Accepted Manuscript MHD mixed convection flow of third grade liquid subject to non-linear thermal radiation and convective condition T. Hayat, Ikram Ullah, B. Ahmed, A. Alsaedi PII: DOI: Reference:
S2211-3797(17)31125-7 http://dx.doi.org/10.1016/j.rinp.2017.07.045 RINP 818
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
26 June 2017 19 July 2017 20 July 2017
Please cite this article as: Hayat, T., Ullah, I., Ahmed, B., Alsaedi, A., MHD mixed convection flow of third grade liquid subject to non-linear thermal radiation and convective condition, Results in Physics (2017), doi: http:// dx.doi.org/10.1016/j.rinp.2017.07.045
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MHD mixed convection flow of third grade liquid subject to non-linear thermal radiation and convective condition T. Hayatd>e , Ikram Ullahd1 , B. Ahmede and A. Alsaedie d
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan e
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University P. O. Box 80203, Jeddah 21589, Saudi Arabia
Abstract: Magnetohydrodynamic (MHD) mixed convection flow of third grade fluid by an exponentially stretching sheet is addressed. Energy expression involves the convective condition and non-linear thermal radiation. Non-linear ordinary di erential systems are first constructed and then solved successfully. Interval of convergence via numerical data and plots are developed and verified. Impact of various influential variables on the dimensionless velocity and temperature are graphically analyzed. The coe!cient of skin friction and local Nusselt number are discussed numerically. Opposite response of velocity and temperature profiles have been noticed for higher mixed convection. Also temperature is an increasing function of thermal radiation.
Keywords: Third grade fluid; Thermal radiation; Mixed convection; Convective boundary condition.
1
Introduction
Mixed convection flows with thermal radiation arise in many practical situations when heat transfer by only natural convection is insu!cient. The combined feature of non-linear thermal radiation and mixed convection is of great significance in organs of human physiology like brain, liver and heart. Mixed convection and radiation maintained the skin vasodilatation and sweating in changing climate situation. When the temperature of surrounding is higher then that of the skin, the body gains heat by radiation and conduction to put body at normal level whereas skin temperature is higher then that of surroundings, the body can release heat by conduction and radiation. Some other involvements of mixed convection are in ocean, nuclear reactor technology, flow movement in the atmosphere, some aspects of electronic cooling etc. Few study dealing with mixed convection can be consulted via refs. [1 7]= 1
Corresponding author. Tel.: +92 51 90642172.
email address:
[email protected]
1
The non-Newtonian fluids such as polymers, cosmetic products, paints, suspension fluids, shampoos, colloidal fluids, mud, ice cream, blood at low shear rate etc important due their wide uses in engineering and industrial processes. The non-Newtonian fluids are not easy to deal when compared with Newtonian fluids. It is due to the diverse behavior of nonNewtonian liquids that cannot be tackled through the simple Navier-Stokes equations. In case of non-Newtonian liquids the shear stresses and shear rate are linked non-linearly . Therefore researchers have suggested many models of non-Newtonian fluids in the literature [8 15]. In general these fluids are classified into integral, di erential and rate types. Third grade fluid model [16 26] is a subclass of di erential type non-Newtonian liquids which describes the shear thinning and thickening e ects. The dynamics of boundary layer flow induced by a stretching surface with heat transfer is attractive topic. Its significant is due to its involvement in food processing, cooling of large metallic plat in a bath, glass fiber production, cooper materials, heat conduction in tissues, heat pump, refrigeration, manufacturing of rubber and plastic sheets wire drawing and many others. Crane [27] initially explore boundary layer flow due to a stretching sheet. Bhattacharyya [28] studied numerically the heat transfer boundary layer flow induced by an exponentially stretching surface. Shooting method is utilized. Mukhopadhyay [29] analyzed the slip e ects on unsteady mixed convective flow and heat transfer bounded by a stretched surface. Hydromagnetic flow and heat transfer for viscoelastic fluid is investigated by Turkyilmazoglu [30]. Heat transfer and velocity slip flow past a porous shrinking surface is explored by Zheng et al. [31]= Hayat et al. [32] found the series solution for unsteady stretched flow of Je rey fluid. Mixed convection flow of viscoelastic fluid in presence of heat transfer is studied by Hayat et al. [33]= Hayat et al. [34] discussed exponentially stretched flow of second grade nanofluid with convective boundary condition. In this paper our main objective is to study the magnetohydrodynamic (MHD) mixed convection flow of third grade fluid caused by an exponentially stretching sheet. The sheet is convectively heated. Energy equation is modeled through thermal radiation and convective condition. Boundary layer approximation are utilized to construct the relevant mathematical formulation. Homotopic analysis technique [35 44] is implemented for the series solutions development. Sketch of various influential variables are presented and examined in detail. The results are demonstrated for the velocity and temperature fields together with skin friction and rate of heat transfer. 2
2
Formulation
We consider the steady 2G mixed convection flow of an incompressible third grade fluid. An exponentially stretched sheet is the main agent for liquid flow. Cartesian system is assumed in such a manner that the {axis is taken along the direction of stretched sheet {
and |axis is transverse to it. Let Xz = X0 h o denotes the velocity of stretched surface along the {direction. A non-uniform magnetic field E({) = E0 exp( 2o{ ) is implemented in the |direction (see Fig. 1). Heat transfer analysis is performed through the convective condition and non-linear thermal radiation. The governing expressions for current flow problems are
x =0
u
v g y
Fig. 1 Flow model. Cx Cy + = 0> C{ C| ¶ µ Cx C 2 x Cx C 2 x C3x C3x + +3 x +y 3 C{C| 2 C{ C| 2 C| C{C| C| µ ¶ 2 E 2 ({) 22 Cx C 2 x 63 Cx C 2 x + j (W W ) + + x> " W C| C{C| C| C| 2
(1)
Cx Cx C 2 x 1 x + = 2+ C{ C| C|
x
CW CW C2W 1 Ctu +y = p 2 = C{ C| C| ( fs ) C|
(2) (3)
The related boundary conditions are {
x = Xz = X0 h o > y = 0> n 3
CW = ki (Wi W ) at | = 0> C|
(4)
x $ 0> W $ W" > as | $ 4>
(5)
where (x> y ) show the velocity components in ({> |) directions, represents the kinematic viscosity, designates the density of the fluid, 1 > 2 and 3 are the material constants, the electrical conductivity, tu signifies the radiative heat flux, fs denotes specific heat flux, W and W" are the surface and surrounding temperatures respectively and p the thermal di usivity of fluid. Through Rosseland’s approximation the radiative heat flux tu is tu =
3 4 1 C(W 4 ) C2W 16 WW W" > = 3p C| 3pWW C| 2
(6)
in which WW shows the Stefan-Boltzman and pWW designates the coe!cient of mean absorption. Invoking Eq. (6) the energy equation can be reduced to the form µ ¶ 3 C2W CW CW CW 1 C 16 WW W" x +y = p 2 = C{ C| C| ( fs ) C| 3pWW C|
(7)
The dimensionless variables are taken in the form < ¡ X0 ¢1@2 { { 0 0 x = X0 h o i ()> y = 2o h 2o (i + i ) > @ ¡ X0 ¢1@2 { h 2o |> > W = W" + (Wi W" )()> = 2o with W = W" (1 + (z 1)) and z =
Wz = W"
(8)
Here z denotes the temperature ratio parameter
with Wz  W" > X0 the reference velocity, o the reference length and z the wall condition. Invoking above expression, the continuity Eq. (1) is now trivially verified while Eqs. (2) (5) and (7) become ³ ´ ³ 2 ´ 2 2 2 i 000 +ii 00 i 0 + 1 3i 000 i 0 i i 0000 2i i 000 9i 00 2 2i 00 + ii 000 +31 i 000 i 00 P 2 i 0 + = 0> (9)
02
02
(1 + 43 Ug) + 43 Ug[(z 1)3 (3 2 + 3 00 ) + 3(z 1)2 (2 + 2 00 )
(10)
02
+3(z 1)( + 00 ) + Pr(i0 i 0 ) = 0>
i = 0> i 0 = 1> 0 = (1 ) at = 0>
(11)
i 0 $ 0> $ 0 as $ 4=
(12)
where l (l = 1> 2) and 1 are the material parameters of order fluids, P for magnetic parameter, Ug for radiation parameter, for mixed convection parameter, Juo for Grashof number, Pr stands for Prandtl number, for Biot number and prime designates di erentiation via = The non-dimensional parameters are 2 =
2 X0 exp( {o ) > o
=
Juo > Re2o
1 = =
1 X0 exp( {o ) > o k n
q
2o , Xz
E 2 ({)o > Xz ({) j W (Wi 3W ")o3 2
P2 =
Juo =
4
Pr =
> p
3 X03 exp( {o ) > o WW 3 16 W" = 3pWW n
1 =
and Ug =
< @ >
(13)
The skin friction coe!cient is given by Fi{ = "
Cx 1 z = + C|
{| ||=0 > 1@2 Xz2
(14)
¶ µ µ ¶3 # 3 Cx C2x Cx Cx C2x +2 x +2 + C{C| C{ C| C| 2 C|
>
(15)
|=0
which in terms of the dimensionless scale is µ µ ¶ ¶ s 7 0 00 1 000 31@2 00 003 i + 1 + 1 i i i ii = Fi{ = 2 (Re) 2 2 =0
(16)
Local Nusselt number is { Qx{ = (Wi W" )
µ
CW C|
¶
1 + (tu )z = s (Re)1@2 (1 + Ug3z )0 (0)> 2 |=0
(17)
where Re designates the Reynolds number defined by Re = Xz o@=
3
Series solutions and convergence
Now to compute the solutions we utilized the homotopy analysis method. The initial guesses, operators and deformation problems are given below.
< i0 () = 1 h > @ () = h3 > > 3
0
(18)
1+
< L¯i = i 000 i 0 > @ L¯ = 00 > >
(19)
< L¯i [E1WW + E2WW h + E3WW h3 ] = 0> @ > L¯ [E4WW h + E5WW h3 ] = 0> i h ¯i [iˆ(> s˘)> ˆ(> s˘)]> (1 s˘)L¯i iˆ(> s˘) i0 () = s˘~i N h i ˆ s˘)> ˆ(> s˘)]> ¯ [i(> (1 s˘)L¯ ˆ(> s˘) 0 () = s˘~ N < iˆ(0> s˘) = 0> iˆ0 (0> s˘) = 1> iˆ0 (4> s˘) = 0> @ ˆ0 (0> s˘) = [1 ˆ(0> s˘)]> ˆ(4> s˘) = 0> >
(20)
(21) (22)
3 Ã !2 Ã !2 4 h i 3ˆ 2ˆ 3 4 4 2 ˆ C iˆ C iˆ C iˆ D C iˆ C iˆ ˆC iˆ ˆ s˜)> ˆ(> s˘) = C i + iˆC i C i ¯i i(; N + 1 C3 i 4 2 9 3 2 3 4 C C C C C C C C C2 3 Ã 4 !2 Ã !2 2ˆ 2ˆ 3ˆ 2ˆ ˆ i i i i C C 3 iˆ C C C 2 Ci D + 31 + P + ˆ> (23) 2 C3 C2 C 2 C 3 C2 C3 C 5
h i 2ˆ ˆ s˘) = (1 + 4 Ug) C ¯ ˆ(> s˘)> i(; N 3 C2 5 µ ³ ´ µ ³ ´ ¶ ¶ 2 2 3 C2ˆ 2 C2ˆ 4 Cˆ C2ˆ 3 ˆ ˆ ˆ ˆ 9 3 Ug(z 1) 3 C + C2 + 3(z 1) 2 C2 + C2 µ³ ´ ¶ +9 2 7 2ˆ C2ˆ C +3(z 1) + ˆ C2 C2 C iˆ C ˆ + Pr iˆ + Pr ˆ > C C
000 ˇp R i () = ip31 () +
p31 X n=0
1
p31 X
ip313n in0000
n=0 p31 X
2 2
ip313n in00 2 1
ip313n in000 + 31
n=0
p31 X
n=0 p31 X
n=0 p31 X
ip313n in000 00 ip313n
n=0
9 p31 X9 4 4 9 00 ˇp R 9 () = (1 + Ug) p31 () + Ug 9 3 3 n=0 7 + Pr
p31 X n=0
ip313n 0n
Pr
p31 X
p31 X
9 1
n X o=0
0 ip313n in000
n=0 p31 X
00 ip313n in00
n=0
2 2
p31 X
00 ip313n in00
n=0
00 0 in3o io000 P 2 ip31 ()>
n P
(25)
o P
00 (z n3o (30o3v 0v + 0o3v v ) v=0 o=0 n o P P 00 +3(z 1)2 p313n n3o (20o3v 0v + 0o3v v ) v=0 o=0 00 0 +3(z 1)((2p313n 0n + 0p313n n ))
1)3 0p313n
p313n in0 >
sented as follows: W ip () = ip () + E1WW + E2WW h + E3WW h3 >
(27)
p () = Wp () + E4WW h + E5WW h3 >
(28)
in which the constants ElWW (l = 1 5) through the boundary conditions are given by < W () Cip WW WW WW WW WW W E2 = E4 = 0> E3 = C |=0 > E1 = E3 ip (0)> @ W > W (0)]= E WW = 1 Cp () | 1+
C
=0
(29)
1+ p
Here s˘ represents the embedding variables and ~i and ~ are the auxiliary parameters. In
homotopic solutions, the rate of deformation and convergence region highly depends upon ~i and ~ = For such interest, the k-curves have been plotted in Figs. 2 & 3 at 16-th order of deformations which provide the admissible ranges of these auxiliary parameters. It is clear from these Figures that the suitable ranges of these parameters are [1=2> 0=2] and 6
6 : : : : : 8
(26)
n=0
W The general solutions (ip ()> p ()) consisting of special solutions (ip ()> Wp ()) are pre-
5
: : 8
(24)
0 ip313n in0 + 3 1
5
6
[1=4> 0=2]= The solutions expressions are convergent in all region of when ~i = 0=5 and ~ = 0=7= Table 1 identifies that 16th order of approximations up to 5 decimal places are essential for the convergent series solutions.
0
5G
E
E H
Tw J
O 3U
0
E
E H
Tw J
f
+ /
T +/
T + /
³f
³T
Fig. 2 : The ~ curve for i ().
Fig. 3 : The ~curve for ()=
Table 1: Convergence solutions when Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z and = 0=3 = = Order of approximations i 00 (0)
4
O 3U
f
+/
5G
0 (0)
1
1=29423 0=22145
5
1=42187 0=21101
10
1=41887 0=20981
16
1=41876 0=20975
20
1=41876 0=20975
25
1=41876 0=20975
30
1=41876 0=20975
35
1=41876 0=20975
Discussion
This portion investigates the impacts of di erent influential variables viz viscoelastic parameters 1 > 2 and 1 , magnetic parameter P> mixed convection parameter > Prandtl number Pr> radiation parameter Ug> temperature ratio parameter z and Biot number on dimensionless velocity i 0 () and temperature () fields. Theses results are sketched via graphs in Figs. (4) (16). E ects of viscoelastic parameter 1 on velocity field i 0 () is displayed 7
in Fig. 4= The velocity profile i 0 () increases for larger values of viscoelastic parameter 1 = Fig. 5 exhibits the velocity field i 0 () for various values of 1 = Here i 0 () is an increasing function of 1 = However the parameter 2 has opposite behavior on the velocity profile when correspond with 1 . Thus a reduction in the momentum boundary layer is observed from Fig. 6= Feature of magnetic parameter P on i 0 () is depicted in Fig. 7= With the increment in P the velocity field reduces close to the surface and it vanishes away from the surface. Obviously magnetic parameter corresponds to an enhancment in Lorentz forces thereby reducing the velocity profile. Fig. 8 depicts the variations in mixed convection parameter on velocity profile. Since is the ratio of buoyancy forces to viscous forces. In fact large values of mixed convection parameter correspond to decrease in viscous forces and so an enhancement in the velocity profile i 0 () is noted= Fig. 9 displays the variations of 1 on the temperature field () = Clearly the viscoelastic parameter increases the elasticity e ects due to which the liquid temperature enhances. Higher values of parameter 1 gives rise the temperature field () (see Fig. 10). It is also worth mentioning that opposite behavior of parameter 1 for velocity and temperature are observed. Fig. 11 presents the variations in magnetic parameter P on the temperature distribution (). It is established fact that the magnetic field intensity tends to produce drag force that restricts the fluid motion and heat up the fluid. As a result there a rise in temperature. Change in mixed convection parameter on () is elucidated in Fig. 12= As (buoyancy e ects) increases, the convective cooling e ects enhances and hence the temperature field () reduces. Fig. 13 is drawn to inspect the e ect of parameter 2 on (). It is found that temperature is decreasing function of parameter 2 . Fig. 14 exhibits the influence of Prandtl number on temperature field (). Larger Prandtl fluids have lower thermal conductivities so heat can spread away from the plate slower than for smaller Pr fluids. Hence rise in Prandtl number substantially decay the temperature and thickness of thermal boundary layer. Fig. 15 illustrates the behavior of Biot number on the temperature filed (). When the values of Biot number increase then rate of convective heat transfer from the hot convectional fluid below the plate surface to the fluid above the plate surface enhances. It leads to an increment in thickness of thermal boundary layer and temperature () = Temperature distribution is reduced via radiation parameter Ug (see Fig. 16). Increase in Ug supplies more heat to the liquid that shows increment in the temperature field. Characteristics of temperature ratio z on () is depicted through Fig. 17= It is declared that the temperature field is enhanced when z gets 8
higher values. Table 1 is computed for various order of approximations of i 00 (0) and 0 (0) when Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z > = 0=3 = = ~i = 0=5 and ~ = 0=7. It is inspected that 16th order of estimations are quite su!cient regarding the ¡ ¢1@2 Fi convergence of homotopic series solutions. Numerical data of surface drag force Re 2
for pertinent flow parameters including 2 > Ug> 1 > 1 > Pr> P> > z and are presented in table 2= Here skin friction increases for higher values of 1 > 2 and P whereas opposite e ects is observed for higher values of and 1 = Table 3 is prepared to understand the variations of
Nusselt number 0 (0) corresponding to distinct parameters. It reveals that Nusselt number enhances via 1 > Ug> Pr> and while it reduces for higher values of P and 2 = Table 4 indicates the comparison of some numerical values of skin friction in present study with the work of Hayat et al. [45]. All the results are found in very good agreement. 0
5G
H
E
Tw J
O 3U
0
E
E H
Tw J
O 3U
H
K
0 5G
H
E
K
Fig. 5= Behavior of i 0 () via 1 =
Fig. 4= Behavior of i 0 () via 1 = Tw J
O 3U
5G
E
E H
Tw J
O 3U
E
I +K /
I +K /
E
I +K /
I +K /
5G
M
K
Fig. 6= Behavior of i 0 () via 2
K
Fig. 7= Behavior of i 0 () via P=
9
0
5G
E
E H
Tw J
3U
0
O
T +K /
I +K /
H
O 3U
E
K
0
5G
E
K
Fig. 8= Behavior of i 0 () via = E Tw J
O 3U
Fig. 9= Behavior of () via 1 =
5G
E
E H
Tw J
O 3U
T +K /
H
M
0 5G
E
K
K
Fig. 11= Behavior of () via P=
Fig. 10= Behavior of () via 1 = E H Tw J
3U
0
5G
E
H
Tw J
O 3U
O
T +K /
T +K /
Tw J
T +K /
E
5G
E
K
K
Fig. 13= Behavior of () via 2 =
Fig. 12= Behavior of () via =
10
0
5G
E
E H
Tw J
3U
T +K /
T +K /
5G
E
E H
Tw O
3U
0
O
J
K
E H
K
Fig. 14= Behavior of () via Pr = 0 E
Tw J
O 3U
Fig. 15= Behavior of () via =
0 5G
E
E H
J
O 3U
5G
T +K /
T +K /
Tw
K
K
Fig. 16= Behavior of () via Ug=
Fig. 17= Behavior of () via z =
11
Table 2: Skin friction
¡ Re ¢1@2 2
Fi when Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2>
P = 0=5> z = 1=2 and = 0=3 = =
Parameters (fixed values)
Parameters
Ug = 2 = 0=1> Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
1
¡ Re ¢1@2 2
0=0 1=29889
0=2 2=33486 0=4 3=36901 Ug = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
2
0=0 1=58314 0=4 2=09146 0=7 2=58448
Ug = 2 = 0=1 = 1 > Pr = 1=5> P = 0=5> z = 1=2 and = 0=3 = =
1
0=0 2=19474 0=2 1=56522 0=4 1=18760
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3=
0=0 1=83314 0=3 1=78079 0=5 1=74747
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> z = 1=2 and = 0=3 = =
P
0=0 1=71818 0=2 1=90328 0=5 2=12816
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3=
0=0 2=14379 0=4 2=13015 0=6 2=11010
2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
Ug
0=2 1=56340 0=6 1=26935 0=9 1=27323
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5 and = 0=3 = =
z
0=0 1=56340 0=3 1=26935 0=4 1=27323
12
Fi
Table 3: Numerical data of surface temperature gradient 0 (0) when Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = = Parameters (fixed values)
Parameters
Ug = 2 = 0=1> Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
1
0 (0) 0=0 0=20972 0=2 0=21008 0=4 0=21038
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> z = 1=2 and = 0=3 = =
P
0=0 0=21127 0=4 0=20580 0=7 0=20351
Ug = 2 = 0=1 = 1 > Pr = 1=5> P = 0=5> z = 1=2 and = 0=3 = =
1
0=0 0=21229 0=2 0=21125 0=4 0=20860
Ug = 2 = 0=1 = 1 > 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
Pr
0=0 0=17350 0=3 0=19137 0=5 0=20355
Ug = 2 = 0=1> Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
1
0=0 0=21164 0=2 0=20548 0=5 0=20318
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3
0=0 0=20945 0=4 0=21058 0=6 0=21130
2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5> z = 1=2 and = 0=3 = =
Ug
0=2 0=00000 0=6 0=20975 0=9 0=29140
Ug = 2 = 0=1 = 1 > Pr = 1=5> 1 = 0=2> P = 0=5 and = 0=3 = =
z
0=0 0=24527 0=3 0=27512 0=4 0=35020
13
Table 4: Comparative study of skin friction with [45] via various P when 1 = 2 = 0 = 1 .
5
Magnetic parameter (P)
Ref.[45]
Present results
0=0
1=281828
1=28181
0=2
1=313294
1=31332
0=4
1=403026
1=40302
Final remarks
Combined e ects of mixed convection and magnetohydrodynamics in flow of third grade fluid and heat transfer analysis by an exponentially starching sheet have been examined. Convective condition are imposed at the boundary. The key points are listed below: • Larger third grade fluid parameter 1 shows an enhancment and reduction in thickness of momentum and thermal boundary layers respectively. • Impact of viscoelastic parameter 1 on both velocity i 0 () and temperature fields are qualitatively similar. • Enhancement in Pr decays the thermal fluid. • Velocity and temperature fields show an opposite behavior for higher . • Temperature is enhanced for higher , Ug and z = • Magnitude of skin
¡ Re ¢1@2 2
Fi is an increasing function of 1 > 2 and P=
• Surface heat transfer rate is enhanced via Pr> > > Ug, z and P=
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