 2010; Rahman, 2010; Rahman et al., 2010; Rahman and Eltayeb, 2011; Alam et al., 2014) have studied several thermal boundary layer problems taking into account the variability of viscosity for both Newtonian and Non-Newtonian fluids in different geometry with various flow conditions. All of these studies confirmed that for the accurate prediction of the thermal characteristics of variable viscosity, the Prandtl number must be treated as a variable rather than a constant. The objective of the present study is to investigate the effects of variable fluid properties on an unsteady hydromagnetic forced convective heat transfer slip flow due to a porous rotating disk taking into account the variable Prandtl number. In this study we extend the work of Alam et al. (2014) to include the effects of magnetic field on slip flow due to a porous rotating disk in an electrically conducting fluid with temperature dependent fluid properties such as density, viscosity and thermal conductivity. By introducing a similarity transformation the governing non-linear partial differential equations are reduced to locally similar ordinary differential equations which are solved numerically by applying shooting method and the results are discussed from the physical point of view.

2. Physical Model and Governing Equations We use a non-rotating cylindrical polar coordinate system r ,  , z  where z is the vertical axis in the cylindrical coordinate system with r and  as the radial and tangential axes respectively. Let us consider a disk which rotates with an angular velocity  about the z-axis. The viscous, compressible, electrically conducting fluid occupies the region z  0 with the rotating disk placed at z  0 . The components of the flow velocity q are u, v, w in the directions of increasing r ,  , z  respectively. The fluid is assumed to be Newtonian, viscous and electrically conducting. The surface of the rotating disk is maintained at a uniform temperature Tw . Far away from the wall; the free stream is kept at a constant temperature T and a constant pressure p  . An external uniform magnetic field is applied perpendicular to the surface (i.e. in the z  direction) of the disk and has a constant magnetic flux density (or, applied magnetic field) B0 everywhere in the fluid. The assumption is valid only when the magnetic Reynolds number is very small (i.e. Re m  1). A uniform suction or injection through the disk is considered. The flow configuration and geometrical coordinates are shown in Fig. 1. We also assume that the fluid properties, viscosity   , thermal conductivity   and density   are the functions of temperature only and obey the following laws (see Jayaraj, 1995; later used by Maleque and Sattar, 2005; Osalusi and Sibanda, 2006; Rahman, 2010; Alam et al., 2014): (1)     T T a ,     T T b ,     T T d , 167

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

where a , b and d are arbitrary components and   ,   and   are the viscosity, thermal conductivity and density of the ambient fluid respectively.

w z

B0

P

T

u v

r



y



x

Tw

Fig. 1. Flow configurations and coordinate system. Therefore under the above assumptions, the governing equations (Navier- Stokes equations and Energy equation), due to unsteady axially symmetric, compressible hydromagnetic laminar flow of a homogeneous fluid take the following form:

 r    ru   rw  0 , t r z 2 p   u    u    u  B02  u u v u            u, ρ  u   w    r r  r  r  r  z  z   r r z   t

v uv v    v    v    v  B 2  v ρ  u   w               0 v , r r z  r  r  r  r  z  z    t p   w  1  w w   w μw    μ w  ,  μ ρ u w    z r  r  r r z  z  r z   t T T   T  κ T   T   T ρc p  u  w    κ  κ  , r z  r  r  r r z  z   t

(2) (3) (4) (5) (6)

where the variables and related quantities are defined in the nomenclature. If mean free path of the fluid particles is comparable to the characteristic dimensions of the flow field domain the assumption of continuum media no longer valid as a consequence Navier-Stokes equation brakes down. In the range 0.1  Kn  10 of Knudsen number, the high order continuum 168

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

equations (Burnett equations) should be used. For the range of 0.001  Kn  0.1 , no-slip boundary conditions cannot be used and should be replaced with the following expression : u (7) U t   *  2    /   , z where U t is the target velocity,  is the target momentum accommodation coefficient and  * is the mean free path. For Kn  0.001 , the no-slip boundary condition is valid; therefore, the velocity at the surface is equal to zero. In this study the slip and the no-slip regimes of the Knudsen number that lies in the range 0  Kn  0.1 are considered. 2.1 Boundary Conditions By using equation (7), the applicable boundary conditions for the present model are as follows: (i) On the surface of the disk ( z  0) : u v (slip flow), u   *  2    /   , v  r   *  2    /   z z w  ww (permeable surface), T  Tw (uniform surface temperature). (8) Matching with the quiescent free stream ( z  ) :

(ii)

u  0 , v  0 , T  T , p  p .

(9)

3. Similarity Transformations To obtain the solutions of the above governing equations (2)-(6) together with the boundary conditions (8)-(9) the following similarity transformations which are little deviated from the usual von-Karman transformations are introduced (see also Alam et al., 2014):

 , u  rF  , v  rG , w  H  ,    T  T  p  p   2  P( ),  ( )  .  Tw  T 



z

(10)

where  is a scale factor and is a function of time as    t  which follows from Sattar and Hossain (1992), Rahman et al. (2012) and Alam et al. (2014). Then substituting (10) into equations (2)-(6) we obtain the following set of dimensionless nonlinear ordinary differential equations: 1 (11) d 1    RH       2RF  RH   0,





F   a 1    F     F   R F 2  G 2  HF 1



1    

1 a  d

1 G   1    G    G  R2FG  HG 1   







1 a  d







1 a  d



1 a  d

 MG 1   

1 H   2P 1  γθ   aγ1  γθ  H θ   λη  RH H  1  γθ     b 1   1 2  Pr     RH 1   d b  0, 1 a



 MF 1   



1 a  d

 0,

 0,

 0,

(12) (13) (14) (15) 169

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189



 2

is the rotational parameter, M  B0  is the magnetic interaction parameter,     c d is the unsteadiness parameter, Pr   p is the ambient Prandtl number and

where R 

2

   dt

2



Tw  T is the relative temperature difference parameter which is positive for a heated T surface, negative for a cooled surface and zero for uniform properties.

 

With reference to the transformation (10), the boundary conditions (8) and (9) transform to

F  F  , G  1  G , H  ws ,   1 at   0 , F  0 , G  0 , p  0 ,   0 as    ,

(16) (17)

*

w

where    2    /   is the slip parameter and ws  w represents a uniform suction when   ws is negative ws  0 and uniform injection when ws is positive ws  0 at the surface of the disk.

4. Variation of Prandtl Number The Prandtl number is a function of both viscosity and thermal conductivity. Since as the viscosity and thermal conductivity varies across the boundary layer, the Prandtl number also varies. The assumption of constant Prandtl number inside the boundary layer may produce unrealistic results (see Pantokratoras, 2005 & 2007; Rahman et al., 2009; Rahman et al., 2010; Rahman and Eltayeb, 2011; Alam et al., 2014). Therefore the Prandtl number related to the variable viscosity and variable thermal conductivity is defined by

 c p   1   a c p a b Pr    Pr 1    . b    1    At the surface   0 of the disk, this can be written as Prw  1   

a b

(18)

Pr .

(19)

From equation (18) it can be seen that for   0 , the variable Prandtl number Pr is equal to the

ambient Prandtl number Pr .For    , i.e. outside the boundary layer,    becomes zero; therefore, Pr equals Pr regardless of the values of  .

Table 1 Values of Pr versus  for Pr  0.64 , a  0.7 , b  0.83 , at   0 .



-0.8

-0.5

-0.2

0.0

0.2

0.3

0.5

0.8

1.0

3.0

5.0

Pr

0.789

0.700

0.659

0.640

0.625

0.619

0.607

0.593

0.585

0.534

0.507

170

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The above Table-1 shows that the variation of the Prandtl number at the surface of the disk for several values of  for a fixed value of the ambient Prandtl number Pr  0.64 and the exponents a  0.7 , b  0.83 . From this Table it is observed that for a positive value of  , Prandtl number at the surface of the disk Prw decreases as  increases. On the other hand, the opposite effect is observed when  is negative. Note that for   1 no physically viable solutions exist.

In light of the above discussion and using equation (18), the non-dimensional temperature equation (15) can be rewritten as



   b 1   1  2  Pr     RH  1   1



a d

 0,

(20)

Equation (20) is the corrected non- dimensional form of the energy equation in which Prandtl number is treated as variable (see also Rahman et al., 2009; Rahman et al., 2010; Rahman and Eltayeb, 2011; Alam et al., 2014).

5. Parameters of Engineering Interest The parameters of engineering interest for the present problem are the skin-friction coefficient and the Nusselt number which indicate physically the wall shear stress and the rate of heat transfer respectively. The action of the variable properties in the fluid adjacent to the disk sets up a tangential shear stress, which opposes the rotation of the disk. As a consequence, it is necessary to provide a torque at the shaft to maintain a steady rotation. The radial shear stress  , and tangential shear stress  t are defined as:

  u

w 

r

a  r     F (0),    1     z  r   z  0  

(21)

  v 1 w  a r  G(0). (22)    1     z r        z 0

 t   

Hence the skin-frictions (Cf   /  2 r 2 ) along the radial and tangential directions are obtained as

Cf r Re1   

a

 F (0) ,

(23)

Cft Re1     G(0) , (24) r where Re  is the rotational Reynolds number. Thus equation (23) and (24) shows that the  radial and tangential skin-frictions coefficient are proportional to F (0) and G(0). a

The rate of heat transfer from the disk surface to the fluid is computed by the application of Fourier’s law as given below

 T  b (T  T ) qw        1    w   (0) .   z  z  0

(25)

Hence the Nusselt number is obtained as 171

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

Nu 

qw

  (Tw  T )

 1     (0) . b

(26)

From equation (26) implies that the Nusselt number is proportional to   (0) .

6. Numerical Procedure The set of equations (11)-(14) and (20) are highly non-linear and coupled and therefore the system cannot be solved analytically. We dropped equation (14) from the system as it can be used for calculating pressure once F and H are known from the rest of the equations. Therefore, the equations (11)-(13) and (20) with boundary conditions (16)-(17) have been solved numerically by using sixth order Runge-Kutta method along with Nachtsheim-Swigert (1965) shooting iteration technique (for detailed discussion of the method see Alam et al., 2006) with M , R ,  ,  ,  , ws and

Pr as prescribed parameters. A step size of   0.01 was selected to be satisfactory for a convergence criterion of 106 in all cases. The value of  was found to each iteration loop by the statement     . The maximum value of  to each group of parameters M , R ,  ,  ,  , ws and Pr determined when the value of the unknown boundary conditions at   0 does not change to a successful loop with an error less than 106. Table 2 Numerical values of F (0) ,  G (0) and   (0) for different values of ws

with

M        0 , R  1.0 and Pr  0.71 .

ws 5 4 3 2 1 0 -1 -2 -3 -4 -5

 G (0)

F (0)

 θ (0)

KD, 2000

Present

KD, 2000

Present

KD, 2000

Present

0.197566 0.243044 0.309147 0.398934 0.489481 0.510233 0.389569 0.242421 0.165582 0.124742 0.999187

0.19756600 0.24304404 0.30914768 0.39893387 0.48948057 0.51022378 0.38954065 0.24241310 0.16558828 0.12475268 0.09991986

0.0154706 0.0289211 0.0602893 0.135952 0.302173 0.615922 1.175222 2.038527 3.012142 4.005180 5.002661

0.01547065 0.02892121 0.06028945 0.13595275 0.30217432 0.61592380 1.17526180 2.03859590 3.01222231 4.00526266 5.00271176

0.00000006 0.00001070 0.000576 0.011013 0.084884 0.325856 0.793048 1.437782 3.135585 2.842381 3.551223

0.00000007 0.00001075 0.00057793 0.01103604 0.08504687 0.32637889 0.79393633 1.43876482 2.13677058 2.84369011 3.55222471

6.1 Code Verification If   0 (i.e. for no-slip property case),   0 (i.e. for steady case), γ  0 (i.e. for constant property case), M  0 (i.e. in absence of MHD) and R  1.0 in the absence of heat transfer, the 172

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present problem exactly coincides with those of Kelson and Desseaux (2000 but herein and after referred as KD, 2000) and Arikoglu and Ozkol (2006 but herein and after referred as AO, 2006).To assess the accuracy of the present code, we have calculated values of F (0) ,  G (0) and   (0) for

different values of ws (see in Table-2) and ε (see in Table-3) and in the absence of heat and mass transfer. Table 2-3 present the comparisons of the data obtained in the present work and those of KD, 2000 and AO, 2006. It is clearly observed that very good agreement between the results exists. This lends confidence to use the present numerical code. It is good to mention that this is a very standard process of validating a numerical code with the existing literature. This process can be found in any ISI journals including American Journal of Heat and Mass Transfer (for example Daniel, 2015; Lin and Hsiao, 2015). Table 3 Numerical values of F (0) ,  G (0) and   (0) for different values of

ε

with

M  ws      0 , R  1.0 and Pr  0.71 .

ε 0.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

 G (0)

F (0)

 θ (0)

AO, 2006

Present

AO, 2006

Present

AO, 2006

Present

0.51023261 0.42145363 0.35258100 0.22384820 0.12792364 0.06101009 0.01858852 0.00681255 0.00236159

0.51022378 0.42144560 0.35257351 0.22384313 0.12792035 0.06100834 0.01858796 0.00861240 0.00236161

0.61592201 0.60583524 0.58367676 0.50280970 0.39492759 0.27337013 0.14338820 0.08103008 0.04378846

0.61592380 0.60583699 0.58367869 0.50281167 0.39492982 0.27337241 0.14339025 0.08103175 0.08378973

0.32586063 0.33349695 0.33678090 0.33465287 0.32043299 0.29299798 0.24440461 0.20504914 0.16882963

0.32637889 0.33402796 0.33734031 0.33519936 0.32099888 0.29357940 0.24466400 0.20570012 0.16953552

7. Results and Discussion Here, we investigate the effects of the pertinent parameters (i.e. magnetic interaction parameter M , rotational parameter R, slip parameter  , unsteadiness parameter , suction/injection parameter ws , relative temperature difference parameter  and variable Prandtl number Pr ) on the flow and heat transfer characteristics and therefore the numerical results are presented graphically as well as tabulated form. In the present analysis the fluid is considered as a flue gas for which ambient Prandtl number, Pr  0.64 and the values of the exponents a , b and d are taken as a  0.7 , b  0.83 and d  1 (see Jayaraj, 1995). The default values of the other parameters throughout the simulation are considered to be M  0.5 , R  1.0 ,   0.2   0.5 ,   0.2 and Pr  0.625 unless otherwise specified. Effects of slip parameter  The effect of the slip parameter  on the non-dimensional velocity and temperature profiles are 173

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

displayed in Figs. 2(a)-(d), respectively keeping all other parameters values fixed. From Fig. 2(a) we see that the radial boundary layer decreases very rapidly with the increase of the slip parameter. Also for large values of  i.e.    (full slip), the rotating disk does not cause rotation of the fluid particles. Because in this range of  the flow becomes entirely potential. Therefore, there will be no motion in the fluid. This can be further explained as follows: the centrifugal force acting on the rotating disk will throw out the fluid that sticks to it. On the other hand, the flow in the axial direction will come forward to compensate for this thrown fluid. But increasing the slip on the surface of the disk reduces the amount of fluid that can stick on it; as a consequence the efficiency of the rotating disk reduced substantially and is unable to transfer its circumferential momentum to the fluid particles. A reduction in the circumferential velocity results in a reduction in the centrifugal force which in turn decreases the inward axial velocity substantially as can be seen from Fig. 2(c). On the other hand, the slip parameter  does not enter directly into the non-dimensional thermal boundary condition. Thus the temperature of the flow field within the boundary layer increases a little with the increase of the slip parameter as can be seen from Fig. 2(d). Fig. 2(e) shows a small decreasing effect of  on the variable Prandtl number throughout the boundary layer. Effects of magnetic interaction parameter M Imposition of a magnetic field to an electrically conducting fluid creates a drag force called the Lorentz force. This force has the tendency to slow down the flow around the disk at the expense of increasing its temperature. The influence of the magnetic field parameter M on the radial, tangential and inward axial velocity (i.e. F, G and  H ) distributions is depicted in Figs. 3(a)-(c) respectively. An increase in M induces a significant decrease in radial and tangential velocity profiles throughout the boundary layer. From Fig. 3(c) it is also apparent that inward axial velocity decreases substantially with the increase of the magnetic field parameter. The magnetic interaction parameter M does not enter directly into the energy equation but its influence come through the momentum equation. From Fig. 3(d), it is observed that the value of non-dimensional temperature profile increases a little with the increasing values of M and this also leads to a small rate of increase in the thermal boundary layer thickness. The variation of the Prandtl number within the boundary layer for different values of the magnetic field parameter is depicted in Fig. 3(e). This figure reveals that variable Prandtl number decreases with the increase of M . Effects of unsteadiness parameter  From Figs. 4(a)-(d) we observe that the radial, tangential, inward axial velocity and temperature profiles decrease with an increasing values of the unsteadiness parameter  . In Fig. 4(e) we see that within the boundary layer for a fixed value of  , variable Prandtl number increases as the unsteadiness parameter  increases while far away from the surface of the disk Pr equals its ambient value Pr . Effects of relative temperature difference parameter  Figs. 5(a)-(d) respectively show the variation of the dimensionless radial, tangential, inward axial velocity and temperature profiles for various values of the relative temperature difference parameter  . From Fig. 5(a), we see that the radial velocity increases for all increasing values of  . The case   0 corresponds to the constant property of the working fluid. From Figs. 5(b)-(c) we 174

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

see that the tangential velocity increases whereas the inward axial velocity decreases with the increasing values of  . It is seen from Fig. 5(d) that the thickness of the thermal boundary layer increases markedly with the increase of  . In Fig. 5(e) we see that the variable Prandtl number Pr decreases very rapidly within the boundary layer for the increase of  . For   0 variable Prandtl number Pr equals the ambient Prandtl number Pr . From this figure we also observe that for fixed of  (> 0), Pr increases as  increases and for    (i. e. outside the boundary layer), it converges to its ambient value Pr . From this figure it is also clear that at the surface of the disk (at   0 ),   0.0, 0.3, 0.6 and 0.8 corresponds to Pr  0.64, 0.619, 0.602 and 0.593 when other parameter values are fixed. Effects of suction/injection parameter ws

The effects of suction/injection parameter ws  on the radial, tangential and inward axial velocity profiles, temperature profiles and variable Prandtl number within the boundary layer have been shown in Figs. 6(a)-(e) respectively. It is seen from Figs. 6(a)-(b) that both the radial and tangential velocity profiles decrease very rapidly as the suction velocity ( ws < 0) intensifies. The maximum of the radial velocity profiles moves towards the surface of the disk. It is also apparent that the thickness of the boundary layer decreases as suction velocity increases. Therefore suction stabilized the boundary layer growth. From Fig. 6(c) it is observe that for strong suction, inward axial velocity is nearly constant. The effect of suction parameter on the thermal boundary layer is found to similar to those of the radial and tangential velocity boundary layers. Thus applying suction, one can control the flow and heat transfer characteristics. In Fig. 6(e) we have plotted variable Prandtl number as a function of  to show the variation of Prandtl number throughout the boundary for several values of the suction parameter. From this figure we see that within the boundary layer for a fixed value of  , variable Prandtl number increases as the suction parameter increases while far away from the surface of the disk Pr equals its ambient value Pr . An opposite effects is observed for the case of fluid injection ( ws > 0). Effects of rotational parameter R The effects of rotational parameter R on the dimensionless radial, tangential and inward axial velocity profiles, temperature profiles and variable Prandtl number within the boundary layer have been shown in Figs. 7(a)-(e) respectively. From these Figs. 7(a)-(d) we observe that the radial and inward axial velocity profiles increase whereas both the tangential velocity and temperature profiles decreases with increasing values of the rotational parameter. In Fig. 7(e) we have plotted variable Prandtl number as a function of  to show the variation of Prandtl number throughout the boundary for several values of the rotational parameter. From this figure we see that within the boundary layer for a fixed value of  , variable Prandtl number increases as the rotational parameter increases while far away from the surface of the disk Pr equals its ambient value Pr . The effects of magnetic interaction parameter M on the radial and tangential skin friction coefficients ( F (0) ,  G (0) ) and the rate of heat transfer (i.e. Nusselt number (   (0) ) are shown in table 4. From this table, we observe that both the radial skin friction coefficient and rate of heat 175

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

transfer decrease whereas the tangential skin friction coefficient increases with the increasing values of the increasing the magnetic interaction parameter M . 0.04

0.9

0.03

F

 = 0.2, 0.7, 1.0, 2.0, 3.0

0.6

 = 0.2, 0.7, 1.0, 2.0, 3.0

G

0.02

0.3

0.01

(a)

(b) 0

0

0

1



2

3

4

1

0

1

2



3

1

 = 0.2, 0.7, 1.0, 2.0, 3.0

0.75

0.9

 = 0.2, 0.7, 1.0, 2.0, 3.0

-H



0.5

0.8 0.25

(c)

0.7

(d) 0



2

4

6

0

0

1

2



3

4

5

0.64

0.635

Pr  = 0.2, 0.7, 1.0, 2.0, 3.0 0.63

(e)

0.625

0

2



4

6

Fig. 2. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profiles and (e) variable Prandtl number for several values of ε.

176

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189 0.9

0.06

(b) (a)

M = 0.0, 0.1, 0.5, 1.0

0.04

0.6

F

M = 0.0, 0.1, 0.5, 1.0

G 0.3

0.02

0

0

1

2



3

0

4

1

0



1

2

3

4

1

M = 0.0, 0.1, 0.5, 1.0

0.75

0.9

M = 0.0, 0.1, 0.5, 1.0

-H

0.5



0.8

0.25

(c)

(d) 0.7

0

2



4

0

6

0

1

2



3

4

5

0.64

0.635

M    

Pr (e)

0.63

0.625

0

1

2



3

4

5

Fig. 3. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profiles and (e) variable Prandtl number for several values of M .

177

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189 0.9

0.05

0.04

 = 0.2, 0.4, 0.6, 0.8

     

0.6

0.03

(b)G

(a) F 0.02

0.3 0.01

0

0 0

1

2

3



4

5

1

0

1

2



3

4

5

1

 = 0.2, 0.4, 0.6, 0.8 0.75 0.9

-H

     

0.5

 0.8 0.25

(c)

(d) 0.7

0

2



4

6

0

8

0

2



4

6

0.64

0.635

Pr

 = 0.2, 0.4, 0.6, 0.8

0.63

(e) 0.625

0

2



4

6

8

Fig. 4. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profiles and (e) variable Prandtl number for several values of λ .

178

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

0.04

0.04

0.03

(a)

0.03

 = 0.0, 0.3, 0.6, 0.8 (b)

0.02

F

 = 0.0, 0.3, 0.6, 0.8

0.02

F

0.01

0

0.01

0

1

2

3



4

0

5

1.2

0

1

2



3

4

5

1

0.75 0.8

 = 0.0,0.3, 0.6, 0.8

-H



0.5

0.4 0.25

(c)

(d)

 = 0.0, 0.3, 0.6, 0.8 0

0

5

10



0

15

0

2



4

6

0.66

0.64

0.62

Pr

 = 0.0, 0.3, 0.6, 0.8

0.6

(e) 0.58

0

2



4

6

Fig. 5. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profiles and (e) variable Prandtl number for several values of γ .

179

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

1 0.12

ws = 1.0, 0.5, 0.0, -0.5, -1.0

0.75

(a)0.08

(b)

F

G

0.04

ws = 1.0, 0.5, 0.0, -0.5, -1.0 0.5

0.25

0 0

0

2

4



0

2

0.75

ws = 1.0, 0.5, 0.0, -0.5, -1.0

0.5

-H

6

1

ws = 1.0, 0.5, 0.0, -0.5, -1.0

1

4



6

0



(c) -0.5

(d)

0.5

0.25

-1 0

5

10



0

15

0

2



4

6

8

0.64

0.635

Pr ws = 1.0, 0.5, 0.0, -0.5, -1.0

0.63

(e)

0.625

0

2



4

6

8

Fig. 6. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profiles and (e) variable Prandtl number for several values of ws .

180

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

0.04

0.9

0.03

R = 0.2, 0.4, 0.6, 0.8

0.6

(a)

R = 0.2, 0.4, 0.6, 0.8

(b) G

0.02

F

0.3 0.01

0

0

1

2



3

4

0

5

0

1

1



2

3

4

1

R = 0.2, 0.4, 0.6, 0.8 0.75

0.75

0.5

0.5

R = 0.2, 0.4, 0.6, 0.8 

-H 0.25

(d)

0.25

(c) 0

0

5



10

0

15

0

2



4

6

0.64

0.635

Pr R = 0.2, 0.4, 0.6, 0.8

0.63

(e) 0.625

0

2



4

6

8

Fig. 7. Variation of (a) radial velocity, (b) tangential velocity, (c) axial velocity, (d) temperature profiles and (e) variable Prandtl number for several values of R .

181

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Table 4 Numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer for different values M.

M

F (0)

 G (0)

  (0)

0.0 0.1 0.2 0.3 0.5 1.0 1.5 2.0

0.14760127 0.13716725 0.12794820 0.11977209 0.10598053 0.08178446 0.06628945 0.05558424

0.90492015 0.93938729 0.97230930 1.00375976 1.06257958 1.18968160 1.29516016 1.38514420

0.73548542 0.73315800 0.73112726 0.72935019 0.72641414 0.72142681 0.71850496 0.71662042

Table 5 Numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer for different values of ε.

ε

F (0)

 G (0)

  (0)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 1.0 3.0 5.0 7.0 8.0 10.0 15.0 16.0

0.21649758 0.14811648 0.10598053 0.07858672 0.05997223 0.04685584 0.03737507 0.01739266 0.00183278 0.00051406 0.00021106 0.00014692 0.00007947 0.00002543 0.00002117

1.34586768 1.18912616 1.06257958 0.95956996 0.87452851 0.80329192 0.74281115 0.57120733 0.26604911 0.17359621 0.12884691 0.11413795 0.09292340 0.06344455 0.05965939

0.72614241 0.72646671 0.72691414 0.72746034 0.72806700 0.72840534 0.72919806 0.71832417 0.71236047 0.71107117 0.71060356 0.71047576 0.71031567 0.71014332 0.71012559

The variation of the radial and tangential skin-frictions and the rate of heat transfer for some selected values of the slip factor ε are shown in table 5. From here we see that skin-friction in both directions decreases with the increase of the slip factor. The largest skin-friction is found for the case of no-slip at the surface. On the other hand the rate of heat transfer increases with the increase of slip factor within the range of 0    1. But outside of this range of ε, the rate of heat transfer decreases with the further increase of the slip factor. Thus the rate of heat transfer can be strongly controlled by controlling the slip on the disk. This result is consistent with the works of Rahman (2010).

182

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The numerical values of fluid temperature for different values of relative temperature difference parameter  and slip parameter  have been shown in table 6 and table 7 respectively. From these tables we observe that fluid temperature  decreases with an increase in the value of  for both cases. Further we also see that for fixed value of  ,  increases with the increasing values of  and  . Table6 Numerical values of temperature obtained for several values of

γ.

η

θ η  γ  0.0

θ η γ  1.0

θ η γ  3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.00000000 0.90887286 0.82369564 0.74436605 0.67074999 0.60268266 0.53997148 0.48240004 0.42973254 0.38171835 0.33809641

1.00000000 0.96332678 0.92679101 0.89043096 0.85428718 0.81840235 0.78282109 0.74758975 0.71275621 0.67836962 0.64448021

1.00000000 0.98412427 0.96817437 0.95215231 0.93606023 0.91990043 0.90367535 088738758 0.87103989 0.85463519 0.83817657

Table7 Numerical values of temperature obtained for several values of

ε.

η

θ η ε  0.0

θ η ε  1.0

θ η ε  3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.00000000 0.92870560 0.68012833 0.79437978 0.73155967 0.67175135 0.61501872 0.56140417 0.51092758 0.46356814 0.41935485

1.00000000 0.92947832 0.86165604 0.79663937 0.73451766 0.67536188 0.61929363 0.56613457 0.51610625 0.46913035 0.42517927

1.00000000 0.93006533 0.86280498 0.79831602 0.73667960 0.67795988 0.62220325 0.56943789 0.51967355 0.47290174 0.42909617

Finally, Table-8 shows that the significance of the relative temperature difference parameter (  ) on the rate of heat transfer for both variable Prandtl number (VPr) and constant Prandtl number (CPr). From this table we observe that in both cases the rate of heat transfer from the surface of the disk to the fluid decreases for all increasing values of  . We also observe that the rate of heat transfer for the variable property case is lower than the constant property case and the relative 183

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

error between them increases significantly with the increase of  . Therefore, consideration of Prandtl number as constant within the boundary layer for variable property is unrealistic. Table 8 Numerical values of  θ (0) for several values of  .

  (0) Absolute error =



(i) C Pr

(ii) V Pr

ii  i  100 ii

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0

0.94129213 0.82510158 0.73306666 0.65861874 0.59715235 0.54567102 0.50203095 0.46451406 0.43203211 0.40357592 0.37853643 0.23178333 0.16660688

0.94129213 0.82096435 0.72641414 0.65052555 0.58783980 0.53525753 0.49114627 0.45321730 0.42082093 0.39239317 0.36729824 0.22206217 0.15838009

0% 0.501421% 0.907492% 1.228813% 1.559493% 1.908382% 2.168129% 2.431952% 2.594988% 2.770916% 2.968853% 4.194072% 4.937845%

8. Conclusions In this paper, we have studied numerically the problem of unsteady hydromagnetic forced convective heat transfer slip flow over a porous rotating disk taking into account the temperature dependent density, viscosity and thermal conductivity. Using a new class of similarity transformation close to von-Karman, the governing non-linear partial differential equations have been transformed into a system of non-linear ordinary differential equations that are locally similar. These are solved numerically by applying Nachtsheim-Swigert shooting iteration technique along with a sixth-order Runge-Kutta integration scheme. Comparison with previously published work for steady case of the problem was performed and found to be in very good agreement. As a result of computations the following major conclusions can be drawn:    

The rate of heat transfer in a fluid of constant property is higher than in a fluid of variable property. For modeling thermal boundary layers with temperature dependent fluid properties (viscosity, thermal conductivity, density), Prandtl number must treated as variable inside the boundary layer. Slip Parameter significantly controls the flow and heat transfer characteristics. Increasing slip parameter decreases fluid whereas increases temperature profile within the boundary layer. 184

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

       

Magnetic field retards the fluid motion. Radial, tangential and inward axial velocity profiles decrease whereas the temperature profiles increase with the increasing values of the magnetic parameter. Suction stabilizes the growth of the boundary layer. Radial and tangential velocities and temperature profiles increase whereas the inward axial velocity decreases with the increasing values of  . Radial and inward axial velocity profiles increase whereas both the tangential velocity and temperature profiles decreases with increasing values of the rotational parameter. Radial, tangential, inward axial velocity and temperature profiles decrease with an increasing values of the unsteadiness parameter. Radial skin friction coefficient and rate of heat transfer decrease whereas the tangential skin friction coefficient increases with the increasing values of the magnetic field parameter M. The rate of heat transfer can be strongly controlled by controlling the slip on the disk.

Nomenclature a b

B0 Cf

: Constant : Constant : Applied magnetic field : Skin-friction coefficients

cp d F G H M Nu p

: Specific heat at constant pressure

p Pr Pr

: Pressure of the ambient fluid

q

qw R

: Constant : Dimensionless radial velocity : Dimensionless tangential velocity : Dimensionless axial velocity : Magnetic interaction parameter : Nusselt number : Pressure within the boundary layer : Variable Prandtl number : Ambient Prandtl number : Velocity vector : Surface heat flux

T Tw

: Rotational parameter : Rotational Reynolds number : Cylindrical radial coordinate : Time : Temperature within the boundary layer : Temperature at the surface of the disk

T

: Temperature of the ambient fluid

Re

r t

185

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189

Ut u v w ws

: Target velocity : Velocity along radial direction : Velocity along tangential direction : Velocity along axial direction : Non-dimensional suction/injection velocity

ww : Dimensional suction/injection velocity : Cylindrical vertical coordinate z Greek symbols

        

: Relative temperature difference parameter : Density of the fluid



: Target momentum accommodation coefficient



: Kinematic viscosity of the ambient fluid : Time dependent length scale : Unsteadiness parameter : Mean free path : Tangential coordinate

  *



r t   

: Density of the ambient fluid : Electrical conductivity : Coefficient of dynamic viscosity : Dynamic viscosity of the ambient fluid : Thermal conductivity : Thermal conductivity of the ambient fluid : Dimensionless similarity variable

: Radial shear stress : Tangential shear stress : Dimensionless temperature : Angular velocity : Slip parameter

References Alam, M. S., Hossain, S. M. C., and Rahman, M. M. (2014). Effects of Temperature Dependent Fluid Properties and Variable Prandtl Number on the Transient Convective Flow Due to a Porous Rotating Disk. Meccanica, 49, 2439-2451. http://dx.doi.org/10.1007/s11012-014-9995-9 Alam, M. S., Rahman, M. M., and Samad, M. A. (2006). Numerical Study of the Combined Free-Forced Convection and Mass Transfer Flow Past a Vertical Porous Plate in a Porous Medium with Heat Generation and Thermal Diffusion. Nonlinear Analysis: Modeling and Control, 11, 331-343. 186

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189 Arikoglu, A., and Ozkol, I. (2006). On the MHD and Slip Flow over a Rotating Disk with Heat Transfer. International Journal of Numerical Methods for Heat and Fluid Flow, 28, 172-184. http://dx.doi.org/10.1108/09615530610644253 Attia, H. A. (1998). Unsteady MHD Flow near a Rotating Porous Disk with Uniform Suction or Injection. Fluid Dynamics Research, 23, 283-290. http://dx.doi.org/10.1016/S0169-5983(98)80011-7 Attia, H. A. (2006). Unsteady Flow and Heat Transfer of Viscous Incompressible Fluid with Temperature dependent Viscosity Due to a Rotating Disk in a Porous Medium. Journal of Physics A. Mathematics General, 39, 979-991. http://dx.doi.org/10.1088/0305-4470/39/4/017 Benton, E. R. (1966). On the Flow Due to a Rotating Disk. Fluid Mechanics, 24(4), 781–800. http://dx.doi.org/10.1017/S0022112066001009 Cochran, W. G. (1934). The Flow Due to a Rotating Disk. Proceedings of the Cambridge Philosophical Society, 30(3), 365–375. http://dx.doi.org/10.1017/S0305004100012561 Daniel, Y. S. (2015). Presence of Heat Generation/Absorption on Boundary Layer Slip Flow of Nanofluid Over A Porous Stretching Sheet. American Journal of Heat and Mass Transfer, 2(1), 15-30. http://dx.doi.org/10.7726/ajhmt.2015.1002 El-Mistikawy, T. M. A., and Attia, H. A. (1990). The Rotating Disk Flow in the Presence of Strong Magnetic Field, in: Proc. Third int. Congress of Fluid Mechanics, Cairo, Egypt, 2-4 January, 3, 1211-1222. El-Mistikawy, T. M. A., Attia, H. A., and Megahed, A. A. (1991). The Rotating Disk Flow in the Presence of Weak Magnetic Field, in: Proceeding of the Fourth Conference on Theoretical and Applied Mechanics, Cairo, Egypt, 5-7 November, 69-82. Hassan, A. L. A., and Attia H. A. (1997). Flow Due to a Rotating Disk with Hall Effect, Physics Letters A, 228, 246-290. Herwig, H., and Klemp, K. (1988). Variable Property Effects of Fully Developed Laminar Flow In Concentric Annuli. ASME Journal of Heat Transfer, 110, 314-320. http://dx.doi.org/10.1115/1.3250486 Jayaraj, S. (1995). Thermophoresis in Laminar Flow over Cold Inclined Plates with Variable Properties. Heat and Mass Transfer, 40, 167-174. http://dx.doi.org/10.1007/BF01476526 Kelson, N., and Desseaux, A. (2000). Note on Porous Rotating Disk Flow. ANZIAM Journal, 42 (E), C837C855. Lin, I.-H. and Hsiao, K. L. (2015). Food Extrusion Energy Conversion Conjugate Ohmic Heat And Mass Transfer For Stagnation Non-Newtonian Fluid Flow with Physical Multimedia Features. American Journal of Heat and Mass Transfer, 2(3), 127-145. http://dx.doi.org/10.7726/ajhmt.2015.1009 Maleque, Kh. A., and Sattar, M. A. (2002).The Effects of Hall Current and Variable Viscosity on an Unsteady MHD Laminar Convective Flow Due to a Rotating Disc. Journal of Energy Heat and Mass Transfer, 24, 335-348. Maleque, Kh. A., and Sattar, M. A. (2005). Steady Laminar Convective Flow with Variable Properties Due to a Porous Rotating Disk. ASME Journal of Heat Transfer, 127, 1406-1409. http://dx.doi.org/10.1115/1.2098860 Maleque, Kh. A., and Sattar, M. A. (2005). The Effects of Variable Fluid Properties and Hall Current on the Steady MHD Laminar Convective Flow Due to a Porous Rotating Disk. International Journal of Heat and Mass Transfer, 48, 4963-4972. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2005.05.017 Miklavcic, M. and Wang, C.Y. (2004). The flow due to a rough rotating disk. Z. Angew. Math. Phys., 55, 235187

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189 246. http://dx.doi.org/10.1007/s00033-003-2096-6 Nachtsheim, P. R., and Swigert, P. (1965). Satisfaction of the Asymptotic Boundary Conditions in Numerical Solution of the System of Non-linear Equations of Boundary Layer Type, NASA TN-D3004. Osalusi, E., and Sibanda, P. (2006). On Variable Laminar Convective Flow Properties Due to a Porous Rotating Disk in a Magnetic Field. Romanian Journal of Physics, 15, 933-944. Osalusi, E., Side, J., and Harris, R. (2008). Thermal-Diffusion and Diffusion-Thermo Effects on Combined Heat and Mass Transfer of a Steady MHD Convective and Slip Flow Due to a Rotating Disk With Viscous Dissipation and Ohmic Heating. International Communications in Heat and Mass Transfer, 35, 908-915. http://dx.doi.org/10.1016/j.icheatmasstransfer.2008.04.011 Pantokratoras, A. (2005). Forced and Mixed Convection Boundary Layer Flow Along a Flat Plate with Variable Viscosity and Variable Prandtl Number, New Results, Heat and Mass Transfer, 41, 1085-1094. http://dx.doi.org/10.1007/s00231-005-0627-8 Pantokratoras, A. (2007). Non-Darcian Forced Convection Heat Transfer over a Flat Plate in a Porous Medium with Variable Viscosity and Variable Prandtl Number. Journal of Porous Media, 10, 201-208. http://dx.doi.org/10.1615/JPorMedia.v10.i2.70 Rahman, ATM. M., Alam, M. S., and Chowdhury, M. K. (2012). Thermophoresis Particle Deposition on Unsteady two-Dimensional Forced Convective Heat and Mass Transfer Flow along a Wedge with Variable Viscosity and Variable Prandtl Number. International Communications in Heat and Mass Transfer, 39, 541-550. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.02.001 Rahman, M. M. (2010). Convective Hydromagnetic Slip Flow with Variable Properties Due to a Porous Rotating Disk. Sultan Qaboos University Journal for Science, 15, 55-79. Rahman, M. M. and Postelnicu, A. (2010). Effects of Thermophoresis on the Forced Convective Laminar Flow of a Viscous Incompressible Fluid over a Rotating Disk. Mechanics Research Communication, 37, 598603. http://dx.doi.org/10.1016/j.mechrescom.2010.07.002 Rahman, M. M., and Eltayeb, I. A. (2011). Convective Slip Flow of Rarefied Fluids Over a Wedge with Thermal Jump and Variable Transport Properties. International Journal of Thermal Science,50, 468-379. http://dx.doi.org/10.1016/j.ijthermalsci.2010.10.020 Rahman, M. M., and Salauddin, K. M. (2010). Study of Hydromagnetic Heat and Mass Transfer Flow over an Inclined Heated Surface with Variable Viscosity and Electric Conductivity. Communication in Nonlinear Science and Numerical Simulation, 15, 2073-2085. http://dx.doi.org/10.1016/j.cnsns.2009.08.012 Rahman, M. M., and Sattar, M. A. (1999). MHD Free Convection and Mass Transfer Flow with Oscillatory Plate Velocity in a Rotating Frame of Reference. Dhaka University Journal of Science, 47(1), 63-73. Rahman, M. M., Aziz, A., and Al-Lawatia, M. (2010). Heat Transfer in Micropolar Fluid along a Inclined Permeable Plate with Variable Fluid Properties. International Journal of Thermal Science, 49, 9931002. http://dx.doi.org/10.1016/j.ijthermalsci.2010.01.002 Rahman, M. M., Rahman, M. A., Samad, M. A., and Alam, M. S. (2009). Heat Transfer in Micropolar Fluid along a Non-linear Stretching Sheet with Temperature Dependent Viscosity and Variable Wall Temperature. International Journal of Thermophysics, 30, 1649-1670. http://dx.doi.org/10.1007/s10765-009-0656-5 Sattar, M. A., and Hossain, M. M. (1992). Unsteady Hydromagnetic Free Convection Flow with Hall Current and Mass Transfer along an Accelerated Porous Plate with Time Dependent Temperature and Concentration. Canadian Journal of Physics, 70, 369-374. http://dx.doi.org/10.1139/p92-061 188

M. S. Alam, Nayan K. Poddar, M. M. Rahman, and K. Vajravelu / American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 3 pp. 165-189 Sparrow, E.M., Beavers, G.S. and Hung, L.Y. (1971). Flow about a Porous-Surface Rotating Disk. International Journal of Heat and Mass Transfer, 14, 993-996. http://dx.doi.org/10.1016/0017-9310(71)90126-8 von Karman, T. (1921). Uber laminare und turbulente reibung. ZAMM, 1(4), 233–235. http://dx.doi.org/10.1002/zamm.19210010401 Zakerullah, M. and Ackroyd, J. A. D. (1979). Laminar Natural Convection Boundary Layers on Horizontal Circular Discs. Journal of Applied Mathematics and Physics, 30,427- 435. http://dx.doi.org/10.1007/BF01588887 Zueco, J. and Rubio, V. (2012). Network Method to Study Magnetohydrodynamic Flow and Heat Transfer about Rotating Disk. Engineering Applications in Computational Fluid Dynamics, 6(3), 336-345. http://dx.doi.org/10.1080/19942060.2012.11015425

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