Impact of thermal radiation on MHD slip flow of a ferrofluid over a non-isothermal wedge

Author’s Accepted Manuscript Impact Of Thermal Radiation On MHD Slip Flow Of A Ferrofluid Over A non-isothermal Wedge A.M. Rashad

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To appear in: Journal of Magnetism and Magnetic Materials Received date: 18 May 2016 Accepted date: 8 August 2016 Cite this article as: A.M. Rashad, Impact Of Thermal Radiation On MHD Slip Flow Of A Ferrofluid Over A non-isothermal Wedge, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.08.056 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Impact Of Thermal Radiation On MHD Slip Flow Of A Ferrofluid Over A non-isothermal Wedge

A.M. Rashad Department of Mathematics, Aswan University, Faculty of Science, Aswan, 81528, Egypt

Abstract This article is concerned with the problem of magnetohydrodynamic (MHD) mixed convection flow of Cobalt-kerosene ferrofluid adjacent a non-isothermal wedge under the influence of thermal radiation and partial slip. Such type of problems are posed by electric generators and biomedical enforcement. The governing equations are solved using the Thomas algorithm with finite-difference type and solutions for a wide range of magnet parameter are presented. It is found that local Nusselt number manifests a considerable diminishing for magnetic parameter and magnifies intensively in case of slip factor, thermal radiation and surface temperature parameters. Further, the skin friction coefficient visualizes a sufficient enhancement for the parameters thermal radiation, surface temperature and magnetic field, but a huge reduction is recorded by promoting the slip factor. Key word: ferrofluid; thermal radiation; velocity slip; mixed convection; MHD.

1. Introduction Magnetohydrodynamics (MHD) flow has a considerable impact in the field of medicine, which is accounted in cancer tumour treatment causing hypothermia, reducing bleeding in case of severe injuries, magnetic resonance imaging and several other diagnostic tests [1,2]. Likewise, the electromagnetic forces not only influence the mechanics of the system but also affect the thermodynamics of the system through thermal radiation. In fact, the consideration of thermal radiation when analyzing the MHD flow and convective heat transfer have great importance in electrical power generation, geothermal extractions, astrophysical flows, solar power technology, space vehicle re-entry and other industrial fields. The coupling of MHD convection flows and radiation is quite complicated and becomes highly nonlinear. various investigators, for instance, [3-9], attempted to examine thermal radiation effects on MHD free and mixed convection flows of viscous fluid past or along heated bodies of different shapes under different boundary conditions. The analysis of nanofluids have received a prominent attention because of their tremendous spectrum of applications including sterilization of medical suspensions, nano1

material processing, automotive coolants, microbial fuel cell technology, polymer coating, intelligent building design, microfluid delivery devices and aerospace tribology. The term nanofluid, first coined by Choi [10], refers to a liquid containing a dispersion of submicron solid particles (nanoparticles) having higher thermal conductivity in a base fluid. It is noticeable that these nanoparticles are taken ultrarefine (i.e. length of order 1-50nm), thus nanofluids seem to conduct more like a single-phase fluid than a solid-liquid suspension. The nanoparticles utilized in nanofluids are usually made of chemically stable metals, oxides, carbides, nitrides, or non-metals, and the base fluid is generally a conductive fluid, such as water, oil (and other lubricants), ethylene glycol (or other coolants), bio-fluids, polymer solutions and other common fluids. Because of the enhanced heat transfer characteristics and useful applications, numerous investigations has been made on nanofluid under various physical circumstances. A comprehensive monographs of the works of to nanofluids along with their different applications is found in the following literatures in [11-18]. To be more particular the magnetic nanofluids (Ferrofluids) which are colloidal suspensions of magnetic nanoparticles like cobalt, magnetite and ferrite scattered in a nonconducting liquids such as kerosene, water, heptane and hydro-carbons. Ferrofluids are pretty helpful in diverse engineering applications such as for example in intelligent biomaterials for wound treatment, medicine drug targeting, megaphones, and bumpers and revolving exception seals. A reliable studies to this fascinating topic along with the theoretical patterns or experimental data is well authenticated in the literature [19-22]. The objective of this study is to determine the influence of partial slip on MHD boundary layer flow from a non-isothermal radiate wedge immersed in the radiating ferrofluid. The simultaneous effect of wide range of MHD with thermal radiation and partial slip for ferrofluid (Cobalt-kerosene) is not investigated in the literature. It would be therefore interesting to examine the effects of strong hydro-magnetic slip on the ferrofluid. The nonlinear equations obtained for stream function formulations are simulated through finitedifference type and results are expressed graphically for skin friction coefficient, Nusselt number and distributions of velocity and temperature. 2. Flow Analysis Consider a wedge surface having an angle Ω with a variable wall temperature Tw(x) immersed in a Cobalt-kerosene ferrofluid with temperature T∞, and uniform velocity U∞(x) driven by the pressure gradient of the corresponding potential flow. As exhibited in Fig. 1, the x-coordinate is taken to be aligned with the flow on wedge surface and the y-coordinate normal to it. The flow is assumed to be laminar and a ferrofluid is chosen to be Newtonian, 2

incompressible and electrically conducting with uniform properties and thermal equilibrium between the kerosene base fluid and the Cobalt ferromagnetic particles are assumed. The magnitude of magnetic field B0 is applied perpendicular to the flow trend, whilst the magnetic Reynolds number is taken small and the induced magnetic field is negligible compared to the applied magnetic field. Consequently, the induced magnetic field is not appreciated for small magnetic Reynolds number. It is assumed that the role of radiation heat flux on x-orientation is negligible in comparison to the flux in the y-orientation. This radiation flux permeates the wedge surface and is absorbed near the ferrofluid of absorption coefficient. Concerning about calefactory of the ferrofluid absorption and the wedge surface through thermal radiation, heat is conveyed from the surface to the surroundings. Thermophysical properties of the ferrofluid are presented in Table 1. Considering the known modeling of Tiwari and Das [11], thus the fundamental equations under the usual Boussinesq approximation for steady MHD flow are(see [23-24]:

u v  0, x y

(1)

u

 dU  u u 1   2u v   U    g (  ) ff (T  T )sin  / 2   ff B02 U   u   , (2) ff  ff  2 x y  ff  dx y 

u

qr T T  2T 1 v   ff 2  . x y y (  C p ) ff y

(3)

The appropriate boundary conditions of the investigation are:

u ( x, 0)  N1 ff

u , Tw ( x)  T  Ax 2 m , y

(4a)

u( x, )  U   ax m , T ( x, )  T ,

(4b)

where x and y are the distances along and normal to the surface, respectively; u and v are the fluid velocity along and normal to the surface, respectively; B0, g, N1 , T are respectively the magnetic induction, acceleration due to gravity, velocity slip coefficient and ferrofluid temperature; U=axm is the potential flow velocity outside the boundary layer, m   / (2   ) , and β is the Hartree pressure gradient parameter which corresponds to    /  for a total angle of the wedge; a, A and m are constants; ff is the thermal expansion coefficient of the ferrofluid; ρff is the ferrofluid density, μff is the dynamic viscosity of the ferrofluid and αff is the thermal diffusivity of the ferrofluid, (ρCp)ff is the heat capacitance of ferrofluid, ff is the electrical conductivity the ferrofluid which are given by [11,25];

3

 ff  (1   )  f  s ,  ff 

   ff

k ff f ,  ff  ,  C p   (1   )  C p     C p  , 2.5 ff f s (1   )  C p  ff

 (1   )    f     s ,

k ff kf



 k  2k   2  k  k  ,  k  2k     k  k  s

f

s

f

f

f

s

s

3  s /  f  1   ff  1 , f  s /  f  2   s /  f  1

(5)

Here subscripts "ff", "f" and "s" stand for the properties of the ferrofluid, base fluid (kerosene) and ferromagnetic particle (cobalt) respectively.  is the solid volume fraction, kff is the thermal conductivity of ferrofluid. The radiative flux in y-orientation is conserved via Rosseland's approximation is defined as [26];

qr  

4 1 T 4 16 1T 3 T  , 3 R y 3 R y

(6)

with σ1 is the Stefan-Boltzmann constant, βR is the absorption coefficient. Introducing the following non-dimensional quantities; 1/2

y U x        ,    f B02 x /  f U  , f ( , )  ,  ( , )  T  T  / Tw  T  , 1/2 x   f  U x    f   Grx / Re2x , Grx  g  f (Tw  T ) x3 /  2f , Re x  U  x /  f , Rd  41T3 /  R k f , H  Tw / T ,

  N1   f  f B02  , Pr  (C p ) f / k f . 1/2

(7)

In view of the Eqn. (7), with the assistance of the defining stream function u   / y and

v   / x , the basic field of Eqs. (1)-(4) with Eqs. (5)-(6) can be expressed in nondimensional form as;

1 f  

 nf 1 m ff   m 1  f 2    1  f     2 sin  / 2 2  f (1     (  s /  f ))  f  f   (1  m)  f   f      

4

(8)





3 knf  4 Rd   1 m f  3     3    ( H  1)  1  2mf   f    (1  m)  f      , (9) Pr k f Pr 3 2    

f ( ,0)  0 , f ( , 0) 

  1/2 f ( , 0) ,  ( ,0)  1 , (1   )2.5

(10a)

f ( , )  1 ,  ( , )  0 .

(10b)

In Eqs. (7)-(9), a prime denotes partial differentiation with respect to , and the parameters 1, 2 and 3 are given by; 1 ( )  3 ( ) 

1      (  ) s / (  ) f  1  ,  (  )  , 2 2.5 1     (s /  f ) (1   ) [1     (  s /  f )] 1

1      (  C p ) s / (  C p ) f    

.

(11)

In the above equation parameters; , , , Grx, Rex, Rd, H, Rd are respectively the mixed convection parameter, velocity slip parameter, magnet interaction parameter, local Grashof number, local Reynolds number, thermal radiation parameter, surface temperature parameter. The quantities of the physical interest are the local skin friction coefficient and local Nusselt number which are an important parameters commonly used in fluid mechanics. The non-dimensional forms of these quantities are defined as;

1 1 C f Re1/2 f ( , 0) , x  2 (1   )2.5 Nux Re

1/2 x

(12)

 knf 4 H 3     Rd   ( , 0) , k  3  f 

(13)

It is noteworthy to mention that by substituting Rd=0, =0, =0 and =0 in Eqs. (8)-(10), the problem is reduced to the MHD forced convection flow of regular fluid over a nonisothermal wedge which is discussed previously in [23-24]. 3. Solution Methodology To get the solution valid over the entire regime for a wide range of magnet parameter 0 < <10, Eqs. (14)-(17) are solved numerically via implicit finite difference method along with the Thomas algorithm [27].The computational domain 0 ≤  ≤ 35, 0 ≤ ≤ 10 is divided into nodes with step sizes 1=0.001 and 1= 0.01 for boundary layer and magnet range, respectively. Convergence of the scheme is assumed when anyone of quantities f' and  their gradients for last two approximations differ from unity by less than 10-6 for all values of  and

5

. Computations are repeated until both the boundary layer and the dimensionless time are reached to the ambient conditions and maximum value, respectively. However, the comparison with already published data is also made to ensure the accuracy of results. 4. Results and Discussion The present model investigated the influence of strong MHD on mixed convection flow of ferrofluid in presence of partial slip and radiation effects. The mathematical formulation in terms of stream function formulations is given in Eqs. (14)-(17) which is solved numerically by using implicit finite difference scheme. The simulated results are presented to visualize the influence of the mixed convection parameter , slip factor , solid volume fraction of ferrofluid , magnet parameter , and radiation parameter Rd, and surface temperature parameter H on the velocity and temperature distributions, local skin friction coefficient and Nusselt number. Figs. 2-4 visualize the impact of the mixed convection parameter  for the buoyancy aiding/opposing flow on the profiles of the ferrofluid velocity f ( , ) , temperature  ( , ) , local skin-friction coefficient 0.5C f Re1/2 and local Nusselt number Nux Rex1/2 , respectively. x It is noteworthy to mention from Figs. 2(a)-2(b) that a rise in mixed convection parameter  promotes the thermal buoyancy impact, which result in more induced flow along the wedge surface. This tends to a considerable boost in the velocity which is achieved at the expense of damp the ferrofluid temperature and weak decrease in the thermal boundary layer thickness. Moreover, the characteristic summits in the velocity distributions move toward the surface by rising . From Figs. 3-4, it is noteworthy to mention that an increment in  has a tendency to enhance both the local skin-friction coefficient and Nusselt number. Such behavior is expected because the evolution in  implies to promote the flow velocity and to damp its temperature, which results in increases in the velocity and temperature gradients at the 1/2 surface. This yields an enhancement in the 0.5C f Re1/2 . Moreover, it is x and Nu x Re x

interesting to note in Figs. 3-4 that the skin-friction coefficient increases considerably by intensifying the magnetic field parameter . This happens because large values of  are responsible to magnify the Lorentz force within boundary layer region which reverses the flow in the opposite direction. As a result Nusselt number diminishes. Figs. 5-10 reveal the impacts of the solid volume fraction of ferrofluid  and slip factor  upon variation velocity and temperature behaviors along with skin-friction coefficient and Nusselt number, respectively. From Figs. 5(a)-5(b), it is manifested that the evolution in the

6

solid volume fraction  tends to promote the velocity near the wedge surface (opposite trend determined far away the surface) and to damp the ferrofluid temperature. This is evident from fact that large values of  correspond to growing the thermal conductivity of ferrofluid (see Table 1) which motivates the heat diffusion so that the heat precipitately diffuse near the wedge surface. From Figs. 6-7, it is depicted that each of skin-friction coefficient and Nusselt number enlarges by amplifying . This happens because large values of  produce a highenergy transmit through the flow related with the irregular movement of the ultrafine particles, and hence produces a considerable enhancement in the shear stress and heat transfer rate. It is apparent form Figs. 8(a)-8(b) that the encourage in slip factor δ results in a prominent increase in the velocity distribution, whilst the trend is reversed for the ferrofluid temperature. This is because the elevation of velocity slip parameter δ has a predilection to accelerate the motion, consequently, the temperature and thermal boundary layer thickness diminishes. Furthermore, It is noteworthy to mention from Figs. 9-10 that the skin-friction coefficient reduces and Nusselt number increases sufficiently by elevating the slip factor δ. Impacts of thermal radiation parameter Rd and surface temperature parameter H on the velocity, temperature distributions and physical quantities 0.5C f Re1/2 and Nux Rex1/2 are x disclosed in Figs 11-13. From Figs. 11(a)-11(b), it is manifested that an increment in the radiation parameter Rd and surface temperature parameter H leads to a considerable elevation in the velocity and temperature distributions with sufficient increase in thermal boundary layer thickness. The reason for this pattern is that, the presence of thermal radiation implies an enormous enlarging in the radiative heat which promotes the thermal state of the ferrofluid creating its temperature to enhance. Moreover, from Figs. 12- 13, it is noteworthy to mention that, the enhancement in Rd and H causes an evolution in the skin friction coefficient and Nusselt number. This corresponds with the physical behaviors that, as Clearfield from Eq. (13), the heat transfer is very higher with the existence of the radiation impact, and hence the shear stress evolves. Moreover, lager value of H implies larger values of wall temperature or lower value of T∞, which in turns that physical quantities 0.5C f Re1/2 and Nux Rex1/2 enhance x as the surface temperature parameter H increases. 5. Conclusions This study is ensued to discuss the impact of the thermal radiation on MHD mixed convection flow of ferrofluid in presence of and partial slip. The governing equations are solved by implicit numerical scheme of finite-difference type. The comparison with already published data is also made to ensure the accuracy of results. The impacts of various

7

significant parameters on flow and heat transfer are accomplished through graphs. From the computational data, it has been concluded that the skin friction coefficient and local Nusselt number enhances sufficiently by promoting the solid volume fraction and mixed convection parameter. Besides, the skin friction coefficient visualizes a consistent behavior for all selected values of parameters thermal radiation, surface temperature and magnetic field, but as expected a huge reduction is recorded by promoting the slip factor. In addition, the Nusselt number shows a considerable diminishing for magnetic parameter and magnifies intensively in case of thermal radiation and surface temperature parameters. References [1] Pai SI. Magnetogasdynamics and plasma dynamics. Berlin: Springer; 1962. [2] Liuta I, Larachi F. Magnetohydrodynamics of trickle bed reactors: mechanistic model, experimental validation and simulations. Chem Eng Sci 2003;58:297-307. [3] A.M. Rashad, Influence of radiation on MHD free convection from a vertical flat plate embedded in porous media with thermophoretic deposition of particles, Commun. Nonlinear Sci. Numer. Simulat., 13 (2008) 2213-2222. [4] Hayat, T., Shafiq, A., Alsaedi, A., and Awais, M. MHD axisymmetric flow of third grade fluid between stretching sheets with heat transfer. Computers & Fluids, 86, 103-108 (2013). [5] Afify, A.A., Uddin, M. J., and Ferdows, M. Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects. Appl. Math. Mech. -Engl. Ed., 35(11), 1375-1386 (2014). [6] M.M. Nejad, K. Javaherdeh , M. Moslemi, MHD mixed convection flow of power law non-Newtonian fluids over an isothermal vertical wavy plate, Journal of Magnetism and Magnetic Materials, Volume 389, 1 September 2015, Pages 66–72 [7] Hayat, T., Shafiq, A., Alsaedi, A., and Shahzad, S. A. Unsteady MHD flow over exponentially stretching sheet with slip conditions. Appl. Math. Mech. -Engl. Ed., 37(2), 193-208 (2016) [8] M. Khan, Hashim, M. Hussain, M. Azam, Magnetohydrodynamic flow of Carreau fluid over a convectively heated surface in the presence of non-linear radiation, Journal of Magnetism and Magnetic Materials, 412(15) 2016, 63-68. [9] M.H.M. Yasin, A. Ishak, I. Pop, MHD heat and mass transfer flow over a permeable stretching/shrinking sheet with radiation effect, Journal of Magnetism and Magnetic Materials, Volume 407, 1 June 2016, Pages 235-240.

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[10] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticle. in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows, ASME FED, vol. 231/MD 66 (1995) 99-105. [11] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J Heat Mass Tran 2007, 50:2002-2018. [12] F.M. Hady, F.S. Ibrahim and S.M. Abdel-Gaied, Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet. Nanoscale Res Lett 2012; 7: 229-236. [13] A.J. Chamkha, S. Abbasbandy, A.M. Rashad, K. Vajravelu, Radiation effects on mixed convection over a wedge embedded in a porous medium filled with a nanofluid, Transport in Porous Medium, 91 (2012) 261-279. [14] O.A. Bég and M. Ferdows, Explicit numerical simulation of magnetohydrodynamic nanofluid flow from an exponential stretching sheet in porous media. Appl Nanosci 2013; 1-15. DOI: 10.1007/s13204-013-0275-0. [15] S.M.M. EL-Kabeir, M. Modather, A.M. Rashad, Effect of thermal radiation on mixed convection flow of a nanofluid about a solid sphere in a saturated porous medium under convective boundary condition, Journal of Porous Media, 18(6) (2015) 569-584. [16] T. Hayat, M. Imtiaz, A. Alsaedi, M.A. Kutbi, MHD three-dimensional flow of nanofluid with velocity slip and nonlinear thermal radiation, Journal of Magnetism and Magnetic Materials, Volume 396, 15 December 2015, Pages 31–37. [17] T. Hayat, T. Muhammad, A. Alsaedi, M.S. Alhuthali, Magnetohydrodynamic threedimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation, Journal of Magnetism and Magnetic Materials, Volume 385, 1 July 2015, Pages 222-229. [18] A.K. Abdul Hakeem, N. Vishnu Ganesh, B. Ganga, Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect, Journal of Magnetism and Magnetic Materials, Volume 381, 1 May 2015, Pages 243-257. [19] B.M. Berkovsky, V.F. Medvdev, M.S. Krakov, Magnetic fluids, engineering applications, (1973) Oxford University Press, Oxford [20] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, London, (1985). [21] M.S. Kandelousi and R. Ellahi, Simulation of ferrofluid flow for magnetic drug targeting, Z. Naturforsch, 1-10, 2015.

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[22] A. Zeeshan, R. Ellahi and M. Hassan, Magnetohydrodynamic flow of water/ethylene glycol based nanofluids with natural convection through a porous medium, Eur. Phys. J. Plus, 129: 261, 2014. [23] K.A. Yih, "MHD forced convection flow adjacent to a non-isothermal wedge", Int. Comm. HeatMass Transfer, Vol. 26, No. 6, pp. 819-827, 1999. [24] A.J. Chamkha, M.A. Quadri, C. Issa, Thermal radiation effects on MHD forced convection flow adjacent to a non-isothermal wedge in the presence of a heat source or sink, Heat Mass Transfer 39 (2003) 305-312. [25] J. Maxwell, A Treatise on Electricity and Magnetism. Second ed., Oxford University Press, Cambridge, UK (1904). [26] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, McGraw-Hill, New York, 1972. [27] F. G. Blottner, Finite-difference methods of solution of the boundary-layer equations, AIAA Journal 8 (1970) 193-205. [28] M. Sheikholeslami, Mofid Gorji-Bandpy, Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field, Power Technol. 256 (2014) 490-498. Table 1. Thermophysical properties of kerosene and cobalt solids [28]. Property

kerosene

cobalt

 (kg m-3)

780

8900

Cp (Jkg-1 K-1)

2090

420

k (W m-1 K-1)

0.149

100

 (K-1)

9.9× 10-4

1.3 × 10-5

 (Simens/m)

6× 10-10

1.602× 107

µ (kg-1 m-1 s-1)

0.00164

-

Table 2. Comparison of the values of the local skin-friction coefficient f (0,0) for various values of m at Rd=0, =0, =0 and =0. m -0.05 0.0 1/3 1.0

Yih [23] 0.213484 0.332057 0.757448 1.232588

Chamkha et al. [24] 0.213802 0.332206 0.757586 1.232710

10

Present results 0.213802 0.332206 0.757586 1.232710

Table 3. Comparison of the values of the local Nusselt number  (0,0) for various values of Pr at Rd=0, =0, =0 and =0. Pr 0.01 0.1 1 10 100

Yih [23] 0.051589 0.140034 0.332057 0.728141 1.571831

Chamkha et al. [24] 0.051830 0.142003 0.332173 0.728310 1.572180

v

Present results 0.051830 0.142003 0.332173 0.728310 1.572180

B0

y u U=ax

m

x

g 

Boundary Layer

Figure 1. Physical model and coordinate system

11

1.2

(a)

1.0

(b) =0.4 =0.1

0.8

0.8

0.6

m=1/3 Rd=5.0 H=1.1 =1.0 =-1,0,1,3,5

f'



1.0

=-1,0,1,3,5

0.6

0.4 =0.4 =0.1

0.4

0.2 0



1

2

m=1/3 0.2 Rd=5.0 H=1.1 =1.0 0.0 3 0.0

0.5

1.0 

1.5

2.0

2.5

Fig. 2: Impact of  on the velocity and temperature distributions.

6

=0.4,=0.1,m=1/3,Rd=5.0,H=1.1 5

0.5CfRex

1/2

4

3

2

1

=-1,0,1,3,5

0 0

2

4

6

8



Fig. 3: Impact of  on the skin friction coefficient.

12

10

20

=0.4,=0.1,m=1/3,Rd=5.0,H=1.1 =-1,0,1,3,5

NuxRex

-1/2

18

16

14

12

10 0

2

4

6

8

10



Fig. 4: Impact of  on the Nusselt number.

1.2

(a)

1.0

(b) =5.0 =0.4

m=1/3 Rd=5.0 H=1.1 =1.0

1.1 0.8 1.0 0.6

=0.05,0.1,0.15,0.2

f'



0.9 =0.05,0.1,0.15,0.2

0.8

0.4 =5.0 =0.4

0.7

0.6

0.5 0

1



2

m=1/3 0.2 Rd=5.0 H=1.1 =1.0 0.0 3 0.0

0.5

1.0 

1.5

2.0

Fig. 5: Impact of  on the velocity and temperature distributions.

13

2.5

7 6

0.5CfRex

1/2

5 4 3

=0.05,0.1,0.15,0.2

2 1

=5.0,=0.4,m=1/3,Rd=5.0,H=1.1 0 0

2

4

6

8

10



Fig. 6: Impact of  on the skin friction coefficient.

20

=5.0,=0.4,m=1/3,Rd=5.0,H=1.1

19

=0.05,0.1,0.15,0.2

18

NuxRex

-1/2

17 16 15 14 13 12 11 0

2

4

6



Fig. 7: Impact of  on the Nusselt number.

14

8

10

1.2

(a)

1.0

1.0

(b) =5.0 =0.1

m=1/3 Rd=5.0 H=1.1 =1.0

0.8

0.8

=0.0,0.2,0.4,0.6,0.8,1.0



f'

0.6 0.6 =0.0,0.2,0.4,0.6,0.8,1.0

0.4 0.4

=5.0 =0.1

0.2

0.0 0



1

2

m=1/3 0.2 Rd=5.0 H=1.1 =1.0 0.0 3 0.0

0.5

1.0 

1.5

2.0

2.5

Fig. 8: Impact of  on the velocity and temperature distributions.

6

=0.0,0.2,0.4,0.6,0.8,1.0 5

0.5CfRex

1/2

4

3

2

1

=5.0,=0.1,m=1/3,Rd=5.0,H=1.1 0 0

2

4

6

8



Fig. 9: Impact of  on the skin friction coefficient.

15

10

20

=0.0,0.2,0.4,0.6,0.8,1.0

NuxRex

-1/2

18

16

14

12

=5.0,=0.1,m=1/3,Rd=5.0,H=1.1 10 0

2

4

6

8

10



Fig. 10: Impact of  on the Nusselt number.

1.4

(a)

1.0

(b) =5.0 =0.4 =0.1

H=1.1,1.3,1.5,1.7,1.9

1.2

0.8

1.0

0.6

H=1.1,1.3,1.5,1.7,1.9

f'



Rd=5.0

m=1/3 =1.0

Rd=0.0 0.8

0.4

=5.0 =0.4 =0.1

0.6

m=1/3 =1.0

0.4 0

1



2

0.2 Rd=5.0 Rd=0.0 0.0 3 0.0 0.5

1.0 

1.5

2.0

Fig. 11: Impacts of Rd and H on the velocity and temperature distributions.

16

2.5

6

H=1.1,1.3,1.5,1.7,1.9

5

0.5CfRex

1/2

4

3

Rd=5.0

2

Rd=0.0 1

=5.0,=0.4,=0.1,m=1/3 0 0

2

4

6

8

10



Fig. 12: Impacts of Rd and H on the skin friction coefficient

36

=5.0,=0.4,=0.1,m=1/3

H=1.1,1.3,1.5,1.7,1.9

32

NuxRex

-1/2

28

Rd=5.0

24 20 16 12 8

Rd=0.0

4 0 0

2

4

6

8



Fig. 13: Impacts of Rd and H on Nusselt number

17

10