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Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet
Ain Shams Engineering Journal (2014) xxx, xxx–xxx
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
ENGINEERING PHYSICS AND MATHEMATICS
Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet Asmat Ara a, Najeeb Alam Khan a b c
b,*
, Hassam Khan b, Faqiha Sultan
c
Department of Mathematical Sciences, Federal Urdu University Arts, Science and Technology, Karachi 75300, Pakistan Department of Mathematical Sciences, University of Karachi, Karachi 75270, Pakistan Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Karachi 75030, Pakistan
Received 27 February 2014; revised 15 May 2014; accepted 12 June 2014
KEYWORDS Boundary layer flow; Eyring–Powell model; Thermal radiation
Abstract The aim of this paper was to examine the steady boundary layer flow of an Eyring–Powell model fluid due to an exponentially shrinking sheet. In addition, the heat transfer process in the presence of thermal radiation is considered. Using usual similarity transformations the governing equations have been transformed into non-linear ordinary differential equations. Homotopy analysis method (HAM) is employed for the series solutions. The convergence of the obtained series solutions is carefully analyzed. Numerical values of the temperature gradient are presented and discussed. It is observed that velocity increases with an increase in mass suction S. In addition, for the temperature profiles opposite behavior is observed for increment in suction. Moreover, the thermal boundary layer thickness decreases due to increase in Prandtl number Pr and thermal radiation R. 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction Viscous boundary layer flow due to a stretching/shrinking sheet is of significant importance due to its vast applications. Aerodynamic extrusion of plastic sheets, glass fiber production, paper production, heat treated materials traveling between a feed roll and a wind-up roll, cooling of an infinite metallic plate in a cooling bath and manufacturing of poly* Corresponding author. Tel.: +92 3333012008. E-mail address: [email protected] (N.A. Khan). Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
meric sheets are some examples for practical applications of non-Newtonian fluid flow over a stretching/shrinking surface. The quality of the final product depends on the rate of heat transfer at the stretching surface. This stretching/shrinking may not necessarily be linear. It may be quadratic, powerlaw, exponential and so on. Over the last few decades, in nearly all investigations on the flow past a stretching/shrinking sheet, the flow occurs due to linear stretching/shrinking velocity of the flat sheet. However, boundary layer flow induced by an exponentially stretching/ shrinking sheet is not studied much. Crane [1] firstly investigated the steady boundary layer flow of the incompressible flow. Gupta and Gupta [2] and Chen and Char [3] extended the work of Crane under various physical conditions. Hayat et al. [4] examined the unsteady three dimensional flow of couple stress fluid over a stretching surface with chemical reaction.
2090-4479 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2014.06.002 Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
2 Hamad [5] found the analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Bachok et al. [6] described stagnationpoint flow over a stretching/shrinking sheet in a nanofluid. Bhattacharyya [7] considered heat transfer analysis in unsteady boundary layer stagnation-point flow toward a shrinking/ stretching sheet. Numerical study of MHD boundary layer flow of a maxwell fluid past a stretching in the presence of nanoparticles has been carried out by Nadeem et al. [8]. Mehmood et al. [9] examined the non-orthogonal flow of a second grade micropolar fluid toward a stretching sheet, they also have taken heat transfer analysis into account. Moreover, Nadeem et al.[10] have recently analyzed the non-orthogonal stagnation point flow of a third order fluid toward a stretching surface in the presence of heat transfer. Magyari et al. [11] obtained the boundary layer flow due to exponentially stretching sheet. Elbashbeshy [12] numerically explained the flow and heat transfer over an exponentially stretching surface considering wall mass suction. The MHD boundary layer flow of a viscous fluid over an exponentially stretching sheet with effects of radiation was studied by Ishak [13]. Al-Odet et al. [14] examined the effect of magnetic field on thermal boundary layer flow on an exponentially stretching continuous surface with an exponentially temperature distribution. Sajid et al. [15] found the influence of thermal radiation on the boundary layer flow past an exponentially stretching sheet and they reported series solutions for the velocity and temperature by employing HAM. Bhattacharyya [16] discussed the boundary layer and heat transfer over an exponentially shrinking sheet. Up to date not much study has been carried out for the two dimensional flow of the Eyring–Powell fluid. Although this fluid model has many advantages over the non-Newtonian fluids models. Firstly, it is extracted from the kinetic theory of liquids rather than the empirical relation. Secondly, for low and high shear rates it correctly reduces to Newtonian behavior. Eyring–Powell fluid model [17] a complete mathematical model proposed by Powell and Eyring in 1944. Hayat et al. [18] analyzed steady flow of an Eyring–Powell fluid over a moving surface with convective boundary conditions. Malik et al. [19] presented boundary layer flow of an Eyring–Powell model fluid due to a stretching cylinder with variable viscosity. Nabil et al. [20] studied numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD Eyring–Powell fluid flow through a porous medium. Javed et al. [21] investigated flow of an Eyring–Powell non-Newtonian fluid over a stretching sheet. Characteristics of heating scheme and mass transfer on the peristaltic flow an Eyring–Powell fluid in an endoscope discussed by Nadeem et al. [22] However, to the best of our knowledge no attempt has been made to study Eyring–Powell fluid over an exponentially shrinking sheet. Thus current work presents a theoretical study Eryning Powell flow of over an exponentially shrinking sheet. A mathematical model has been prepared in the presence of radiation effects. We developed series solutions for the resulting problems by using the homotopy analysis method [23–27]. Results for the velocity and temperature are constructed. Convergence criteria for the derived series solutions are established. The velocity and temperature are analyzed to gain thorough insight toward the physics of the problem for various parameters of interest. Numerical values of the temperature gradient are presented and discussed.
A. Ara et al. 2. Mathematical formulation Consider the steady two dimensional boundary layer flow of an Eyring–Powell fluid with heat transfer in the presence of thermal radiation over an exponentially shrinking sheet (see Fig. 1). The stress tensor of an Eyring–Powell model [28] is expressed as A ¼ pI þ s
ð1Þ
where extra stress tensor sij is given by @ui 1 1 @ui : sij ¼ l þ sinh1 @xj b d @xj
ð2Þ
In accordance with the boundary layer approximations, the governing equations for the flow are @u @v þ ¼0 @x @y
ð3Þ
u
ð4Þ
2 2 2 @u @u 1 @p 1 @ u 1 @u @ u þv ¼ þ mþ 3 2 @x @y q @x qbd @y @y2 2qbd @y @p ¼0 @y
ð5Þ
Eqs. (4) and (5) show that the pressure is independent of y. Since the lateral velocity is zero far away from the sheet and the pressure is uniform, then Eq. (4) takes the form 2 2 @u @u 1 @2u 1 @u @ u ð6Þ u þv ¼ mþ 3 2 @x @y qbd @y @y2 2qbd @y The equations representing temperature with heat radiation may be written in usual notation as below: u
@T @T j @2T 1 @qr þv ¼ @x @y qCp @y2 qCp @y
ð7Þ
where u and v are the velocity components, m is the kinamatic viscosity, q is the fluid density, b and d are the fluid parameters of Eyring–Powell model, d has the dimension of (time)1. j is the fluid thermal conductivity and Cp is the specific heat con4 , where stant. The radiative heat flux, qr is given by qr ¼ 4r@T 3K@y r is Stefan–Boltzman constant and K is Rosseland mean absorption coefficient. The boundary conditions are given by u ¼ Uw ðxÞ; v ¼ vw at y ¼ 0; u ! 0 as y ! 1
Fig. 1
ð8Þ
Physical model of the fluid.
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet x 2L
T ¼ Tw ¼ T1 þ T0 e at y ¼ 0; T ! T1 as y ! 1
ð9Þ
Tw is temperature of the sheet and T1 is the free stream temperature, both are assumed to be constant with Tw > T1 and T0 is the constant which measures the rate of temperature increase along the sheet. The shrinking velocity Uw is given by x
Uw ðxÞ ¼ ceL
ð10Þ
where c > 0 is shrinking constant. Here vw is the constant with v0 < 0 for mass suction and v0 > 0 for mass injection. Now, the stream function w(x, y) has been introduced as u¼
@w @w and v ¼ @y @x
ð11Þ
The continuity Eq. (1) is identically satisfied by Eq. (11), the momentum and energy equations have been reduced to the forms 2 2 3 @w @ 2 w @w @ 2 w 1 @3w 1 @ w @ w ¼ mþ @y @x@y @x @y@y qbd @y3 2qbd3 @y2 @y3 ð12Þ
3
The non-linear coupled differential Eqs. (16) and (17) along with the boundary conditions Eqs. (18) and (19) have been solved by employing HAM. 3. Series solutions The auxiliary linear operators and the initial approximations for the homotopy analysis solutions are chosen as solutions of Eqs. (16) and (17) with the boundary conditions Eqs. (18) and (19). f0 ¼ 1 þ eg þ S; h0 ¼ eg L1 ðfÞ ¼ f 000 f 0 L2 ðhÞ ¼ h00 h
ð22Þ ð23Þ ð24Þ
We note that the auxiliary linear operators in the above equations satisfies the following properties L1 ½c1 þ c2 eg þ c3 eg ¼ 0;
L2 ½c4 eg þ c5 eg ¼ 0
ð25Þ
The associated zeroth-order deformation problems are ~ pÞ f0 ðgÞ ¼ p ~ pÞ ð1 pÞL1 ½fðg; hN1 ½fðg; ~ pÞ h0 ðgÞ ¼ p ~ pÞ; ~ ð1 pÞL2 ½hðg; hN2 ½fðg; h ðg; pÞ 0 0 ~ ~ ~ fð0; pÞ ¼ S; f ð0; pÞ ¼ 1; f ð1; pÞ ¼ 0;
ð26Þ
The boundary conditions in Eq. (8) for the velocity components take the form
~ hð0; pÞ ¼ 0;
ð29Þ
@w ¼ Uw ðxÞ; @y
The boundary conditions for this deformation are based in Eqs. (16) and (17), the non-linear operators N1 and N2 are introduced as
@w @T @w @T j @2T 1 @qr ¼ @y @x @x @y Cp q @y2 qCp @y
@w ¼ vw @x
at y ¼ 0;
ð13Þ
@w ! 0 as y ! 1; @y ð14Þ
The dimensionless variables for w and T are pffiffiffiffiffiffiffiffiffiffi T T1 x w ¼ 2cmLfðgÞe2L ; hðgÞ ¼ Tw T1
ð15Þ
pffiffiffiffiffi ffi x c 2L where g is the similarity variable defined as g ¼ y 2mL e
On utilizing the above relation, Eqs. (12) and (13) are reduced to ð1 þ NÞf 000 þ ff 00 2f 02 Nkf 002 f 000 ¼ 0 4 1 þ R h00 þ Prð fh0 f 0 hÞ ¼ 0 3 3
3x
3~ 2~ ~ pÞÞ ¼ ð1 þ NÞ @ fðg; pÞ fðg; ~ pÞ @ fðg; pÞ N1 ðfðg; @g3 @g2 !2 !2 2 3 ~ pÞ ~ pÞ @ fðg; ~ pÞ @ fðg; @ fðg; 2 Nk 2 @g @g @g3
~ pÞ; ~ hðg; pÞÞ ¼ N2 ðfðg;
ð16Þ
ð28Þ
ð30Þ
2~ 4 @ hðg; pÞ 1þ R 3 @g2 ~ ~ pÞ @ fðg; ~ pÞ @ hðg; pÞ ~ þ Pr fðg; hðg; pÞ @g @g
ð17Þ
1 c eL where N ¼ dbl ; k ¼ 4d are the fluid parameters and Pr ¼ 2 Lm 3
~ hð1; pÞ ¼ 0
ð27Þ
!
ð31Þ
Cp l j
is Prandtl number and R ¼ 4TjK1 r is radiation parameter. The boundary conditions applicable to the present fluid flow are
Here, p is an embedding parameter, h is the non-zero auxiliary parameter. For p = 0 and p = 1, we have
fðgÞ ¼ S; f 0 ðgÞ ¼ 1 at g ¼ 0; f 0 ðgÞ ! 0 as g ! 1 hðgÞ ¼ 1 at g ¼ 0; hðgÞ ! 0 as g ! 1
~ 0Þ ¼ f0 ðgÞ; fðg; ~ 1Þ ¼ fðgÞ fðg; ~ 0Þ ¼ h0 ðgÞ; ~ hðg; hðg; 1Þ ¼ hðgÞ
ð18Þ ð19Þ
where S ¼ pv0ffiffiffi is a mass transfer parameter. S > 0(v0 < 0) mc 2L
corresponds to mass suction and S < 0(v0 > 0) corresponds to mass injection. The local skin-friction coefficient or frictional drag coefficient on the surface of the exponentially shrinking sheet is sxy y¼0 Cf ¼ ð20Þ x 2 qðc eL Þ In non-dimensional form above quantity can be given as 1
Re2x Cf ¼ ð1 þ NÞf 00 ð0Þ 1
x
N 003 kf ð0Þ 3
where Re2x ¼ 2Lcm eL is the local Reynolds number.
ð21Þ
ð32Þ ð33Þ
~ pÞ; ~ Further, when p increases from 0 to 1, fðg; hðg; pÞ vary from ~ ~ f0 ðgÞ; h0 ðgÞ, to f(g), h(g). By using Taylor’s series expansion, one can write 1 X 1 @ m fðg; pÞ fm ðgÞpm ; fm ðgÞ ¼ jp¼0 m! @pm m¼1 1 X 1 @ m hðg; pÞ hm ðgÞpm ; hm ðgÞ ¼ jp¼0 hðg; pÞ ¼ h0 ðgÞ þ m! @pm m¼1
fðg; pÞ ¼ f0 ðgÞ þ
ð34Þ ð35Þ
and the convergence of Eqs. (26) and (27) strictly depend upon . The values of h h are selected in such a manner that Eqs. (26) and (27) are convergent at Therefore, mth order deformation problems are given by
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
4
A. Ara et al.
Lf ½fm ðgÞ vm fm1 ðgÞ ¼ hRm f ðgÞ
ð36Þ
hRm h ðgÞ
ð37Þ Lh ½hm ðgÞ vm hm1 ðgÞ ¼ fm ð0Þ ¼ S; fm0 ð0Þ ¼ 1; fm0 ð1Þ ¼ 0; hm ð0Þ ¼ 0; hm ð1Þ ¼ 0 000 Rm f ðgÞ ¼ ð1 þ NÞfm1 þ
m1 m1 X X 00 fm1k f00k 2 fm1k fk00 k¼0
k¼0
m1 m1 X X 000 00 00 Nk fm1k fkl fkl k¼0
Rm h ðgÞ ¼ vm ¼
m1 m1 X X 4 1 þ h00m1 þ Pr h0m1 fm Pr f0m1 hm 3 k¼0 k¼0
0;
m61
ð39Þ ð40Þ
1;
m>1 1 X fðgÞ ¼ f0 ðgÞ þ fm ðgÞ hm ðgÞ
ð42Þ
m¼1
The general solution can be written as fm ðgÞ ¼ fm ðgÞ þ c1 þ c2 eg þ c3 eg
ð43Þ
hm ðgÞ ¼ hm ðgÞ þ c4 eg þ c5 eg
ð44Þ
In which fm ðgÞ and and h(g).
hm ðgÞ
represent the special solutions for f(g)
4. Convergence analysis and discussion of results It is noticed that Eqs. (36) and (37) have the auxiliary parameters h, and this parameter has a key role to adjust and control the convergence of series solutions. The curves have been sketched at the 7th-order of approximations to determine the suitable ranges for h. Figs. 2 and 3 show the acceptable range of auxiliary parameter h to converge the series solution presented in Eqs. (26) and (27) by keeping the values of pertinent parameters fixed. The admissible values for the parameter h for f00 (0) are 3.5 6 h 6 0.2 for S = 2.4 from Fig. 2 and for h0 (0) are 1.5 6 h < 0 from Fig. 3. In order to get a clear insight into the physical problem, numerical computations have been carried out for obtaining the condition by using HAM, under which the steady flow over exponentially shrinking sheet is
Fig. 2
Fig. 3
h curve for h00 (0) when N = 0.1 k = 0.1 and Pr = 0.5.
ð41Þ
m¼1 1 X
hðgÞ ¼ h0 ðgÞ þ
ð38Þ
k¼0
00
h curve for f (0) when N = 0.1 k = 0.1.
possible. The shrinking rate in the exponential case is much faster than of the linear case. Thus, the amount of velocity generated due to exponential shrinking is greater than that of linear shrinking. Fig. 4 illustrates the variations in the velocity profiles for the various values of mass suction parameter S. It clearly indicates that thickness of the velocity boundary layer decreases with the increasing values of S. In order to validate the accuracy of the analytical solution the comparison is made with the results obtained with earlier work in [16] by taking N = 0 and k = 0 in Fig. 5 and 6 respectively. Fig. 5 reperesents the behavior of velocity profiles f0 (g) for different values of fluid parameter k. It is observed that the increase in k enhances the velocity profiles. The behavior of f0 (g) by changing the Eyring–Powell parameter N is observed in Fig. 6. It is noticed that the increase in N reduces the velocity profiles. In Fig. 7 the temperature profiles h(g) have been shown for the different values of S. the temperature at a point decreases for an increase of S. Finally, the effects of the Prandtl number Pr on the dimensionless temperature profile are offered in Fig. 8. It is observed that temperature is decreasing with in increase of Pr. Also, it is worth mentioning that the thermal boundary layer thickness reduces considerably due to increase in Pr, since the Prandtl number is inversely proportional to the thermal conductivity. Thus, the fluid with higher Prandtl number has lower thermal conductivities which led the heat
Fig. 4 Velocity profiles of f0 (g) for various values of S keeping k = 0.1 and N = 0.1 fixed.
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet
5
Fig. 5 Velocity profiles of f0 (g) for various values of k keeping S = 2.4 and N = 0.2 fixed.
Fig. 8 Temperature profiles h(g) for Pr keeping k = 0.01, R = 0.5, N = 0.2 and S = 2.5.
Fig. 6 Velocity profiles of f0 (g) for various values of N keeping S = 2.4 and k = 1 fixed.
Fig. 9 Temperature profiles h(g) for R keeping k = 1, Pr = 0.5, N = 0.2 and S = 2.5.
1
Table 1 Numerical values of skin-friction coefficient Re2x Cf for different values of N and k. fl
k/N fi 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.28741 1.28741 1.28741 1.28741 1.28741 1.28741 1.28741 1.28741 1.28741
0.2 0.13362 0.11786 0.10112 0.08330 0.06427 0.04392 0.00141 0.02680 0.05433
0.4
0.6
0.8
1.0
0.31007 0.29554 0.27968 0.26230 0.24316 0.22198 0.17196 0.14206 0.10794
0.42261 0.41241 0.40119 0.38877 0.37498 0.35955 0.32250 0.29996 0.27387
0.50186 0.49543 0.48835 0.48053 0.47183 0.46212 0.43888 0.42483 0.40868
0.56254 0.55876 0.55460 0.55000 0.54492 0.53926 0.52586 0.51786 0.50877
Fig. 7 Temperature profiles h(g) for various values of S keeping k = 1, R = 0.2, N = 0.2 and Pr = 0.5.
diffusion slows down. On the other hand, for the case of lower Pr fluids the heat diffusion gets faster, with higher thermal conductivities. The influence of the thermal radiation on temperature is depicted in Fig. 9. It is interesting to note that thermal radiation has a major influence on the temperature distribution in the fluid. We observed that the fluid temperature increases by increasing thermal radiation. This is due to
the fact that increase in the values of the thermal radiation parameter increases radiation in the boundary layer, and hence increases the values of the temperature profiles in the thermal boundary layer. In Table 1 the dimensionless velocity gradient on the sheet is approximated for various values of N and k. We observed that skin friction coefficient is reduced by sufficiently large values of N and k. In Table 2 numerical values
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
6
A. Ara et al. Table 2 Numerical values of temperature gradient h0 (0) for different values of N, k, Pr, S and R. N
k
Pr
R
S
h0 (0)
0 0.2 0.4 0.6 0.2
0.2
0.1
0.5
2.4
0.20120 0.21648 0.21647 0.21620 0.13754 0.13775 0.21682 0.24191 0.38037 0.81102 0.13764 0.13271 0.13045 0.21648 0.20475 0.19935
0.1 0.3 0.5 0.2
0.2 0.5 01 0.1
0.5 01 1.5 0.5
2.4 2.6 2.8
of surface heat transfer h0 (0) have been computed for different values of Pr, R, k, S and N. It is apparent that the magnitude of surface heat transfer is an increasing function of Pr and k, while decreasing one for the different values of S, N and R. 5. Conclusions The boundary layer flow and heat transfer over an exponentially shrinking sheet have been studied. Series solutions are obtained for the velocity and temperature profiles. It is observed that velocity increases with an increase in mass suction S. In addition, for the temperature profiles opposite behavior is seen for increment in suction. Moreover, the thermal boundary layer thickness decreases due to increase in Prandtl number Pr and thermal radiation R. References [1] Crane LJ. Flow past a stretching plate. Z Angew Math Phys 1970;21:645–7. [2] Gupta PS, S Gupta A. Heat and mass transfer on a stretching sheet with suction or blowing. J Chem Eng 1977;55:744–6. [3] Chen CK, Char MI. Heat transfer of a continuous, stretching surface with suction or blowing. J Maths Anal Appl 1988;135: 568–80. [4] Hayat T, Awais M, Safdar A, Hendi AA. Unsteady three dimensional flow of couple stress fluid over a stretching surface with chemical reaction. Nonlinear Anal: Mod Contr 2012;17(1): 47–59. [5] Hamad MAA. Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Int Commun Heat Mass Trans 2011;38:487–92. [6] Bachok N, Ishak A, Pop I. Stagnation-point flow over a stretching/shrinking sheet in a nanofluid. Nanoscale Res Let 2011;6:623. [7] Bhattacharyya K. Heat transfer analysis in unsteady boundary layer stagnation-point flow towards a shrinking/stretching sheet. Al Shams Eng J 2012.
[8] Nadeem S, Ul Haq Rizwan, Khan ZH. Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles. J Taiwan Inst Chem Eng 2013;45: 121–6. [9] Mehmood R, Nadeem S, Akbar NS. Non-orthogonal stagnation point flow of a micropolar second grade fluid towards a stretching surface with heat transfer. J Taiwan Inst Chem Eng 2013;44: 86–595. [10] Nadeem S, Mehmood R, Akbar NS. Influence of heat transfer on the non-orthogonal stagnation point flow of a third order fluid towards a stretching surface. Heat Transfer Asian Res 2013;42: 319–26. [11] Magyari E, Keller B. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J Phys D: Appl Phys 1999;32:577–85. [12] Elbashbeshy EM. Flow and heat transfer over an exponentially stretching surface considering wall mass suction. Arch Mech 2001;53:643. [13] Ishak A. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. Sains Malays 2011;40:391–5. [14] Al Odat MQ, Damesh R, Azab TAA. Thermal boundary layer on an exponentially stretching continuous surface in the presence of magnetic field effect. Int J Appl Mech Eng 2006;11:289. [15] Sajid M, Hayat T. Influence of thermal radiation on the boundary layer flow past an exponentially stretching. Int Commun Heat Mass Transfer 2008;35:347. [16] Bhattacharyya K. Boundary layer and heat transfer over an exponentially shrinking sheet. Chin Phys Lett 2011;28(7):074701. [17] Powell RE, Eyring H. Mechanisms for the relaxation theory of viscosity. Nature 1944;154:427–8. [18] Hayat T, Iqbal Z, Qasim M, Obaidat S. Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions. Int J Heat Mass Transfer 2012;55:1817–22. [19] Malik MY, Hussain A, Nadeem S. Boundary layer flow of an Eyring–Powell model fluid due to a stretching cylinder with variable viscosity. Scientia Iranica, Trans B: Mech Eng 2013;20(2): 313–21. [20] Eldabe NTM, Sallam SN, Abou-zeid MY. Numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD non-Newtonian fluid flow through a porous medium. J Egy Math Soc 2012;20:139–51. [21] Javed T, Abbas NAliZ, Sajid M. Flow of an Eyring–Powell nonnewtonian fluid over a stretching sheet. Chem Eng Commun 2013;200:327–36. [22] Akbar NS, Nadeem S. Characteristics of heating scheme and mass transfer on the peristaltic flow for an Eyring–Powell fluid in an endoscope. Int J Heat Mass Transfer 2012;55:375–83. [23] Khan NA, Riaz F. Off centered stagnation point flow of a couple stress fluid towards a rotating disk. Sci World J 2013 [Art. ID: 163585]. [24] Khan NA, Riaz F, Sultan F. Effects of chemical reaction and magnetic field on a couple stress fluid over a non-linearly stretching sheet. Eur J Phys – Plus 2014:18, 129. [25] Khan NA, Jamil M, Khan Nadeem A. Effects of slip factors on unsteady stagnation point flow and heat transfer towards a stretching sheet: an analytical study. Heat Transfer Res 2012;43(8):779–94. [26] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. ChapmanHall/CRC; 2002. [27] Liao SJ. Homotopy analysis method in non-linear differential equations. Springer; 2011. [28] Steff JF. Rheological methods in food process engineering. 2nd ed. East Lansing (Mich): Freeman Press; 1996.
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
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ENGINEERING PHYSICS AND MATHEMATICS
Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet Asmat Ara a, Najeeb Alam Khan a b c
b,*
, Hassam Khan b, Faqiha Sultan
c
Department of Mathematical Sciences, Federal Urdu University Arts, Science and Technology, Karachi 75300, Pakistan Department of Mathematical Sciences, University of Karachi, Karachi 75270, Pakistan Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Karachi 75030, Pakistan
Received 27 February 2014; revised 15 May 2014; accepted 12 June 2014
KEYWORDS Boundary layer flow; Eyring–Powell model; Thermal radiation
Abstract The aim of this paper was to examine the steady boundary layer flow of an Eyring–Powell model fluid due to an exponentially shrinking sheet. In addition, the heat transfer process in the presence of thermal radiation is considered. Using usual similarity transformations the governing equations have been transformed into non-linear ordinary differential equations. Homotopy analysis method (HAM) is employed for the series solutions. The convergence of the obtained series solutions is carefully analyzed. Numerical values of the temperature gradient are presented and discussed. It is observed that velocity increases with an increase in mass suction S. In addition, for the temperature profiles opposite behavior is observed for increment in suction. Moreover, the thermal boundary layer thickness decreases due to increase in Prandtl number Pr and thermal radiation R. 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction Viscous boundary layer flow due to a stretching/shrinking sheet is of significant importance due to its vast applications. Aerodynamic extrusion of plastic sheets, glass fiber production, paper production, heat treated materials traveling between a feed roll and a wind-up roll, cooling of an infinite metallic plate in a cooling bath and manufacturing of poly* Corresponding author. Tel.: +92 3333012008. E-mail address: [email protected] (N.A. Khan). Peer review under responsibility of Ain Shams University.
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meric sheets are some examples for practical applications of non-Newtonian fluid flow over a stretching/shrinking surface. The quality of the final product depends on the rate of heat transfer at the stretching surface. This stretching/shrinking may not necessarily be linear. It may be quadratic, powerlaw, exponential and so on. Over the last few decades, in nearly all investigations on the flow past a stretching/shrinking sheet, the flow occurs due to linear stretching/shrinking velocity of the flat sheet. However, boundary layer flow induced by an exponentially stretching/ shrinking sheet is not studied much. Crane [1] firstly investigated the steady boundary layer flow of the incompressible flow. Gupta and Gupta [2] and Chen and Char [3] extended the work of Crane under various physical conditions. Hayat et al. [4] examined the unsteady three dimensional flow of couple stress fluid over a stretching surface with chemical reaction.
2090-4479 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2014.06.002 Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
2 Hamad [5] found the analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Bachok et al. [6] described stagnationpoint flow over a stretching/shrinking sheet in a nanofluid. Bhattacharyya [7] considered heat transfer analysis in unsteady boundary layer stagnation-point flow toward a shrinking/ stretching sheet. Numerical study of MHD boundary layer flow of a maxwell fluid past a stretching in the presence of nanoparticles has been carried out by Nadeem et al. [8]. Mehmood et al. [9] examined the non-orthogonal flow of a second grade micropolar fluid toward a stretching sheet, they also have taken heat transfer analysis into account. Moreover, Nadeem et al.[10] have recently analyzed the non-orthogonal stagnation point flow of a third order fluid toward a stretching surface in the presence of heat transfer. Magyari et al. [11] obtained the boundary layer flow due to exponentially stretching sheet. Elbashbeshy [12] numerically explained the flow and heat transfer over an exponentially stretching surface considering wall mass suction. The MHD boundary layer flow of a viscous fluid over an exponentially stretching sheet with effects of radiation was studied by Ishak [13]. Al-Odet et al. [14] examined the effect of magnetic field on thermal boundary layer flow on an exponentially stretching continuous surface with an exponentially temperature distribution. Sajid et al. [15] found the influence of thermal radiation on the boundary layer flow past an exponentially stretching sheet and they reported series solutions for the velocity and temperature by employing HAM. Bhattacharyya [16] discussed the boundary layer and heat transfer over an exponentially shrinking sheet. Up to date not much study has been carried out for the two dimensional flow of the Eyring–Powell fluid. Although this fluid model has many advantages over the non-Newtonian fluids models. Firstly, it is extracted from the kinetic theory of liquids rather than the empirical relation. Secondly, for low and high shear rates it correctly reduces to Newtonian behavior. Eyring–Powell fluid model [17] a complete mathematical model proposed by Powell and Eyring in 1944. Hayat et al. [18] analyzed steady flow of an Eyring–Powell fluid over a moving surface with convective boundary conditions. Malik et al. [19] presented boundary layer flow of an Eyring–Powell model fluid due to a stretching cylinder with variable viscosity. Nabil et al. [20] studied numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD Eyring–Powell fluid flow through a porous medium. Javed et al. [21] investigated flow of an Eyring–Powell non-Newtonian fluid over a stretching sheet. Characteristics of heating scheme and mass transfer on the peristaltic flow an Eyring–Powell fluid in an endoscope discussed by Nadeem et al. [22] However, to the best of our knowledge no attempt has been made to study Eyring–Powell fluid over an exponentially shrinking sheet. Thus current work presents a theoretical study Eryning Powell flow of over an exponentially shrinking sheet. A mathematical model has been prepared in the presence of radiation effects. We developed series solutions for the resulting problems by using the homotopy analysis method [23–27]. Results for the velocity and temperature are constructed. Convergence criteria for the derived series solutions are established. The velocity and temperature are analyzed to gain thorough insight toward the physics of the problem for various parameters of interest. Numerical values of the temperature gradient are presented and discussed.
A. Ara et al. 2. Mathematical formulation Consider the steady two dimensional boundary layer flow of an Eyring–Powell fluid with heat transfer in the presence of thermal radiation over an exponentially shrinking sheet (see Fig. 1). The stress tensor of an Eyring–Powell model [28] is expressed as A ¼ pI þ s
ð1Þ
where extra stress tensor sij is given by @ui 1 1 @ui : sij ¼ l þ sinh1 @xj b d @xj
ð2Þ
In accordance with the boundary layer approximations, the governing equations for the flow are @u @v þ ¼0 @x @y
ð3Þ
u
ð4Þ
2 2 2 @u @u 1 @p 1 @ u 1 @u @ u þv ¼ þ mþ 3 2 @x @y q @x qbd @y @y2 2qbd @y @p ¼0 @y
ð5Þ
Eqs. (4) and (5) show that the pressure is independent of y. Since the lateral velocity is zero far away from the sheet and the pressure is uniform, then Eq. (4) takes the form 2 2 @u @u 1 @2u 1 @u @ u ð6Þ u þv ¼ mþ 3 2 @x @y qbd @y @y2 2qbd @y The equations representing temperature with heat radiation may be written in usual notation as below: u
@T @T j @2T 1 @qr þv ¼ @x @y qCp @y2 qCp @y
ð7Þ
where u and v are the velocity components, m is the kinamatic viscosity, q is the fluid density, b and d are the fluid parameters of Eyring–Powell model, d has the dimension of (time)1. j is the fluid thermal conductivity and Cp is the specific heat con4 , where stant. The radiative heat flux, qr is given by qr ¼ 4r@T 3K@y r is Stefan–Boltzman constant and K is Rosseland mean absorption coefficient. The boundary conditions are given by u ¼ Uw ðxÞ; v ¼ vw at y ¼ 0; u ! 0 as y ! 1
Fig. 1
ð8Þ
Physical model of the fluid.
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet x 2L
T ¼ Tw ¼ T1 þ T0 e at y ¼ 0; T ! T1 as y ! 1
ð9Þ
Tw is temperature of the sheet and T1 is the free stream temperature, both are assumed to be constant with Tw > T1 and T0 is the constant which measures the rate of temperature increase along the sheet. The shrinking velocity Uw is given by x
Uw ðxÞ ¼ ceL
ð10Þ
where c > 0 is shrinking constant. Here vw is the constant with v0 < 0 for mass suction and v0 > 0 for mass injection. Now, the stream function w(x, y) has been introduced as u¼
@w @w and v ¼ @y @x
ð11Þ
The continuity Eq. (1) is identically satisfied by Eq. (11), the momentum and energy equations have been reduced to the forms 2 2 3 @w @ 2 w @w @ 2 w 1 @3w 1 @ w @ w ¼ mþ @y @x@y @x @y@y qbd @y3 2qbd3 @y2 @y3 ð12Þ
3
The non-linear coupled differential Eqs. (16) and (17) along with the boundary conditions Eqs. (18) and (19) have been solved by employing HAM. 3. Series solutions The auxiliary linear operators and the initial approximations for the homotopy analysis solutions are chosen as solutions of Eqs. (16) and (17) with the boundary conditions Eqs. (18) and (19). f0 ¼ 1 þ eg þ S; h0 ¼ eg L1 ðfÞ ¼ f 000 f 0 L2 ðhÞ ¼ h00 h
ð22Þ ð23Þ ð24Þ
We note that the auxiliary linear operators in the above equations satisfies the following properties L1 ½c1 þ c2 eg þ c3 eg ¼ 0;
L2 ½c4 eg þ c5 eg ¼ 0
ð25Þ
The associated zeroth-order deformation problems are ~ pÞ f0 ðgÞ ¼ p ~ pÞ ð1 pÞL1 ½fðg; hN1 ½fðg; ~ pÞ h0 ðgÞ ¼ p ~ pÞ; ~ ð1 pÞL2 ½hðg; hN2 ½fðg; h ðg; pÞ 0 0 ~ ~ ~ fð0; pÞ ¼ S; f ð0; pÞ ¼ 1; f ð1; pÞ ¼ 0;
ð26Þ
The boundary conditions in Eq. (8) for the velocity components take the form
~ hð0; pÞ ¼ 0;
ð29Þ
@w ¼ Uw ðxÞ; @y
The boundary conditions for this deformation are based in Eqs. (16) and (17), the non-linear operators N1 and N2 are introduced as
@w @T @w @T j @2T 1 @qr ¼ @y @x @x @y Cp q @y2 qCp @y
@w ¼ vw @x
at y ¼ 0;
ð13Þ
@w ! 0 as y ! 1; @y ð14Þ
The dimensionless variables for w and T are pffiffiffiffiffiffiffiffiffiffi T T1 x w ¼ 2cmLfðgÞe2L ; hðgÞ ¼ Tw T1
ð15Þ
pffiffiffiffiffi ffi x c 2L where g is the similarity variable defined as g ¼ y 2mL e
On utilizing the above relation, Eqs. (12) and (13) are reduced to ð1 þ NÞf 000 þ ff 00 2f 02 Nkf 002 f 000 ¼ 0 4 1 þ R h00 þ Prð fh0 f 0 hÞ ¼ 0 3 3
3x
3~ 2~ ~ pÞÞ ¼ ð1 þ NÞ @ fðg; pÞ fðg; ~ pÞ @ fðg; pÞ N1 ðfðg; @g3 @g2 !2 !2 2 3 ~ pÞ ~ pÞ @ fðg; ~ pÞ @ fðg; @ fðg; 2 Nk 2 @g @g @g3
~ pÞ; ~ hðg; pÞÞ ¼ N2 ðfðg;
ð16Þ
ð28Þ
ð30Þ
2~ 4 @ hðg; pÞ 1þ R 3 @g2 ~ ~ pÞ @ fðg; ~ pÞ @ hðg; pÞ ~ þ Pr fðg; hðg; pÞ @g @g
ð17Þ
1 c eL where N ¼ dbl ; k ¼ 4d are the fluid parameters and Pr ¼ 2 Lm 3
~ hð1; pÞ ¼ 0
ð27Þ
!
ð31Þ
Cp l j
is Prandtl number and R ¼ 4TjK1 r is radiation parameter. The boundary conditions applicable to the present fluid flow are
Here, p is an embedding parameter, h is the non-zero auxiliary parameter. For p = 0 and p = 1, we have
fðgÞ ¼ S; f 0 ðgÞ ¼ 1 at g ¼ 0; f 0 ðgÞ ! 0 as g ! 1 hðgÞ ¼ 1 at g ¼ 0; hðgÞ ! 0 as g ! 1
~ 0Þ ¼ f0 ðgÞ; fðg; ~ 1Þ ¼ fðgÞ fðg; ~ 0Þ ¼ h0 ðgÞ; ~ hðg; hðg; 1Þ ¼ hðgÞ
ð18Þ ð19Þ
where S ¼ pv0ffiffiffi is a mass transfer parameter. S > 0(v0 < 0) mc 2L
corresponds to mass suction and S < 0(v0 > 0) corresponds to mass injection. The local skin-friction coefficient or frictional drag coefficient on the surface of the exponentially shrinking sheet is sxy y¼0 Cf ¼ ð20Þ x 2 qðc eL Þ In non-dimensional form above quantity can be given as 1
Re2x Cf ¼ ð1 þ NÞf 00 ð0Þ 1
x
N 003 kf ð0Þ 3
where Re2x ¼ 2Lcm eL is the local Reynolds number.
ð21Þ
ð32Þ ð33Þ
~ pÞ; ~ Further, when p increases from 0 to 1, fðg; hðg; pÞ vary from ~ ~ f0 ðgÞ; h0 ðgÞ, to f(g), h(g). By using Taylor’s series expansion, one can write 1 X 1 @ m fðg; pÞ fm ðgÞpm ; fm ðgÞ ¼ jp¼0 m! @pm m¼1 1 X 1 @ m hðg; pÞ hm ðgÞpm ; hm ðgÞ ¼ jp¼0 hðg; pÞ ¼ h0 ðgÞ þ m! @pm m¼1
fðg; pÞ ¼ f0 ðgÞ þ
ð34Þ ð35Þ
and the convergence of Eqs. (26) and (27) strictly depend upon . The values of h h are selected in such a manner that Eqs. (26) and (27) are convergent at Therefore, mth order deformation problems are given by
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
4
A. Ara et al.
Lf ½fm ðgÞ vm fm1 ðgÞ ¼ hRm f ðgÞ
ð36Þ
hRm h ðgÞ
ð37Þ Lh ½hm ðgÞ vm hm1 ðgÞ ¼ fm ð0Þ ¼ S; fm0 ð0Þ ¼ 1; fm0 ð1Þ ¼ 0; hm ð0Þ ¼ 0; hm ð1Þ ¼ 0 000 Rm f ðgÞ ¼ ð1 þ NÞfm1 þ
m1 m1 X X 00 fm1k f00k 2 fm1k fk00 k¼0
k¼0
m1 m1 X X 000 00 00 Nk fm1k fkl fkl k¼0
Rm h ðgÞ ¼ vm ¼
m1 m1 X X 4 1 þ h00m1 þ Pr h0m1 fm Pr f0m1 hm 3 k¼0 k¼0
0;
m61
ð39Þ ð40Þ
1;
m>1 1 X fðgÞ ¼ f0 ðgÞ þ fm ðgÞ hm ðgÞ
ð42Þ
m¼1
The general solution can be written as fm ðgÞ ¼ fm ðgÞ þ c1 þ c2 eg þ c3 eg
ð43Þ
hm ðgÞ ¼ hm ðgÞ þ c4 eg þ c5 eg
ð44Þ
In which fm ðgÞ and and h(g).
hm ðgÞ
represent the special solutions for f(g)
4. Convergence analysis and discussion of results It is noticed that Eqs. (36) and (37) have the auxiliary parameters h, and this parameter has a key role to adjust and control the convergence of series solutions. The curves have been sketched at the 7th-order of approximations to determine the suitable ranges for h. Figs. 2 and 3 show the acceptable range of auxiliary parameter h to converge the series solution presented in Eqs. (26) and (27) by keeping the values of pertinent parameters fixed. The admissible values for the parameter h for f00 (0) are 3.5 6 h 6 0.2 for S = 2.4 from Fig. 2 and for h0 (0) are 1.5 6 h < 0 from Fig. 3. In order to get a clear insight into the physical problem, numerical computations have been carried out for obtaining the condition by using HAM, under which the steady flow over exponentially shrinking sheet is
Fig. 2
Fig. 3
h curve for h00 (0) when N = 0.1 k = 0.1 and Pr = 0.5.
ð41Þ
m¼1 1 X
hðgÞ ¼ h0 ðgÞ þ
ð38Þ
k¼0
00
h curve for f (0) when N = 0.1 k = 0.1.
possible. The shrinking rate in the exponential case is much faster than of the linear case. Thus, the amount of velocity generated due to exponential shrinking is greater than that of linear shrinking. Fig. 4 illustrates the variations in the velocity profiles for the various values of mass suction parameter S. It clearly indicates that thickness of the velocity boundary layer decreases with the increasing values of S. In order to validate the accuracy of the analytical solution the comparison is made with the results obtained with earlier work in [16] by taking N = 0 and k = 0 in Fig. 5 and 6 respectively. Fig. 5 reperesents the behavior of velocity profiles f0 (g) for different values of fluid parameter k. It is observed that the increase in k enhances the velocity profiles. The behavior of f0 (g) by changing the Eyring–Powell parameter N is observed in Fig. 6. It is noticed that the increase in N reduces the velocity profiles. In Fig. 7 the temperature profiles h(g) have been shown for the different values of S. the temperature at a point decreases for an increase of S. Finally, the effects of the Prandtl number Pr on the dimensionless temperature profile are offered in Fig. 8. It is observed that temperature is decreasing with in increase of Pr. Also, it is worth mentioning that the thermal boundary layer thickness reduces considerably due to increase in Pr, since the Prandtl number is inversely proportional to the thermal conductivity. Thus, the fluid with higher Prandtl number has lower thermal conductivities which led the heat
Fig. 4 Velocity profiles of f0 (g) for various values of S keeping k = 0.1 and N = 0.1 fixed.
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet
5
Fig. 5 Velocity profiles of f0 (g) for various values of k keeping S = 2.4 and N = 0.2 fixed.
Fig. 8 Temperature profiles h(g) for Pr keeping k = 0.01, R = 0.5, N = 0.2 and S = 2.5.
Fig. 6 Velocity profiles of f0 (g) for various values of N keeping S = 2.4 and k = 1 fixed.
Fig. 9 Temperature profiles h(g) for R keeping k = 1, Pr = 0.5, N = 0.2 and S = 2.5.
1
Table 1 Numerical values of skin-friction coefficient Re2x Cf for different values of N and k. fl
k/N fi 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.28741 1.28741 1.28741 1.28741 1.28741 1.28741 1.28741 1.28741 1.28741
0.2 0.13362 0.11786 0.10112 0.08330 0.06427 0.04392 0.00141 0.02680 0.05433
0.4
0.6
0.8
1.0
0.31007 0.29554 0.27968 0.26230 0.24316 0.22198 0.17196 0.14206 0.10794
0.42261 0.41241 0.40119 0.38877 0.37498 0.35955 0.32250 0.29996 0.27387
0.50186 0.49543 0.48835 0.48053 0.47183 0.46212 0.43888 0.42483 0.40868
0.56254 0.55876 0.55460 0.55000 0.54492 0.53926 0.52586 0.51786 0.50877
Fig. 7 Temperature profiles h(g) for various values of S keeping k = 1, R = 0.2, N = 0.2 and Pr = 0.5.
diffusion slows down. On the other hand, for the case of lower Pr fluids the heat diffusion gets faster, with higher thermal conductivities. The influence of the thermal radiation on temperature is depicted in Fig. 9. It is interesting to note that thermal radiation has a major influence on the temperature distribution in the fluid. We observed that the fluid temperature increases by increasing thermal radiation. This is due to
the fact that increase in the values of the thermal radiation parameter increases radiation in the boundary layer, and hence increases the values of the temperature profiles in the thermal boundary layer. In Table 1 the dimensionless velocity gradient on the sheet is approximated for various values of N and k. We observed that skin friction coefficient is reduced by sufficiently large values of N and k. In Table 2 numerical values
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002
6
A. Ara et al. Table 2 Numerical values of temperature gradient h0 (0) for different values of N, k, Pr, S and R. N
k
Pr
R
S
h0 (0)
0 0.2 0.4 0.6 0.2
0.2
0.1
0.5
2.4
0.20120 0.21648 0.21647 0.21620 0.13754 0.13775 0.21682 0.24191 0.38037 0.81102 0.13764 0.13271 0.13045 0.21648 0.20475 0.19935
0.1 0.3 0.5 0.2
0.2 0.5 01 0.1
0.5 01 1.5 0.5
2.4 2.6 2.8
of surface heat transfer h0 (0) have been computed for different values of Pr, R, k, S and N. It is apparent that the magnitude of surface heat transfer is an increasing function of Pr and k, while decreasing one for the different values of S, N and R. 5. Conclusions The boundary layer flow and heat transfer over an exponentially shrinking sheet have been studied. Series solutions are obtained for the velocity and temperature profiles. It is observed that velocity increases with an increase in mass suction S. In addition, for the temperature profiles opposite behavior is seen for increment in suction. Moreover, the thermal boundary layer thickness decreases due to increase in Prandtl number Pr and thermal radiation R. References [1] Crane LJ. Flow past a stretching plate. Z Angew Math Phys 1970;21:645–7. [2] Gupta PS, S Gupta A. Heat and mass transfer on a stretching sheet with suction or blowing. J Chem Eng 1977;55:744–6. [3] Chen CK, Char MI. Heat transfer of a continuous, stretching surface with suction or blowing. J Maths Anal Appl 1988;135: 568–80. [4] Hayat T, Awais M, Safdar A, Hendi AA. Unsteady three dimensional flow of couple stress fluid over a stretching surface with chemical reaction. Nonlinear Anal: Mod Contr 2012;17(1): 47–59. [5] Hamad MAA. Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Int Commun Heat Mass Trans 2011;38:487–92. [6] Bachok N, Ishak A, Pop I. Stagnation-point flow over a stretching/shrinking sheet in a nanofluid. Nanoscale Res Let 2011;6:623. [7] Bhattacharyya K. Heat transfer analysis in unsteady boundary layer stagnation-point flow towards a shrinking/stretching sheet. Al Shams Eng J 2012.
[8] Nadeem S, Ul Haq Rizwan, Khan ZH. Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles. J Taiwan Inst Chem Eng 2013;45: 121–6. [9] Mehmood R, Nadeem S, Akbar NS. Non-orthogonal stagnation point flow of a micropolar second grade fluid towards a stretching surface with heat transfer. J Taiwan Inst Chem Eng 2013;44: 86–595. [10] Nadeem S, Mehmood R, Akbar NS. Influence of heat transfer on the non-orthogonal stagnation point flow of a third order fluid towards a stretching surface. Heat Transfer Asian Res 2013;42: 319–26. [11] Magyari E, Keller B. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J Phys D: Appl Phys 1999;32:577–85. [12] Elbashbeshy EM. Flow and heat transfer over an exponentially stretching surface considering wall mass suction. Arch Mech 2001;53:643. [13] Ishak A. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. Sains Malays 2011;40:391–5. [14] Al Odat MQ, Damesh R, Azab TAA. Thermal boundary layer on an exponentially stretching continuous surface in the presence of magnetic field effect. Int J Appl Mech Eng 2006;11:289. [15] Sajid M, Hayat T. Influence of thermal radiation on the boundary layer flow past an exponentially stretching. Int Commun Heat Mass Transfer 2008;35:347. [16] Bhattacharyya K. Boundary layer and heat transfer over an exponentially shrinking sheet. Chin Phys Lett 2011;28(7):074701. [17] Powell RE, Eyring H. Mechanisms for the relaxation theory of viscosity. Nature 1944;154:427–8. [18] Hayat T, Iqbal Z, Qasim M, Obaidat S. Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions. Int J Heat Mass Transfer 2012;55:1817–22. [19] Malik MY, Hussain A, Nadeem S. Boundary layer flow of an Eyring–Powell model fluid due to a stretching cylinder with variable viscosity. Scientia Iranica, Trans B: Mech Eng 2013;20(2): 313–21. [20] Eldabe NTM, Sallam SN, Abou-zeid MY. Numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD non-Newtonian fluid flow through a porous medium. J Egy Math Soc 2012;20:139–51. [21] Javed T, Abbas NAliZ, Sajid M. Flow of an Eyring–Powell nonnewtonian fluid over a stretching sheet. Chem Eng Commun 2013;200:327–36. [22] Akbar NS, Nadeem S. Characteristics of heating scheme and mass transfer on the peristaltic flow for an Eyring–Powell fluid in an endoscope. Int J Heat Mass Transfer 2012;55:375–83. [23] Khan NA, Riaz F. Off centered stagnation point flow of a couple stress fluid towards a rotating disk. Sci World J 2013 [Art. ID: 163585]. [24] Khan NA, Riaz F, Sultan F. Effects of chemical reaction and magnetic field on a couple stress fluid over a non-linearly stretching sheet. Eur J Phys – Plus 2014:18, 129. [25] Khan NA, Jamil M, Khan Nadeem A. Effects of slip factors on unsteady stagnation point flow and heat transfer towards a stretching sheet: an analytical study. Heat Transfer Res 2012;43(8):779–94. [26] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. ChapmanHall/CRC; 2002. [27] Liao SJ. Homotopy analysis method in non-linear differential equations. Springer; 2011. [28] Steff JF. Rheological methods in food process engineering. 2nd ed. East Lansing (Mich): Freeman Press; 1996.
Please cite this article in press as: Ara A et al., Radiation effect on boundary layer flow of an Eyring–Powell fluid over an exponentially shrinking sheet, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.06.002