- Email: [email protected]
Numerical solution to the flow of a second grade fluid over a stretching sheet using the method of quasi-linearization
Applied Mathematics and Computation 149 (2004) 165–173 www.elsevier.com/locate/amc
Numerical solution to the flow of a second grade fluid over a stretching sheet using the method of quasi-linearization M. Massoudi
a,*
, C.E. Maneschy
b
a
b
US Department of Energy, National Energy Technology Laboratory, P.O. Box 10940, Pittsburgh, PA, 15236-0940, USA Mechanical Engineering Department, CT- Universidade Federal do Para, 01 66075-900 Belem, PA, Brazil
Abstract In this short work we look at the flow of a second grade fluid due to a stretching sheet coinciding with the plane y ¼ 0. Two equal and opposite forces are applied along the x-axis (horizontal plane) in such a way that the origin remains fixed and the velocity of stretching is proportional to the distance from the origin. This problem was studied by Rajagopal et al. [Rheol. Acta 23 (1984) 213; Meccanica 19 (1984) 158] where a perturbation scheme was used to obtain the solution. Using the same similarity transformation given by Rajagopal et al., we solve the full equation numerically, using the method of quasi-linearization of Bellman and Kalaba [Quasilinearization and NonLinear Boundary Value Problems, American Elsevier, New York, 1965]. Results will be given for the velocity distribution and the shear stress at the wall. Ó 2003 Elsevier Inc. All rights reserved.
1. Introduction The flow of a Newtonian fluid over a stretching sheet has been studied extensively in the last few decades. This is a problem of interest when a polymer sheet is extruded continuously from a die (cf. [15]). Very often it can be
*
Corresponding author. E-mail address: [email protected] (M. Massoudi).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00963-3
166
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
assumed that the sheet is inextensible, but there are situations in which it becomes necessary to deal with a stretching sheet. Sakiadis [22] studied the boundary layer flow of a Newtonian fluid on a moving surface. The effect of suction and injection was studied later by Erickson et al. [28]. Crane [6] studied the heat conduction where the main stream velocity in the outer flow is proportional to the distance from the stagnation point. In all these studies, the fluid was assumed to be Newtonian. However, many polymers and plastic materials exhibit non-linear behavior and as such many investigators have studied this problem using various non-Newtonian fluid models. In some cases a power-law-type model has been used to characterize the fluid. One of the early attempts to extend this study to a non-Newtonian fluid was due to Fox et al. [9] where the fluid was characterized by a power-law model. Recently, Hassanien et al. [13] have studied the heat transfer characteristics of a powerlaw fluid in such a flow. One of the shortcomings of a power-law-type model is that the elastic properties of the fluid (such as normal-stress differences in shear flow) cannot be predicted. Perhaps the simplest non-Newtonian model that can represent the normal stress effects is the second grade fluid model. Siddappa and Abel [23] studied this boundary layer flow using a constitutive equation proposed by Walters [29] with suction at the surface. Later, Bujurke et al. [5] assumed cubic profile for velocity and temperature fields and used a second order fluid model. A mathematical analysis was performed by Rajagopal et al. [19,20] where they studied the flow of a second grade fluid. This model has subsequently been used by other researchers [1,7,10–12,16–18,26,27]. More complicated fluid models such as an Oldroyd-B type has also been studied by Bhatnagar et al. [4]. Rajagopal et al. [19,20] used a perturbation method to obtain numerical solution for the velocity profiles. In this paper, the same model is used and the full equation is solved numerically.
2. Equations of motion The stress in a second grade fluid is given by [21,24] T ¼ p1 þ lA1 þ a1 A2 þ a2 A21 ;
ð1Þ
where p is the pressure and l is the shear viscosity (assumed to be constant in this problem), a1 and a2 are material parameters called the normal stress moduli. The tensors A1 and A2 are given by, T
A1 ¼ grad v þ ðgrad vÞ ; A2 ¼
d T A1 þ A1 ðgrad vÞ þ ðgrad vÞ A1 ; dt
ð2Þ ð3Þ
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
167
where v is the velocity vector and d=dt is the material time derivative, which is defined as follows: d oðÞ ðÞ ¼ þ ½gradðÞv: dt ot
ð4Þ
The thermodynamics and stability of the model in Eq. (1) were studied in detail by Dunn and Fosdick [8]. Thermodynamics compatibility in the sense that all motions of the fluid meet the Clausius–Duhem inequality (generally interpreted as a statement of the second law of thermodynamics) and the assumption that the specific Helmholtz free energy of the fluid be a minimum when the fluid is locally at rest requires that (cf. [8]), l P 0;
a1 P 0;
a1 þ a2 ¼ 0:
ð5Þ
It is assumed that the fluid is modeled by Eq. (1) subject to restrictions given by (5). The flow of such a fluid over a stretching sheet coinciding with plane y ¼ 0 will be studied. The governing equations of motion are the conservation of mass and linear momentum. These are div v ¼ 0;
ð6Þ
and q
dv ¼ div T þ qb: dt
ð7Þ
The steady two-dimensional boundary layer equations for such a model were derived by Beard and Walters [2] as ou ov þ ¼ 0; ox oy u
ð8Þ
2 ou ou o2 u o ou ou o2 v o3 u þv ¼m 2þk u 2 þ þ v ; ox oy oy ox oy oy oy 2 oy 3
ð9Þ
where m ¼ l=q and k ¼ a1 =q. The boundary conditions are u ¼ cx; v ¼ 0 u!0
at y ¼ 0 c > 0;
as y ! 1:
ð10Þ ð11Þ
Since the free stream velocity is zero, the flow is caused by the stretching of the sheet. The importance of suction or injection at the wall, i.e., when v is not zero at the wall has been discussed for many interesting problems (cf. [14 or 25]). Rajagopal et al. [19,20] showed that by using the similarity transformation u ¼ cxf 0 ðgÞ;
v ¼ ðmcÞ1=2 f ðgÞ;
ð12Þ
168
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
where g¼
c 1=2 m
ð13Þ
y
the conservation of mass, Eq. (8), is automatically satisfied and Eq. (9) reduces to 2
f 02 ff 00 ¼ f 000 þ k½2f 0 f 000 ðf 00 Þ ff iv ;
ð14Þ
where prime designates differentiation with respect to g. The boundary conditions now become, f 0 ð0Þ ¼ 1;
f ð0Þ ¼ 0;
f 0 ð1Þ ¼ 0:
ð15Þ
Eq. (4) is a fourth order equation and in addition to the three boundary conditions, given by Eq. (15), one more is required. Rajagopal and Gupta [20] suggested using the condition that ou !0 oy
as y ! 1;
ð16Þ
which, because of Eq. (2), becomes f 00 ð1Þ ! 0
as g ! 1:
ð17Þ
3. Numerical method Eq. (14), subjected to the boundary conditions (15) and (17) constitutes the final differential equation to be solved. Following the quasi-linearization method developed by Bellman and Kalaba [3], Eq. (14) can be written in the form below: ! ! _ 00 _0 _0 1 1 1 2 f 2 f f iv 000 0 00 000 f f 2f_ þ f f_ k f_ k f k f_ f f 000 f 000 ; ð18Þ 2 f 00 2f 0 f 000 þ ¼ f k kf where f and f_ represent the current and previous approximation to the solution, respectively. Eq. (18) is a linear non-homogeneous equation in the unknown function f . Its solution can be written as a linear combination of two linear independent solutions of the homogeneous problem, fH1 and fH2 , and a particular solution, fp , such that f ¼ C1 fH1 þ C2 fH2 þ fp :
ð19Þ
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
169
In order to satisfy boundary conditions (15) and (17) and the linear independence requirement, the three solutions above are assumed to have the following values at y ¼ 0: 0 00 000 fH1 ; fH1 ¼ ð0; 0; 1; 0Þ; ; fH1 ; fH1 0 00 000 fH2 ; fH2 ; fH2 ; fH2 ¼ ð0; 0; 0; 1Þ; ð20Þ h i 0 00 000 fp ; fp ; fp ; fp ¼ ð0; 1; 0; 0Þ: For each of the homogeneous and particular solutions, Eq. (18) is numerically integrated by using the Runge–Kutta integration method. The values found at the end points are used to find the pair of constants C1 and C2 so that the boundary conditions at y ¼ y1 are met. Therefore, 0 0 C1 fH1 ðy1 Þ þ C2 fH2 ðy1 Þ þ fp0 ðy1 Þ ¼ 0; 00 00 C2 fH1 ðy1 Þ þ C2 fH1 ðy1 Þ þ fp00 ðy1 Þ ¼ 0:
ð21Þ
Once the values for the constants are found the function f can be obtained from Eq. (19) for all discretization points 0 ¼ y1 < < yj < < yN ¼ y1 . This process is repeated until the magnitude of the difference between two consecutive iterations falls below a prescribed tolerance, at all points. In this work, a tolerance limit of 107 was used. As the zeroth approximation to the solution, the function f0 ðyÞ ¼ y;
ð22Þ
satisfying all boundary conditions was assumed.
4. Results and discussion As a result of the similarity transformation, Eqs. (12) and (13), the only parameter of interest for the present problem is k which is related to a1 . The values of f and f 00 as a function of g are plotted in Figs. 1 and 2. These functions are related to the velocity components u and v, through Eqs. (12) and (13). It can be seen that the magnitude of the vertical velocity component increases significantly for high values of k. From Fig. 2 it is observed that f increases as k increases. The results for small values of k can be compared to the results obtained by the perturbation analysis of Rajagopal et al. [19]. The agreement is very good. A quantity of interest is the magnitude of the skin friction coefficient which is related to the shear stress at the wall ou ou ou o2 u Txy jy¼0 ¼ l þ a1 2 : ð23Þ þu oy ox oy oxoy y¼0
170
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
Fig. 1. Function f for different values of k.
Fig. 2. First derivative of f for different values of k.
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
Fig. 3. Second derivative of f for different values of k.
Fig. 4. Third derivative of f for different values of k.
171
172
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
Table 1 Values of Txy k
Txy
0.01 0.10 1.00 10.00
)1.02489 )1.23453 )2.83024 )10.21376
Figs. 3 and 4 show the variation of f 00 and f 000 with k. These functions are necessary to obtain the values of the shear stress given in Eq. (23). Some values of the shear stress at the wall are given in Table 1. References [1] P.D. Ariel, Axisymmetric flow of second grade fluid past a stretching sheet, Int. J. Engng. Sci 39 (2001) 529. [2] D.W. Beard, K. Walters, Elastico-viscous boundary layer flows, Proc. Comb. Phil. Soc. 60 (1964) 667. [3] R.E. Bellman, R.E. Kalaba, Quasilinearization and Non-Linear Boundary Value Problems, American Elsevier, New York, 1965. [4] R.K. Bhatnagar, G. Gupta, K.R. Rajagopal, Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity, Int. J. Non-Linear Mech. 30 (1995) 391. [5] N.M. Bujurke, S.N. Biradar, P. Hiremath, Second-order fluid past a stretching sheet with heat transfer, ZAMP 38 (1987) 654. [6] L.J. Crane, Flow past a stretching sheet, ZAMP 21 (1970) 645. [7] B.S. Dandapat, A.S. Gupta, Flow and heat transfer in a viscoelastic fluid over a stretching sheet, Int. J. Non-Linear Mech. 24 (1989) 215. [8] J.E. Dunn, R.L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal. 56 (1974) 191. [9] V.G. Fox, L.E. Erickson, L.T. Fan, The laminar boundary layer on a moving continuous flat sheet immersed in a non-Newtonian fluid, AIChE J. 15 (1969) 327. [10] V.K. Garg, Improved shooting techniques for linear boundary value problems, Comput. Meth. Appl. Mech. Eng. 22 (1980) 87. [11] V.K. Garg, K.R. Rajagopal, Flow of a non-Newtonian fluid past a wedge, Acta Mech. 88 (1991) 113. [12] V.K. Garg, K.R. Rajagopal, Stagnation point flow of a non-Newtonian fluid, Mech. Res. Comm. 17 (1990) 415. [13] I.A. Hassanien, A.A. Abdullah, R.S.R. Gorla, Flow and heat transfer in a power-law fluid over a non-isothermal stretching sheet, Math. Comput. Modelling 28 (1998) 105. [14] M. Massoudi, M. Ramezan, Effect of injection or suction on the Falkner–Skan flows of second grade fluids, Int. J. Non-Linear Mech. 24 (1989) 221. [15] P.D. McCormack, L. Crane, Physical Fluid Dynamics, Academic Press, New York, 1973. [16] B. Nageswara Rao, Flow of a fluid of second grade fluid over a stretching sheet, Int. J. NonLinear Mech. 31 (1996) 547. [17] G. Pontrelli, Flow of a fluid of second grade over a stretching sheet, Int. J. Non-Linear Mech. 30 (1995) 287. [18] K.R. Rajagopal, T.Y. Na, A.S. Gupta, A non-similar boundary layer on a stretching sheet in a non-Newtonian fluid with uniform free stream, J. Math. Phys. Sci. 21 (1987) 189.
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
173
[19] K.R. Rajagopal, A.S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta 23 (1984) 213. [20] K.R. Rajagopal, T.Y. Na, A.S. Gupta, An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate, Meccanica 19 (1984) 158. [21] R.S. Rivlin, J.L. Ericksen, Stress deformation relation for isotropic materials, J. Rational Mech. Anal. 4 (1995) 325. [22] B.C. Sakiadis, Boundary-layer behaviour on continuous solid surface, AIChE J. 7 (1961) 26. [23] B. Siddappa, M.S. Abel, Visco-elastic boundary layer flow past a stretching plate with suction and heat transfer, Rheol. Acta 25 (1986) 319. [24] C. Truesdell, W. Noll, Non-linear Field Theories of Mechanics, second ed., Springer-Verlag, NY, 1992. [25] S. Turbatu, H.M. Srivastava, Some boundary-layer growth problems associated with a flat plate with suction or injection, Appl. Math. Comp. 122 (2001) 365. [26] K. Vajravelu, T. Roper, Flow and heat transfer in a second grade fluid over a stretching sheet, Int. J. Non-Linear Mech. 34 (1999) 1031. [27] M. Yurusoy, M. Pakdemirli, Exact solutions of boundary layer equations of special nonNewtonian fluid over a stretching sheet, Mech. Res. Comm. 26 (1999) 171. [28] L.E. Erickson, L.T. Fan, V.G. Fox, Heat and mass transfer on a moving continuous flat plate with suction or injection, Ind. Engng. Chem. Fund. 5 (1996) 19. [29] K. Walters, The motion of an elastico-viscous liquid contained between coaxial cylinders (II), Q. J. Mech. Appl. Math. 13 (1960) 444.
Numerical solution to the flow of a second grade fluid over a stretching sheet using the method of quasi-linearization M. Massoudi
a,*
, C.E. Maneschy
b
a
b
US Department of Energy, National Energy Technology Laboratory, P.O. Box 10940, Pittsburgh, PA, 15236-0940, USA Mechanical Engineering Department, CT- Universidade Federal do Para, 01 66075-900 Belem, PA, Brazil
Abstract In this short work we look at the flow of a second grade fluid due to a stretching sheet coinciding with the plane y ¼ 0. Two equal and opposite forces are applied along the x-axis (horizontal plane) in such a way that the origin remains fixed and the velocity of stretching is proportional to the distance from the origin. This problem was studied by Rajagopal et al. [Rheol. Acta 23 (1984) 213; Meccanica 19 (1984) 158] where a perturbation scheme was used to obtain the solution. Using the same similarity transformation given by Rajagopal et al., we solve the full equation numerically, using the method of quasi-linearization of Bellman and Kalaba [Quasilinearization and NonLinear Boundary Value Problems, American Elsevier, New York, 1965]. Results will be given for the velocity distribution and the shear stress at the wall. Ó 2003 Elsevier Inc. All rights reserved.
1. Introduction The flow of a Newtonian fluid over a stretching sheet has been studied extensively in the last few decades. This is a problem of interest when a polymer sheet is extruded continuously from a die (cf. [15]). Very often it can be
*
Corresponding author. E-mail address: [email protected] (M. Massoudi).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00963-3
166
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
assumed that the sheet is inextensible, but there are situations in which it becomes necessary to deal with a stretching sheet. Sakiadis [22] studied the boundary layer flow of a Newtonian fluid on a moving surface. The effect of suction and injection was studied later by Erickson et al. [28]. Crane [6] studied the heat conduction where the main stream velocity in the outer flow is proportional to the distance from the stagnation point. In all these studies, the fluid was assumed to be Newtonian. However, many polymers and plastic materials exhibit non-linear behavior and as such many investigators have studied this problem using various non-Newtonian fluid models. In some cases a power-law-type model has been used to characterize the fluid. One of the early attempts to extend this study to a non-Newtonian fluid was due to Fox et al. [9] where the fluid was characterized by a power-law model. Recently, Hassanien et al. [13] have studied the heat transfer characteristics of a powerlaw fluid in such a flow. One of the shortcomings of a power-law-type model is that the elastic properties of the fluid (such as normal-stress differences in shear flow) cannot be predicted. Perhaps the simplest non-Newtonian model that can represent the normal stress effects is the second grade fluid model. Siddappa and Abel [23] studied this boundary layer flow using a constitutive equation proposed by Walters [29] with suction at the surface. Later, Bujurke et al. [5] assumed cubic profile for velocity and temperature fields and used a second order fluid model. A mathematical analysis was performed by Rajagopal et al. [19,20] where they studied the flow of a second grade fluid. This model has subsequently been used by other researchers [1,7,10–12,16–18,26,27]. More complicated fluid models such as an Oldroyd-B type has also been studied by Bhatnagar et al. [4]. Rajagopal et al. [19,20] used a perturbation method to obtain numerical solution for the velocity profiles. In this paper, the same model is used and the full equation is solved numerically.
2. Equations of motion The stress in a second grade fluid is given by [21,24] T ¼ p1 þ lA1 þ a1 A2 þ a2 A21 ;
ð1Þ
where p is the pressure and l is the shear viscosity (assumed to be constant in this problem), a1 and a2 are material parameters called the normal stress moduli. The tensors A1 and A2 are given by, T
A1 ¼ grad v þ ðgrad vÞ ; A2 ¼
d T A1 þ A1 ðgrad vÞ þ ðgrad vÞ A1 ; dt
ð2Þ ð3Þ
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
167
where v is the velocity vector and d=dt is the material time derivative, which is defined as follows: d oðÞ ðÞ ¼ þ ½gradðÞv: dt ot
ð4Þ
The thermodynamics and stability of the model in Eq. (1) were studied in detail by Dunn and Fosdick [8]. Thermodynamics compatibility in the sense that all motions of the fluid meet the Clausius–Duhem inequality (generally interpreted as a statement of the second law of thermodynamics) and the assumption that the specific Helmholtz free energy of the fluid be a minimum when the fluid is locally at rest requires that (cf. [8]), l P 0;
a1 P 0;
a1 þ a2 ¼ 0:
ð5Þ
It is assumed that the fluid is modeled by Eq. (1) subject to restrictions given by (5). The flow of such a fluid over a stretching sheet coinciding with plane y ¼ 0 will be studied. The governing equations of motion are the conservation of mass and linear momentum. These are div v ¼ 0;
ð6Þ
and q
dv ¼ div T þ qb: dt
ð7Þ
The steady two-dimensional boundary layer equations for such a model were derived by Beard and Walters [2] as ou ov þ ¼ 0; ox oy u
ð8Þ
2 ou ou o2 u o ou ou o2 v o3 u þv ¼m 2þk u 2 þ þ v ; ox oy oy ox oy oy oy 2 oy 3
ð9Þ
where m ¼ l=q and k ¼ a1 =q. The boundary conditions are u ¼ cx; v ¼ 0 u!0
at y ¼ 0 c > 0;
as y ! 1:
ð10Þ ð11Þ
Since the free stream velocity is zero, the flow is caused by the stretching of the sheet. The importance of suction or injection at the wall, i.e., when v is not zero at the wall has been discussed for many interesting problems (cf. [14 or 25]). Rajagopal et al. [19,20] showed that by using the similarity transformation u ¼ cxf 0 ðgÞ;
v ¼ ðmcÞ1=2 f ðgÞ;
ð12Þ
168
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
where g¼
c 1=2 m
ð13Þ
y
the conservation of mass, Eq. (8), is automatically satisfied and Eq. (9) reduces to 2
f 02 ff 00 ¼ f 000 þ k½2f 0 f 000 ðf 00 Þ ff iv ;
ð14Þ
where prime designates differentiation with respect to g. The boundary conditions now become, f 0 ð0Þ ¼ 1;
f ð0Þ ¼ 0;
f 0 ð1Þ ¼ 0:
ð15Þ
Eq. (4) is a fourth order equation and in addition to the three boundary conditions, given by Eq. (15), one more is required. Rajagopal and Gupta [20] suggested using the condition that ou !0 oy
as y ! 1;
ð16Þ
which, because of Eq. (2), becomes f 00 ð1Þ ! 0
as g ! 1:
ð17Þ
3. Numerical method Eq. (14), subjected to the boundary conditions (15) and (17) constitutes the final differential equation to be solved. Following the quasi-linearization method developed by Bellman and Kalaba [3], Eq. (14) can be written in the form below: ! ! _ 00 _0 _0 1 1 1 2 f 2 f f iv 000 0 00 000 f f 2f_ þ f f_ k f_ k f k f_ f f 000 f 000 ; ð18Þ 2 f 00 2f 0 f 000 þ ¼ f k kf where f and f_ represent the current and previous approximation to the solution, respectively. Eq. (18) is a linear non-homogeneous equation in the unknown function f . Its solution can be written as a linear combination of two linear independent solutions of the homogeneous problem, fH1 and fH2 , and a particular solution, fp , such that f ¼ C1 fH1 þ C2 fH2 þ fp :
ð19Þ
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
169
In order to satisfy boundary conditions (15) and (17) and the linear independence requirement, the three solutions above are assumed to have the following values at y ¼ 0: 0 00 000 fH1 ; fH1 ¼ ð0; 0; 1; 0Þ; ; fH1 ; fH1 0 00 000 fH2 ; fH2 ; fH2 ; fH2 ¼ ð0; 0; 0; 1Þ; ð20Þ h i 0 00 000 fp ; fp ; fp ; fp ¼ ð0; 1; 0; 0Þ: For each of the homogeneous and particular solutions, Eq. (18) is numerically integrated by using the Runge–Kutta integration method. The values found at the end points are used to find the pair of constants C1 and C2 so that the boundary conditions at y ¼ y1 are met. Therefore, 0 0 C1 fH1 ðy1 Þ þ C2 fH2 ðy1 Þ þ fp0 ðy1 Þ ¼ 0; 00 00 C2 fH1 ðy1 Þ þ C2 fH1 ðy1 Þ þ fp00 ðy1 Þ ¼ 0:
ð21Þ
Once the values for the constants are found the function f can be obtained from Eq. (19) for all discretization points 0 ¼ y1 < < yj < < yN ¼ y1 . This process is repeated until the magnitude of the difference between two consecutive iterations falls below a prescribed tolerance, at all points. In this work, a tolerance limit of 107 was used. As the zeroth approximation to the solution, the function f0 ðyÞ ¼ y;
ð22Þ
satisfying all boundary conditions was assumed.
4. Results and discussion As a result of the similarity transformation, Eqs. (12) and (13), the only parameter of interest for the present problem is k which is related to a1 . The values of f and f 00 as a function of g are plotted in Figs. 1 and 2. These functions are related to the velocity components u and v, through Eqs. (12) and (13). It can be seen that the magnitude of the vertical velocity component increases significantly for high values of k. From Fig. 2 it is observed that f increases as k increases. The results for small values of k can be compared to the results obtained by the perturbation analysis of Rajagopal et al. [19]. The agreement is very good. A quantity of interest is the magnitude of the skin friction coefficient which is related to the shear stress at the wall ou ou ou o2 u Txy jy¼0 ¼ l þ a1 2 : ð23Þ þu oy ox oy oxoy y¼0
170
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
Fig. 1. Function f for different values of k.
Fig. 2. First derivative of f for different values of k.
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
Fig. 3. Second derivative of f for different values of k.
Fig. 4. Third derivative of f for different values of k.
171
172
M. Massoudi, C.E. Maneschy / Appl. Math. Comput. 149 (2004) 165–173
Table 1 Values of Txy k
Txy
0.01 0.10 1.00 10.00
)1.02489 )1.23453 )2.83024 )10.21376
Figs. 3 and 4 show the variation of f 00 and f 000 with k. These functions are necessary to obtain the values of the shear stress given in Eq. (23). Some values of the shear stress at the wall are given in Table 1. References [1] P.D. Ariel, Axisymmetric flow of second grade fluid past a stretching sheet, Int. J. Engng. Sci 39 (2001) 529. [2] D.W. Beard, K. Walters, Elastico-viscous boundary layer flows, Proc. Comb. Phil. Soc. 60 (1964) 667. [3] R.E. Bellman, R.E. Kalaba, Quasilinearization and Non-Linear Boundary Value Problems, American Elsevier, New York, 1965. [4] R.K. Bhatnagar, G. Gupta, K.R. Rajagopal, Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity, Int. J. Non-Linear Mech. 30 (1995) 391. [5] N.M. Bujurke, S.N. Biradar, P. Hiremath, Second-order fluid past a stretching sheet with heat transfer, ZAMP 38 (1987) 654. [6] L.J. Crane, Flow past a stretching sheet, ZAMP 21 (1970) 645. [7] B.S. Dandapat, A.S. Gupta, Flow and heat transfer in a viscoelastic fluid over a stretching sheet, Int. J. Non-Linear Mech. 24 (1989) 215. [8] J.E. Dunn, R.L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal. 56 (1974) 191. [9] V.G. Fox, L.E. Erickson, L.T. Fan, The laminar boundary layer on a moving continuous flat sheet immersed in a non-Newtonian fluid, AIChE J. 15 (1969) 327. [10] V.K. Garg, Improved shooting techniques for linear boundary value problems, Comput. Meth. Appl. Mech. Eng. 22 (1980) 87. [11] V.K. Garg, K.R. Rajagopal, Flow of a non-Newtonian fluid past a wedge, Acta Mech. 88 (1991) 113. [12] V.K. Garg, K.R. Rajagopal, Stagnation point flow of a non-Newtonian fluid, Mech. Res. Comm. 17 (1990) 415. [13] I.A. Hassanien, A.A. Abdullah, R.S.R. Gorla, Flow and heat transfer in a power-law fluid over a non-isothermal stretching sheet, Math. Comput. Modelling 28 (1998) 105. [14] M. Massoudi, M. Ramezan, Effect of injection or suction on the Falkner–Skan flows of second grade fluids, Int. J. Non-Linear Mech. 24 (1989) 221. [15] P.D. McCormack, L. Crane, Physical Fluid Dynamics, Academic Press, New York, 1973. [16] B. Nageswara Rao, Flow of a fluid of second grade fluid over a stretching sheet, Int. J. NonLinear Mech. 31 (1996) 547. [17] G. Pontrelli, Flow of a fluid of second grade over a stretching sheet, Int. J. Non-Linear Mech. 30 (1995) 287. [18] K.R. Rajagopal, T.Y. Na, A.S. Gupta, A non-similar boundary layer on a stretching sheet in a non-Newtonian fluid with uniform free stream, J. Math. Phys. Sci. 21 (1987) 189.
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