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Unsteady stagnation point flow of second grade fluid with variable free stream
Alexandria Engineering Journal (2014) xxx, xxx–xxx
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Unsteady stagnation point flow of second grade fluid with variable free stream a,b
T. Hayat a b c
, M. Qasim c, S.A. Shehzad
a,*
, A. Alsaedi
b
Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, Comsats Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000, Pakistan
Received 5 November 2013; revised 13 February 2014; accepted 23 February 2014
KEYWORDS Unsteady permeable stretching sheet; Second grade fluid; Stagnation point flow; Heat transfer
Abstract This article discusses the stagnation-point flow of second grade fluid over an unsteady stretching surface in the presence of variable free stream. Flow analysis has been carried out with heat transfer analysis. The resulting partial differential equations have been converted into ordinary differential equations by employing the suitable transformations. Computations of dimensionless velocity and temperature fields have been performed by using homotopy analysis method (HAM). Graphs are plotted to examine the behaviors of arising physical parameters on the dimensionless velocity and temperature. Numerical values of skin-friction coefficient and local Nusselt number are computed and examined. ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
1. Introduction The range of non-Newtonian fluids is very large because of their occurrence in the engineering and industrial processes. Non-Newtonian fluids have been investigated by the several researchers under various conditions (see [1–10]). Such fluids are specifically quite common in the process of manufacturing coated sheets, foods, optical fibers, drilling muds, plastic polymers, etc. It is well known that all the non-Newtonian fluids * Corresponding author. Tel./fax: +92 51 90642172. E-mail address: [email protected] (S.A. Shehzad). Peer review under responsibility of Faculty of Engineering, Alexandria University.
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cannot be described by a single constitutive relationship in view of their diverse characteristics. Hence several models of non-Newtonian fluids have been suggested. The governing equations in the non-Newtonian fluids in general are much complicated, more nonlinear and higher order than the Navier–Stokes equations. The non-Newtonian fluids have been mainly classified into three types which are called the differential, the rate and the integral. Out of these, the differential type fluids have been attracted much by the researchers. A simplest subclass of differential type model is called a second grade fluid which we aim to study here. The stagnation point flow and heat transfer over a stretching sheet is further important in the process of polymer extrusion, paper production, insulating materials, glass drawing, continuos casting, fine fiber matts and many others. Considerable progress has been made regarding the stretching and stagnation point flows in the past. Chiam [11] studied the two-dimensional stagnation-point flow of a viscous fluid
1110-0168 ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. http://dx.doi.org/10.1016/j.aej.2014.02.004 Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
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T. Hayat et al.
toward a linear stretching surface. Mahapatra and Gupta [12] analyzed the effects of heat transfer in the stagnation point flow toward a stretching surface. The steady stagnation point flow of an incompressible micropolar fluid over a stretching surface is studied by Nazar et al. [13]. Sadeghy et al. [14] numerically studied the stagnation point flow of an upper convected Maxwell fluid. Ishak et al. [15] investigated the mixed convection stagnation point flow of an incompressible viscous fluid toward a vertical permeable stretching sheet. The effect of thermal radiation on the mixed convection boundary layer magnetohydrodynamic stagnation point flow in a porous space was investigated by Hayat et al. [16]. Mahapatra et al. [17] discussed the steady two-dimensional oblique stagnation point flow of an incompressible viscoelastic fluid toward a stretching sheet. Labropulu and Li [18] examined the steady two dimensional stagnation point flow in the presence of slip effects. Literature survey depicts that much attention has been given to the stretching flows in steady situations. There are very few attempts which shed light on the time-dependent stretching flow problems [19–24]. Sharma and Singh [25] numerically studied the flow about a stagnation point over an unsteady stretching sheet in the presence of variable free stream. The purpose of present study is to investigate the stagnation point flow of a second grade fluid in the presence of variable free stream. The paper is structured as follows. The problem is formulated in section two. Sections three and four deal with the solutions by the homotopy analysis technique [26–33] and their convergence respectively. Results and discussion are given in section five. Section six consists of concluding remarks. 2. Mathematical formulation
Fig. 1
Physical model.
where Vw represents the mass transfer at surface with Vw > 0 for injection and Vw < 0 for suction and v0 is a constant measuring the strength of applied suction/injection. Further the stretching velocity Uw ðx; tÞ and surface temperature Tw ðx; tÞ are taken as follows cx Uw ðx; tÞ ¼ ; 1 et bx ; Tw ðx; tÞ ¼ T1 þ 1 et ax Uðx; tÞ ¼ ð6Þ 1 et in which a; c and e are the constants with a > 0 and e P 0 ðwith et < 1Þ and both a and e have dimension time-1. We introduce the following transformations [23]: rffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Uw T T1 y; w ¼ mxUw fðgÞ; hðgÞ ¼ g¼ ; ð7Þ mx Tw T1 and the velocity components
Consider the unsteady stagnation point flow of an incompressible second grade fluid over a porous stretching surface with variable free stream. We select x-axis along the surface and y-axis normal to it. In addition, the heat transfer is considered. The boundary layer equations which can govern the present flow problem are @u @v þ ¼ 0; @x @y
ð1Þ 2
3
3
3
@ u @ u þ u @x@y 2 @t@y2 @u @u @u @U @U @ 2 u a1 5 ð2Þ þu þv ¼ þU þm 2 þ 4 2 2 q þ @u @ u þ @u @ v þ v @ 3 u @t @x @y @t @x @y @x @y2
@y @y2
@T @T @T @2T qcp þu þv ¼k 2 ; @t @x @y @y
@y3
ð3Þ
where u and v being the velocity components along the x- and y-axes, U the free stream velocity, a1 the second grade parameter, q the fluid density, m the kinematic viscosity, T the fluid temperature, q the fluid density and cp the specific heat. The associated boundary conditions of the present flow analysis are (see Fig. 1): u ¼ Uw ðx; tÞ; v ¼ Vw ðx; tÞ; T ¼ Tw ðx; tÞ at y ¼ 0; u ! Uðx; tÞ ¼; T ! T1 as y ! 1:
ð4Þ
with Vw defined Vw ¼
v0 ð1 etÞ1=2
;
ð5Þ
u¼
@w @w v¼ ; @y @x
ð8Þ
where w is a stream function. The continuity equation is identically satisfied and the resulting problems for f and h long with the boundary conditions are 1 f 0 0 0 f 02 þ ff 00 A f 0 þ g f 00 2 0000 1 0000 a2 a þ 2 þ A ¼ 0; ð9Þ þ a 2f 0 f 0 0 0 f 002 ff þ A 2f 000 þ gf c 2 c 1 00 0 0 ð10Þ h þ Prðfh f 0 hÞ PrA h þ gh ; 2 fð0Þ ¼ S; f 0 ð0Þ ¼ 1; f 0 ð1Þ ! a=c; hð0Þ ¼ 1; hð1Þ ! 0: ð11Þ Here A ¼ e=c is the unsteadiness parameter, a ¼ ca1 =lð1 etÞ lc the dimensionless second grade parameter, Pr ¼ kp the Prandtl number and primes indicate the differentiation with respect to g. The skin friction coefficient Cf and local Nusselt number Nux are defined below sw Cf ¼ 2 ; quw xqw ; ð12Þ Nu ¼ kðTw T1 Þ where the skin-friction sw and wall heat flux qw are defined as [33]:
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
Unsteady stagnation point flow of second grade fluid with variable free stream sw ¼ l
@u þ a1 @y
qw ¼ k
@2u @u @u @2u @2u þ2 þu þv 2 ; @y@t @x @y @x@y @y y¼0
@T : @y y¼0
ð13Þ
ð14Þ
The skin-friction coefficient and the local Nusselt number in dimensionless forms can be written as A Re1=2 Cf ¼ f 00 ðgÞþa 3f 0 ðgÞf 00 ðgÞfðgÞf 000 ðgÞþ ð3f 00 ðgÞþgf 000 ðgÞÞ ; x 2 g¼0
ð15Þ
R1=2 Nux ¼ h 0 ð0Þ: ex
ð16Þ
3. Solution expressions For the HAM solutions we consider fðgÞ and hðgÞ by the set of base functions fgk expðngÞjk P 0; n P 0g
ð17Þ
and write
3
!2 3^ 2^ ^ pÞ @ @ fðg; pÞ fðg; ^ pÞ ¼ ^ pÞ @ fðg; pÞ N f ½fðg; þ fðg; 3 @g @g @g2 ! ^ pÞ @ fðg; ^ pÞ g @ 2 fðg; a2 a A þ þ 2 þA c 2 @g2 c @g 2 !2 ! 2^ 3^ 4^ fðg; pÞ fðg; pÞ fðg; pÞ @ @ 1 @ A 2 þ g þ a4 @g2 @g3 @g4 2 # ^ pÞ @ 3 fðg; ^ pÞ @ fðg; ^ pÞ @ 4 fðg; ^ pÞ @ fðg; þ2 ; 3 @g @g @g @g4 hðg; pÞ @2^ ^ pÞ; ^ hðg; pÞ ¼ N h ½fðg; @g2 ! ^ ^ pÞ @ fðg; ^ pÞ @ hðg; pÞ ^ hðg; pÞ þ Pr fðg; @g @g ! 1 @^ hðg; pÞ hðg; pÞ þ g : PrA ^ 2 @g
ð18Þ
^ 0Þ ¼ f0 ðgÞ; fðg; ^ 1Þ ¼ fðgÞ; fðg; ^ hðg; 1Þ ¼ hðgÞ; hðg; 0Þ ¼ h0 ðgÞ; ^
ð19Þ
^ pÞ and ^ and when p increases from 0 to 1 then fðg; hðg; pÞ deforms from f0 ðgÞ; h0 ðgÞ to fðgÞ and hðgÞ respectively. Expand^ pÞ and ^ ing fðg; hðg; pÞ one can write
n¼0 k¼0 1 X 1 X bkm;n gk expðngÞ; hm ðgÞ ¼ n¼0 k¼0
in which am;n and bm;n are the coefficients. The initial guesses f0 and h0 and the auxiliary linear operators are taken in the following forms: a a f0 ðgÞ ¼ g þ 1 ½ð1 expðgÞ; ð20Þ c c
^ pÞ ¼ f0 ðgÞ þ fðg;
1 X
h0 ðgÞ ¼ expðgÞ:
ð21Þ
d3 f df Lf ¼ 3 ; dg dg
ð22Þ
d2 h h: dg2
ð23Þ
^ hðg; pÞ ¼ h0 ðgÞ þ
ð34Þ
hm ðgÞpm ;
ð35Þ
; p¼0
1 @m^ hðg; pÞ hm ðgÞ ¼ m! @pm
:
ð36Þ
p¼0
The auxiliary parameters hf and hh have been chosen in such a way that the series (34) and (35) converge at p ¼ 1. By setting p ¼ 0, we have 1 X fm ðgÞ;
Note that the above operators satisfy the following properties Lf ½C1 þ C2 expðgÞ þ C3 expðgÞ ¼ 0;
ð24Þ
hðgÞ ¼ h0 ðgÞ þ
Lh ½C4 expðgÞ þ C5 expðgÞ ¼ 0;
ð25Þ
ð37Þ
m¼1 1 X
hm ðgÞ:
ð38Þ
m¼1
in which Ci ði ¼ 1 5Þ are the arbitrary constants. If p 2 ½0; 1 is an embedding parameter, hf and hh the nonzero auxiliary parameters then the zeroth-order deformation problems can be written as ^ pÞ f0 ðgÞ ¼ phf N f ½ fðg; ^ pÞ; ^hðg; pÞ; ð1 pÞLf ½ fðg;
ð26Þ
^ pÞ; ^hðg; pÞ; ð1 pÞLh ½^hðg; pÞ h0 ðgÞ ¼ phh N h ½ fðg;
ð27Þ
^ pÞ @ fðg; ^ fðg; pÞjg¼0 ¼ S; @g
ð28Þ
^ hðg; pÞjg¼0 ¼ 1; ^hðg; pÞjg¼1 ¼ 0;
ð33Þ
fm ðgÞpm ;
^ pÞ 1 @ m fðg; fm ðgÞ ¼ m! @pm
fðgÞ ¼ f0 ðgÞ þ
g¼0
ð32Þ
m¼1 1 X m¼1
^ pÞ @ fðg; ¼ 1; @g
ð31Þ
For p ¼ 0 and p ¼ 1, we have
1 X 1 X fm ðgÞ ¼ akm;n gk expðngÞ;
Lh ¼
ð30Þ
¼ a=c; g¼1
ð29Þ
The mth order deformation problems can be constructed as follows Lf ½fm ðgÞ vm fm1 ðgÞ ¼ hf Rfm ðgÞ; Lh ½hm ðgÞ v hm1 ðgÞ ¼ hh Rh ðgÞ; m m fm ð0Þ ¼ 0; fm00 ð0Þ ¼ 0;fm00 ð1Þ ¼ 0;
ð39Þ ð40Þ
hm ð0Þ ¼ 0;
ð41Þ hm ð1Þ ¼ 0; /m ð0Þ ¼ 0; /m ð1Þ ¼ 0; 1 1 0000 000 00 00 000 þ aA 2fm1 Rfm ðgÞ ¼ fm1 A fm1 þ gfm1 þ gfm1 2 2 " # 2 00 m1 X fm1k fk 00 fm1k fk 00 a a
00 2 þ A ð1 vm Þ þ ; 0000 00 c c þa 2fm1k fk0 0 0 fm1k fk 00 fm1k fk k¼0
1 00 00 Rhm ðgÞ ¼ hm1 A Pr hm1 þ ghm1 2 m1 X
00 fm1k hk 00 fm1k hk þ Pr
ð42Þ
ð43Þ
k¼0
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T. Hayat et al.
0; m 6 1 vm ¼ : 1; m > 1
ð44Þ
Table 1 Convergence of HAM solutions for different order of approximations when a ¼ 0:2; A ¼ 0:3; S ¼ 0:5; Pr ¼ 1:0 and a=c ¼ 0:2.
ð45Þ ð46Þ
Order of convergence
f 00 ð0Þ
h 0 ð0Þ
1 5 10 15 20 25 30
0.979200 0.959231 0.959203 0.959204 0.959204 0.959204 0.959204
1.156667 1.459734 1.458497 1.458505 1.458503 1.458503 1.458503
The general solutions of Eqs. (39)–(41) are fm ðgÞ ¼ fm ðgÞ þ C1 þ C2 expðgÞ þ C3 expðgÞ; hm ðgÞ ¼ h m ðgÞ þ C4 expðgÞ þ C5 expðgÞ; in which
f m ðgÞ
and
h m ðgÞ
C2 ¼ C4 ¼ 0; C1 ¼ C3 fm ð0Þ; C3 ¼
are the special solutions and
@fm ðgÞ ; @g g¼0
C5 ¼ hm ð0Þ:
ð47Þ
4. Convergence of the derived series solutions It can be clearly seen that the series solutions (37) and (38) contain the non-zero auxiliary parameters hf and hh . These parameters are useful in adjusting and controlling the convergence of developed solutions. For the admissible values of hf and hh of the functions f 00 ð0Þ and h 00ð0Þ , the hf and hh -curves are plotted for 15 th-order of approximations. Fig. 2 clearly indicates that the range for the admissible values of hf and hh is 1:25 6 hf 6 0:25; 1:30 6 hh 6 0:3. The series given by (37) and (38) converge in the whole region of g when hf ¼ hh ¼ 0:7 (see Table 1). 5. Results and discussion In this section, our main interest is to discuss the variation of the emerging parameters such as a=c; a; A; S and Pr on the dimensionless velocity components, temperature field, the skin friction coefficient and the local Nusselt number. The analysis of such variation is made by Figs. 3–14. Fig. 3 shows the effects of a=c on the velocity component f 0 . When a=c ¼ 0 then there is no stagnation point flow. Velocity f 0 increases when the parameter a=c increases. It is noticed that the flow has a boundary layer structure for values of a=c > 1 and thickness of boundary layer decreases with an increase in a=c. Further Fig. 3 clearly depicts that when the stretching velocity of the surface is greater than the stagnation velocity of the external stream (i.e. a=c < 1), the flow has inverted boundary layer
Fig. 2 The h-curves of the functions f00 ð0Þ and h0 ð0Þ at 15th order of HAM approximation.
Fig. 3
Influence of a=c on f 0 .
structure. Fig. 4 is drawn for the several values of a and a=c when S ¼ 1:0 and A ¼ 0:3. It is seen that the velocity f 0 is greater for second grade fluid when compared to a viscous fluid. Figs. 5 and 6 depict the effects of a on f 0 in suction and injection cases, respectively. In both cases, the velocity profile is increased with an increase in a. It is also seen that the boundary layer thickness is enhanced for the increasing values of a. However, in injection case such increase is more pronounced than the suction case. The behaviors of A and S on the velocity profile are seen in Figs. 7 and 8. Both A and
Fig. 4
Influence of a and a=c on f0 .
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
Unsteady stagnation point flow of second grade fluid with variable free stream
Fig. 5
Influence of a on f 0 for suction.
5
Fig. 8
Influence of S on f 0 . α=0.2, S=1.0, A=0.3
Fig. 9 Fig. 6
Influence of a on f 0 for injection.
α=0.2, A=0.3, a/c=0.3
Fig. 10
Fig. 7
Influence of A on f 0 .
S decrease the velocity profile. Sucking fluid particles through porous wall reduce the growth of the fluid boundary layer. This is in accordance with the fact that suction causes
Influence of a=c on f.
Influence of S on f.
reduction in the boundary layer thickness. Fig. 9 is plotted to examine the variation in vertical component of velocity f. We have seen that f increases by increasing a=c. Fig. 10 presents the effect of S on f. It is also found that f increases when S increases. The effects of a=c and S on the temperature field are similar in a qualitative way (see Figs. 11 and 12). The variation of Pr on the temperature field is sketched in Figs. 13 and 14 for suction and injection cases, respectively. As expected h is
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
6
T. Hayat et al. α=0.2, S=1.0, A=0.3, Pr=1.0
α=0.2, a/c=0.3, A=0.3, S=−1.0
Pr Pr Pr Pr
Fig. 11
Influence of a=c on h.
Influence of Pr on h for suction.
Table 2 Values of the skin friction coefficient Re1=2 Cf and x Nux for some values of the local Nusselt number Re1=2 x a; A; S and a=c when Pr ¼ 1:0.
α=0.2 ,a/c=0.3, A=0.3, Pr=1.0
Fig. 12
Fig. 14
a
A
S
a=c
Re1=2 Cf x
Rex1=2 Nux
0.0 0.1 0.2 0.1
0.3
1.0
0.2
1.0
0.2
0.1
0.2 0.4 0.7 1.0 0.5
0.3
0.1
0.5
0.0 0.5 1.0 1.0
1.469456 1.627582 1.792047 1.602816 1.652567 1.728709 1.806307 1.172283 1.341659 1.506370 1.968956 1.833141 1.677760 1.506372
1.753764 1.782792 1.801183 1.757819 1.807621 1.881074 1.952925 1.237817 1.522888 1.855564 1.785723 1.808871 1.832274 1.855564
Influence of a=c on h.
0.0 0.1 0.2 0.3
α=0.2, a/c=0.3, A=0.3 ,S=1.0 Pr Pr Pr Pr
Fig. 13
Influence of Pr on h for suction.
decreasing when Pr increases. However, such decrease is larger for the suction when compared with the injection case (Figs. 14). From Table 2 it is noticed that the magnitude of the skin-friction coefficient increases for larger values of a, A; a=c and S. The values of skin-friction coefficient increase by increasing A. We found that for a fixed values of other parameters, the local Nusselt number increases when the values of a increases. Table 3 shows the comparison values with
Table 3 Comparison values of Re1=2 Nux with [23,34,35] for x the different values of Pr when a ¼ A ¼ a=c ¼ S ¼ 0:0. Pr
[23] Rex1=2 Nux
[34] Rex1=2 Nux
[35] Rex1=2 Nux
Present study Re1=2 Nux x
0.01 0.72 1.0 3.0
0.02944 1.08855 1.33334 2.50971
0.0294 1.0885 1.3333 2.5097
0.02942 1.08853 1.33334 2.50972
0.029425 1.088575 1.33327 2.509465
the previous published results in a limiting case. One can see that our present results have an excellent agreement with the previous published results. 6. Concluding remarks Here we studied the stagnation point flow of a second fluid with heat transfer analysis in the presence of variable free stream. The governing nonlinear problem has been computed via homotopy analysis method. The main points of this investigation can be summarized as follows:
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
Unsteady stagnation point flow of second grade fluid with variable free stream The velocity component f 0 is a decreasing function of a=c < 1. The effects of A and S on the velocity profile f 0 are similar in a qualitative sense. The velocity f 0 increases when a increases. The momentum boundary layer thickness is a decreasing function of A. Both f 0 and h are decreasing functions of S. The temperature h yields a decrease when Pr increases. The local Nusselt number is an increasing function of a; S and a=c.
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[16] T. Hayat, Z. Abbas, I. Pop, S. Asghar, Effects of radiation and magnetic field on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium, Int. J. Heat Mass Transfer 53 (2010) 466–474. [17] T.R. Mahapatra, S. Dholey, A.S. Gupta, Oblique stagnation point flow of an incompressible viscoelastic fluid towards a stretching surface, Int. J. Non-Linear Mech. 42 (2007) 484–499. [18] F. Labropulu, D. Li, Stagnation-point flow of a second grade fluid with slip, Int. J. Non-Linear Mech. 43 (2008) 941–947. [19] A. Ishak, R. Nazar, I. Pop, Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature, Nonlinear Anal.: Real World Appl. 10 (2009) 2909–2913. [20] R. Tsai, K.H. Huang, J.S. Huang, Flow and heat transfer over an unsteady stretching surface with non-uniform heat source, Int. Commun. Heat Mass Transfer 35 (2008) 1340–1343. [21] T. Hayat, M. Qasim, Z. Abbas, Radiation and mass transfer effects on the magnetohydrodynamic unsteady flow induced by a stretching sheet, Z. Naturforsch. 65 (2010) 231–239. [22] S. Mukhopadhyay, H.I. Andersson, Effects of slip and heat transfer analysis of flow over an unsteady stretching surface, Heat Mass Transfer 45 (2009) 1447–1452. [23] S. Mukhopadhyay, Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in a porous medium, Int. J. Heat Mass Transfer 52 (2009) 3261–3265. [24] T. Hayat, M. Qasim, Radiation and magnetic filed effects on the unsteady mixed convection flow of a second grade fluid over a vertical stretching sheet, Int. J. Numer. Meth. Fluids. doi:10.002/fld.2285. [25] P.R. Sharma, G. Singh, Unsteady flow about a stagnation point on a stretching sheet in presence of variable free stream, Thammasat Int. J. Sci. Technol. 13 (2008) 11–16. [26] S.J. Liao, The Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems, Ph.D. Dissertation, Shanghai Jiao Tong University, Shanghai, 1992 (in English). [27] T. Hayat, M. Qasim, S. Mesloub, MHD flow and heat transfer over permeable stretching sheet with slip conditions. Int. J. Numer. Meth. Fluids. http://dx.doi.org/10.1002/fld.2294. [28] F. Shao-Dong, C. Li-Qun, Homotopy analysis approach to periodic solutions of a nonlinear jerk equation, Chinese Phys. Lett. 26 (2009). [29] T. Hayat, M. Qasim, Z. Abbas, Homotopy solution for unsteady three-dimensional MHD flow and mass transfer in a porous space, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2375–2387. [30] H. Vosughi, E. Shivanian, S. Abbasbandy, A new analytical technique to solve Volterra’s integral equations, Math. Methods Appl. Sci. 34 (2011) 1243–1253. [31] M.M. Rashidi, S.A.M. Pour, Analytic approximate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis method, Nonlinear Anal.: Model. Control 15 (2010) 83–95. [32] T. Hayat, S.A. Shehzad, M. Qasim, S. Obaidat, Steady flow of Maxwell fluid with convective boundary conditions, Z. Naturforsch. 66a (2011) 417–422. [33] Z. Abbas, T. Hayat, M. Sajid, S. Asghar, Unsteady flow of a second grade fluid film over an unsteady stretching sheet, Math. Comput. Model. 48 (2008) 518–526. [34] L.J. Grubka, K.M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, J. Heat Trans. – Trans. ASME 107 (1985) 248–250. [35] C.H. Chen, Laminar mixed convection adjacent to vertical, continuously stretching sheets, Heat Mass Transfer 33 (1998) 471–476.
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Unsteady stagnation point flow of second grade fluid with variable free stream a,b
T. Hayat a b c
, M. Qasim c, S.A. Shehzad
a,*
, A. Alsaedi
b
Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, Comsats Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000, Pakistan
Received 5 November 2013; revised 13 February 2014; accepted 23 February 2014
KEYWORDS Unsteady permeable stretching sheet; Second grade fluid; Stagnation point flow; Heat transfer
Abstract This article discusses the stagnation-point flow of second grade fluid over an unsteady stretching surface in the presence of variable free stream. Flow analysis has been carried out with heat transfer analysis. The resulting partial differential equations have been converted into ordinary differential equations by employing the suitable transformations. Computations of dimensionless velocity and temperature fields have been performed by using homotopy analysis method (HAM). Graphs are plotted to examine the behaviors of arising physical parameters on the dimensionless velocity and temperature. Numerical values of skin-friction coefficient and local Nusselt number are computed and examined. ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
1. Introduction The range of non-Newtonian fluids is very large because of their occurrence in the engineering and industrial processes. Non-Newtonian fluids have been investigated by the several researchers under various conditions (see [1–10]). Such fluids are specifically quite common in the process of manufacturing coated sheets, foods, optical fibers, drilling muds, plastic polymers, etc. It is well known that all the non-Newtonian fluids * Corresponding author. Tel./fax: +92 51 90642172. E-mail address: [email protected] (S.A. Shehzad). Peer review under responsibility of Faculty of Engineering, Alexandria University.
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cannot be described by a single constitutive relationship in view of their diverse characteristics. Hence several models of non-Newtonian fluids have been suggested. The governing equations in the non-Newtonian fluids in general are much complicated, more nonlinear and higher order than the Navier–Stokes equations. The non-Newtonian fluids have been mainly classified into three types which are called the differential, the rate and the integral. Out of these, the differential type fluids have been attracted much by the researchers. A simplest subclass of differential type model is called a second grade fluid which we aim to study here. The stagnation point flow and heat transfer over a stretching sheet is further important in the process of polymer extrusion, paper production, insulating materials, glass drawing, continuos casting, fine fiber matts and many others. Considerable progress has been made regarding the stretching and stagnation point flows in the past. Chiam [11] studied the two-dimensional stagnation-point flow of a viscous fluid
1110-0168 ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. http://dx.doi.org/10.1016/j.aej.2014.02.004 Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
2
T. Hayat et al.
toward a linear stretching surface. Mahapatra and Gupta [12] analyzed the effects of heat transfer in the stagnation point flow toward a stretching surface. The steady stagnation point flow of an incompressible micropolar fluid over a stretching surface is studied by Nazar et al. [13]. Sadeghy et al. [14] numerically studied the stagnation point flow of an upper convected Maxwell fluid. Ishak et al. [15] investigated the mixed convection stagnation point flow of an incompressible viscous fluid toward a vertical permeable stretching sheet. The effect of thermal radiation on the mixed convection boundary layer magnetohydrodynamic stagnation point flow in a porous space was investigated by Hayat et al. [16]. Mahapatra et al. [17] discussed the steady two-dimensional oblique stagnation point flow of an incompressible viscoelastic fluid toward a stretching sheet. Labropulu and Li [18] examined the steady two dimensional stagnation point flow in the presence of slip effects. Literature survey depicts that much attention has been given to the stretching flows in steady situations. There are very few attempts which shed light on the time-dependent stretching flow problems [19–24]. Sharma and Singh [25] numerically studied the flow about a stagnation point over an unsteady stretching sheet in the presence of variable free stream. The purpose of present study is to investigate the stagnation point flow of a second grade fluid in the presence of variable free stream. The paper is structured as follows. The problem is formulated in section two. Sections three and four deal with the solutions by the homotopy analysis technique [26–33] and their convergence respectively. Results and discussion are given in section five. Section six consists of concluding remarks. 2. Mathematical formulation
Fig. 1
Physical model.
where Vw represents the mass transfer at surface with Vw > 0 for injection and Vw < 0 for suction and v0 is a constant measuring the strength of applied suction/injection. Further the stretching velocity Uw ðx; tÞ and surface temperature Tw ðx; tÞ are taken as follows cx Uw ðx; tÞ ¼ ; 1 et bx ; Tw ðx; tÞ ¼ T1 þ 1 et ax Uðx; tÞ ¼ ð6Þ 1 et in which a; c and e are the constants with a > 0 and e P 0 ðwith et < 1Þ and both a and e have dimension time-1. We introduce the following transformations [23]: rffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Uw T T1 y; w ¼ mxUw fðgÞ; hðgÞ ¼ g¼ ; ð7Þ mx Tw T1 and the velocity components
Consider the unsteady stagnation point flow of an incompressible second grade fluid over a porous stretching surface with variable free stream. We select x-axis along the surface and y-axis normal to it. In addition, the heat transfer is considered. The boundary layer equations which can govern the present flow problem are @u @v þ ¼ 0; @x @y
ð1Þ 2
3
3
3
@ u @ u þ u @x@y 2 @t@y2 @u @u @u @U @U @ 2 u a1 5 ð2Þ þu þv ¼ þU þm 2 þ 4 2 2 q þ @u @ u þ @u @ v þ v @ 3 u @t @x @y @t @x @y @x @y2
@y @y2
@T @T @T @2T qcp þu þv ¼k 2 ; @t @x @y @y
@y3
ð3Þ
where u and v being the velocity components along the x- and y-axes, U the free stream velocity, a1 the second grade parameter, q the fluid density, m the kinematic viscosity, T the fluid temperature, q the fluid density and cp the specific heat. The associated boundary conditions of the present flow analysis are (see Fig. 1): u ¼ Uw ðx; tÞ; v ¼ Vw ðx; tÞ; T ¼ Tw ðx; tÞ at y ¼ 0; u ! Uðx; tÞ ¼; T ! T1 as y ! 1:
ð4Þ
with Vw defined Vw ¼
v0 ð1 etÞ1=2
;
ð5Þ
u¼
@w @w v¼ ; @y @x
ð8Þ
where w is a stream function. The continuity equation is identically satisfied and the resulting problems for f and h long with the boundary conditions are 1 f 0 0 0 f 02 þ ff 00 A f 0 þ g f 00 2 0000 1 0000 a2 a þ 2 þ A ¼ 0; ð9Þ þ a 2f 0 f 0 0 0 f 002 ff þ A 2f 000 þ gf c 2 c 1 00 0 0 ð10Þ h þ Prðfh f 0 hÞ PrA h þ gh ; 2 fð0Þ ¼ S; f 0 ð0Þ ¼ 1; f 0 ð1Þ ! a=c; hð0Þ ¼ 1; hð1Þ ! 0: ð11Þ Here A ¼ e=c is the unsteadiness parameter, a ¼ ca1 =lð1 etÞ lc the dimensionless second grade parameter, Pr ¼ kp the Prandtl number and primes indicate the differentiation with respect to g. The skin friction coefficient Cf and local Nusselt number Nux are defined below sw Cf ¼ 2 ; quw xqw ; ð12Þ Nu ¼ kðTw T1 Þ where the skin-friction sw and wall heat flux qw are defined as [33]:
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
Unsteady stagnation point flow of second grade fluid with variable free stream sw ¼ l
@u þ a1 @y
qw ¼ k
@2u @u @u @2u @2u þ2 þu þv 2 ; @y@t @x @y @x@y @y y¼0
@T : @y y¼0
ð13Þ
ð14Þ
The skin-friction coefficient and the local Nusselt number in dimensionless forms can be written as A Re1=2 Cf ¼ f 00 ðgÞþa 3f 0 ðgÞf 00 ðgÞfðgÞf 000 ðgÞþ ð3f 00 ðgÞþgf 000 ðgÞÞ ; x 2 g¼0
ð15Þ
R1=2 Nux ¼ h 0 ð0Þ: ex
ð16Þ
3. Solution expressions For the HAM solutions we consider fðgÞ and hðgÞ by the set of base functions fgk expðngÞjk P 0; n P 0g
ð17Þ
and write
3
!2 3^ 2^ ^ pÞ @ @ fðg; pÞ fðg; ^ pÞ ¼ ^ pÞ @ fðg; pÞ N f ½fðg; þ fðg; 3 @g @g @g2 ! ^ pÞ @ fðg; ^ pÞ g @ 2 fðg; a2 a A þ þ 2 þA c 2 @g2 c @g 2 !2 ! 2^ 3^ 4^ fðg; pÞ fðg; pÞ fðg; pÞ @ @ 1 @ A 2 þ g þ a4 @g2 @g3 @g4 2 # ^ pÞ @ 3 fðg; ^ pÞ @ fðg; ^ pÞ @ 4 fðg; ^ pÞ @ fðg; þ2 ; 3 @g @g @g @g4 hðg; pÞ @2^ ^ pÞ; ^ hðg; pÞ ¼ N h ½fðg; @g2 ! ^ ^ pÞ @ fðg; ^ pÞ @ hðg; pÞ ^ hðg; pÞ þ Pr fðg; @g @g ! 1 @^ hðg; pÞ hðg; pÞ þ g : PrA ^ 2 @g
ð18Þ
^ 0Þ ¼ f0 ðgÞ; fðg; ^ 1Þ ¼ fðgÞ; fðg; ^ hðg; 1Þ ¼ hðgÞ; hðg; 0Þ ¼ h0 ðgÞ; ^
ð19Þ
^ pÞ and ^ and when p increases from 0 to 1 then fðg; hðg; pÞ deforms from f0 ðgÞ; h0 ðgÞ to fðgÞ and hðgÞ respectively. Expand^ pÞ and ^ ing fðg; hðg; pÞ one can write
n¼0 k¼0 1 X 1 X bkm;n gk expðngÞ; hm ðgÞ ¼ n¼0 k¼0
in which am;n and bm;n are the coefficients. The initial guesses f0 and h0 and the auxiliary linear operators are taken in the following forms: a a f0 ðgÞ ¼ g þ 1 ½ð1 expðgÞ; ð20Þ c c
^ pÞ ¼ f0 ðgÞ þ fðg;
1 X
h0 ðgÞ ¼ expðgÞ:
ð21Þ
d3 f df Lf ¼ 3 ; dg dg
ð22Þ
d2 h h: dg2
ð23Þ
^ hðg; pÞ ¼ h0 ðgÞ þ
ð34Þ
hm ðgÞpm ;
ð35Þ
; p¼0
1 @m^ hðg; pÞ hm ðgÞ ¼ m! @pm
:
ð36Þ
p¼0
The auxiliary parameters hf and hh have been chosen in such a way that the series (34) and (35) converge at p ¼ 1. By setting p ¼ 0, we have 1 X fm ðgÞ;
Note that the above operators satisfy the following properties Lf ½C1 þ C2 expðgÞ þ C3 expðgÞ ¼ 0;
ð24Þ
hðgÞ ¼ h0 ðgÞ þ
Lh ½C4 expðgÞ þ C5 expðgÞ ¼ 0;
ð25Þ
ð37Þ
m¼1 1 X
hm ðgÞ:
ð38Þ
m¼1
in which Ci ði ¼ 1 5Þ are the arbitrary constants. If p 2 ½0; 1 is an embedding parameter, hf and hh the nonzero auxiliary parameters then the zeroth-order deformation problems can be written as ^ pÞ f0 ðgÞ ¼ phf N f ½ fðg; ^ pÞ; ^hðg; pÞ; ð1 pÞLf ½ fðg;
ð26Þ
^ pÞ; ^hðg; pÞ; ð1 pÞLh ½^hðg; pÞ h0 ðgÞ ¼ phh N h ½ fðg;
ð27Þ
^ pÞ @ fðg; ^ fðg; pÞjg¼0 ¼ S; @g
ð28Þ
^ hðg; pÞjg¼0 ¼ 1; ^hðg; pÞjg¼1 ¼ 0;
ð33Þ
fm ðgÞpm ;
^ pÞ 1 @ m fðg; fm ðgÞ ¼ m! @pm
fðgÞ ¼ f0 ðgÞ þ
g¼0
ð32Þ
m¼1 1 X m¼1
^ pÞ @ fðg; ¼ 1; @g
ð31Þ
For p ¼ 0 and p ¼ 1, we have
1 X 1 X fm ðgÞ ¼ akm;n gk expðngÞ;
Lh ¼
ð30Þ
¼ a=c; g¼1
ð29Þ
The mth order deformation problems can be constructed as follows Lf ½fm ðgÞ vm fm1 ðgÞ ¼ hf Rfm ðgÞ; Lh ½hm ðgÞ v hm1 ðgÞ ¼ hh Rh ðgÞ; m m fm ð0Þ ¼ 0; fm00 ð0Þ ¼ 0;fm00 ð1Þ ¼ 0;
ð39Þ ð40Þ
hm ð0Þ ¼ 0;
ð41Þ hm ð1Þ ¼ 0; /m ð0Þ ¼ 0; /m ð1Þ ¼ 0; 1 1 0000 000 00 00 000 þ aA 2fm1 Rfm ðgÞ ¼ fm1 A fm1 þ gfm1 þ gfm1 2 2 " # 2 00 m1 X fm1k fk 00 fm1k fk 00 a a
00 2 þ A ð1 vm Þ þ ; 0000 00 c c þa 2fm1k fk0 0 0 fm1k fk 00 fm1k fk k¼0
1 00 00 Rhm ðgÞ ¼ hm1 A Pr hm1 þ ghm1 2 m1 X
00 fm1k hk 00 fm1k hk þ Pr
ð42Þ
ð43Þ
k¼0
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T. Hayat et al.
0; m 6 1 vm ¼ : 1; m > 1
ð44Þ
Table 1 Convergence of HAM solutions for different order of approximations when a ¼ 0:2; A ¼ 0:3; S ¼ 0:5; Pr ¼ 1:0 and a=c ¼ 0:2.
ð45Þ ð46Þ
Order of convergence
f 00 ð0Þ
h 0 ð0Þ
1 5 10 15 20 25 30
0.979200 0.959231 0.959203 0.959204 0.959204 0.959204 0.959204
1.156667 1.459734 1.458497 1.458505 1.458503 1.458503 1.458503
The general solutions of Eqs. (39)–(41) are fm ðgÞ ¼ fm ðgÞ þ C1 þ C2 expðgÞ þ C3 expðgÞ; hm ðgÞ ¼ h m ðgÞ þ C4 expðgÞ þ C5 expðgÞ; in which
f m ðgÞ
and
h m ðgÞ
C2 ¼ C4 ¼ 0; C1 ¼ C3 fm ð0Þ; C3 ¼
are the special solutions and
@fm ðgÞ ; @g g¼0
C5 ¼ hm ð0Þ:
ð47Þ
4. Convergence of the derived series solutions It can be clearly seen that the series solutions (37) and (38) contain the non-zero auxiliary parameters hf and hh . These parameters are useful in adjusting and controlling the convergence of developed solutions. For the admissible values of hf and hh of the functions f 00 ð0Þ and h 00ð0Þ , the hf and hh -curves are plotted for 15 th-order of approximations. Fig. 2 clearly indicates that the range for the admissible values of hf and hh is 1:25 6 hf 6 0:25; 1:30 6 hh 6 0:3. The series given by (37) and (38) converge in the whole region of g when hf ¼ hh ¼ 0:7 (see Table 1). 5. Results and discussion In this section, our main interest is to discuss the variation of the emerging parameters such as a=c; a; A; S and Pr on the dimensionless velocity components, temperature field, the skin friction coefficient and the local Nusselt number. The analysis of such variation is made by Figs. 3–14. Fig. 3 shows the effects of a=c on the velocity component f 0 . When a=c ¼ 0 then there is no stagnation point flow. Velocity f 0 increases when the parameter a=c increases. It is noticed that the flow has a boundary layer structure for values of a=c > 1 and thickness of boundary layer decreases with an increase in a=c. Further Fig. 3 clearly depicts that when the stretching velocity of the surface is greater than the stagnation velocity of the external stream (i.e. a=c < 1), the flow has inverted boundary layer
Fig. 2 The h-curves of the functions f00 ð0Þ and h0 ð0Þ at 15th order of HAM approximation.
Fig. 3
Influence of a=c on f 0 .
structure. Fig. 4 is drawn for the several values of a and a=c when S ¼ 1:0 and A ¼ 0:3. It is seen that the velocity f 0 is greater for second grade fluid when compared to a viscous fluid. Figs. 5 and 6 depict the effects of a on f 0 in suction and injection cases, respectively. In both cases, the velocity profile is increased with an increase in a. It is also seen that the boundary layer thickness is enhanced for the increasing values of a. However, in injection case such increase is more pronounced than the suction case. The behaviors of A and S on the velocity profile are seen in Figs. 7 and 8. Both A and
Fig. 4
Influence of a and a=c on f0 .
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Unsteady stagnation point flow of second grade fluid with variable free stream
Fig. 5
Influence of a on f 0 for suction.
5
Fig. 8
Influence of S on f 0 . α=0.2, S=1.0, A=0.3
Fig. 9 Fig. 6
Influence of a on f 0 for injection.
α=0.2, A=0.3, a/c=0.3
Fig. 10
Fig. 7
Influence of A on f 0 .
S decrease the velocity profile. Sucking fluid particles through porous wall reduce the growth of the fluid boundary layer. This is in accordance with the fact that suction causes
Influence of a=c on f.
Influence of S on f.
reduction in the boundary layer thickness. Fig. 9 is plotted to examine the variation in vertical component of velocity f. We have seen that f increases by increasing a=c. Fig. 10 presents the effect of S on f. It is also found that f increases when S increases. The effects of a=c and S on the temperature field are similar in a qualitative way (see Figs. 11 and 12). The variation of Pr on the temperature field is sketched in Figs. 13 and 14 for suction and injection cases, respectively. As expected h is
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
6
T. Hayat et al. α=0.2, S=1.0, A=0.3, Pr=1.0
α=0.2, a/c=0.3, A=0.3, S=−1.0
Pr Pr Pr Pr
Fig. 11
Influence of a=c on h.
Influence of Pr on h for suction.
Table 2 Values of the skin friction coefficient Re1=2 Cf and x Nux for some values of the local Nusselt number Re1=2 x a; A; S and a=c when Pr ¼ 1:0.
α=0.2 ,a/c=0.3, A=0.3, Pr=1.0
Fig. 12
Fig. 14
a
A
S
a=c
Re1=2 Cf x
Rex1=2 Nux
0.0 0.1 0.2 0.1
0.3
1.0
0.2
1.0
0.2
0.1
0.2 0.4 0.7 1.0 0.5
0.3
0.1
0.5
0.0 0.5 1.0 1.0
1.469456 1.627582 1.792047 1.602816 1.652567 1.728709 1.806307 1.172283 1.341659 1.506370 1.968956 1.833141 1.677760 1.506372
1.753764 1.782792 1.801183 1.757819 1.807621 1.881074 1.952925 1.237817 1.522888 1.855564 1.785723 1.808871 1.832274 1.855564
Influence of a=c on h.
0.0 0.1 0.2 0.3
α=0.2, a/c=0.3, A=0.3 ,S=1.0 Pr Pr Pr Pr
Fig. 13
Influence of Pr on h for suction.
decreasing when Pr increases. However, such decrease is larger for the suction when compared with the injection case (Figs. 14). From Table 2 it is noticed that the magnitude of the skin-friction coefficient increases for larger values of a, A; a=c and S. The values of skin-friction coefficient increase by increasing A. We found that for a fixed values of other parameters, the local Nusselt number increases when the values of a increases. Table 3 shows the comparison values with
Table 3 Comparison values of Re1=2 Nux with [23,34,35] for x the different values of Pr when a ¼ A ¼ a=c ¼ S ¼ 0:0. Pr
[23] Rex1=2 Nux
[34] Rex1=2 Nux
[35] Rex1=2 Nux
Present study Re1=2 Nux x
0.01 0.72 1.0 3.0
0.02944 1.08855 1.33334 2.50971
0.0294 1.0885 1.3333 2.5097
0.02942 1.08853 1.33334 2.50972
0.029425 1.088575 1.33327 2.509465
the previous published results in a limiting case. One can see that our present results have an excellent agreement with the previous published results. 6. Concluding remarks Here we studied the stagnation point flow of a second fluid with heat transfer analysis in the presence of variable free stream. The governing nonlinear problem has been computed via homotopy analysis method. The main points of this investigation can be summarized as follows:
Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004
Unsteady stagnation point flow of second grade fluid with variable free stream The velocity component f 0 is a decreasing function of a=c < 1. The effects of A and S on the velocity profile f 0 are similar in a qualitative sense. The velocity f 0 increases when a increases. The momentum boundary layer thickness is a decreasing function of A. Both f 0 and h are decreasing functions of S. The temperature h yields a decrease when Pr increases. The local Nusselt number is an increasing function of a; S and a=c.
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Please cite this article in press as: T. Hayat et al., Unsteady stagnation point flow of second grade fluid with variable free stream, Alexandria Eng. J. (2014), http://dx.doi.org/10.1016/j.aej.2014.02.004