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Unsteady boundary layer flow due to a stretching surface in a rotating fluid
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 31 (2004) 121–128 www.elsevier.com/locate/mechrescom
Unsteady boundary layer flow due to a stretching surface in a rotating fluid R. Nazar a, N. Amin a, I. Pop b
b,*
a Department of Mathematics, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia Department of Applied Mathematics, Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania
Received 24 February 2003; received in revised form 2 July 2003
Abstract The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Stretching surface; Rotating fluid; Unsteady viscous flow
1. Introduction The description of flow and heat transfer in the boundary layer induced by a stretching surface has many important applications in manufacturing processes in industry such as the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film and condensation processes, the cooling and/or drying of paper and textiles, and glass filer production, to name just a few of these applications. In particular, in the extrusion of a polymer in a melt-spinning process, the extrusion from the die is generally drawn and simultaneously stretched into a sheet, which is then solidified throughout quenching or gradual cooling by direct contact with water. In all these cases, a study of the flow field and heat transfer can be of a significant importance because the quality of the final product depends to a large extent on the skin friction coefficient and the surface heat transfer rate. Recent papers by Wang et al. (1997), Magyari and Keller (2000), Chen (2000) and Mahapatra and Gupta (2002), and the book by Pop and Ingham (2001) show considerable research activities in this area.
*
Corresponding author. Tel.: +40-264-594315; fax: +40-264-591906. E-mail address: [email protected] (I. Pop).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.09.004
122
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
The steady state flow due to two-dimensional stretching of a surface in a quiescent fluid, has been first studied by Crane (1970). Axisymmetric and three-dimensional cases were studied by Brady and Acrivos (1981), and Wang (1984). The two-dimensional stretching of a sheet is particularly interesting because Crane (1970) has shown that this problem possesses a closed form solution. The steady state flow due to a stretching surface in a rotating fluid, has also been considered by Wang (1988). It is worth mentioning that these problems belong to an important class of exact solutions of the Navier–Stokes equations described by Wang (1991). However, relatively little work has been done on the unsteady boundary layer flow due to stretching of a surface in a quiescent fluid. To our best knowledge, there are only two papers, namely, by Pop and Na (1996) and Wang et al. (1997), which deal with unsteady boundary layer flow due to impulsive starting from rest of a stretching sheet in a viscous fluid. Practical situations cited above of boundary layer flows due to stretching surfaces call for a complete analysis of the fluid dynamics that would be a three-dimensional time-dependent flow. The aim of this paper is therefore, to extend the problem of the stretching of a surface in a steady rotating fluid considered by Wang (1988) to the case when the induced rotating fluid is due to a suddenly stretched surface. When there is an impulsive change in the velocity field, the inviscid flow is developed instantaneously, but the flow in the viscous layer near the wall is developed slowly and becomes fully developed steady flow after some times. For small time, the flow is dominated by the viscous forces, the pressure gradient and the convective acceleration. The problem is formulated in such a manner that for t ¼ 0 (initial unsteady flow) and t ! 1 (final steady state flow), the governing equations reduce to those of Rayleigh equations and Wang (1988) equations, respectively. Thus, it is convenient to choose a time scale n so that the region of time integration may become finite. Such transformations have been found by Williams and Rhyne (1980), and have been used recently by Seshadri et al. (2002) for the problem of unsteady mixed convection flow along a heated vertical plate.
2. Formulation Consider the two-dimensional stretching of a surface in a rotating fluid as shown in Fig. 1. At time t ¼ 0, the surface at z ¼ 0 is impulsively stretched in the x direction in a rotating fluid. Due to the Coriolis force, the fluid motion is three-dimensional. Let (u; v; w) be the velocity components in the direction of the Cartesian axes (x; y; z), respectively, with the axes rotating at an angular velocity X in the z direction. The unsteady Navier–Stokes equations governing the flow are ou ov ow þ þ ¼0 ox oy oz
ð1Þ
z Ω
y x
Fig. 1. Physical model and coordinate system.
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
123
ou ou ou ou 1 op þ u þ v þ w 2Xv ¼ þ mr2 u ot ox oy oz q ox
ð2Þ
ov ov ov ov 1 op þ u þ v þ w 2Xu ¼ þ mr2 v ot ox oy oz q oy
ð3Þ
ow ow ow ow 1 op þu þv þw ¼ þ mr2 w ot ox oy oz q oz
ð4Þ
where p is the pressure, q is the density, m is the kinematic viscosity and r2 denotes the three-dimensional Laplacean. Let the surface be impulsively stretched in the x direction such that the initial and boundary conditions are t<0:u¼v¼w¼0
for any x; y; z
t P 0 : u ¼ ax; v ¼ w ¼ 0 at z ¼ 0 u ! 0; v ! 0; w ! 0 as z ! 1
ð5Þ
where a (>0) has the dimension of [t1 ] and represents the stretching rate. We now introduce the following similarity variables g ¼ ða=mÞ1=2 n1=2 z; w ¼ ðamÞ
1=2 1=2
n
u ¼ axf 0 ðn; gÞ;
v ¼ axhðn; gÞ; s
f ðn; gÞ;
n¼1e ;
s ¼ at
ð6Þ
Then, Eqs. (2)–(4) become 1 of 0 f 000 þ ð1 nÞgf 00 þ nðff 00 f 02 þ 2khÞ ¼ nð1 nÞ 2 on
ð7Þ
1 oh h00 þ ð1 nÞgh0 þ nðfh0 f 0 h 2kf 0 Þ ¼ nð1 nÞ 2 on
ð8Þ
where k ¼ X=a. The boundary conditions (5) become f ðn; 0Þ ¼ 0;
f 0 ðn; 0Þ ¼ 1;
hðn; 0Þ ¼ 0
f 0 ðn; 1Þ ¼ 0;
hðn; 1Þ ¼ 0
ð9Þ
The wall shear stresses sxw and syw in the x and y directions are related to the non-dimensional skin friction coefficient in x and y directions, Cfx and Cfy , respectively, according to sxw m ou x Cf ¼ ¼ 2 2 qðaxÞ ðaxÞ oz z¼0 ð10Þ y sw m ov Cfy ¼ ¼ 2 2 qðaxÞ ðaxÞ oz z¼0 Using variables (6), we obtain ¼ n1=2 f 00 ðn; 0Þ; Cfx Re1=2 x
Cfy Re1=2 ¼ n1=2 h0 ðn; 0Þ x
where Rex ¼ ðaxÞx=m is the local Reynolds number.
ð11Þ
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R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
3. Solution A numerical solution of Eqs. (7) and (8) subjected to the boundary conditions (9) is obtained for 0 6 n 6 1 using the Keller-box method described in the book by Cebeci and Bradshaw (1984). 3.1. Initial unsteady solution at n ¼ 0 For n ¼ 0 (initial unsteady flow), corresponding to s ¼ 0, we have from (7) and (8) 1 f 000 þ gf 00 ¼ 0 2
ð12Þ
1 h00 þ gh0 ¼ 0 2
ð13Þ
subject to f ð0Þ ¼ 0;
f 0 ð0Þ ¼ 1;
hð0Þ ¼ 0;
f 0 ð1Þ ¼ 0;
hð1Þ ¼ 0
ð14Þ
The solution of these equations is 2 2 f ðgÞ ¼ g erfcðg=2Þ þ pffiffiffi ð1 eg =4 Þ p
ð15Þ
hðgÞ ¼ 0
ð16Þ
where erfcðzÞ is the complementary error function defined as Z 1 2 2 erfcðzÞ ¼ pffiffiffi es ds p z
ð17Þ
3.2. Steady state solution at n ¼ 1 For n ¼ 1 (final steady flow), corresponding to s ! 1, Eqs. (7) and (8) give f 000 þ ff 00 f 02 þ 2kh ¼ 0
ð18Þ
h00 þ fh0 f 0 h 2kf 0 ¼ 0
ð19Þ
subject to (14). 3.3. Solution for small n (or s) On the other hand, approximate solutions of Eqs. (7) and (8) subjected to the boundary conditions (9), which is valid in the region n 1, equivalent to small time s 1 solution, can be expressed as f ðn; gÞ ¼ f0 ðgÞ þ f1 ðgÞn þ f2 ðgÞn2 þ h:o:t: hðn; gÞ ¼ h0 ðgÞ þ h1 ðgÞn þ h2 ðgÞn2 þ h:o:t:
ð20Þ
where 1 f0000 þ gf000 ¼ 0 2 f0 ð0Þ ¼ 0; f00 ð0Þ ¼ 1;
ð21Þ f00 ð1Þ ¼ 0
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
1 1 f1000 þ gf100 f10 ¼ gf000 f0 f000 þ f002 2 2 1 0 00 h1 þ gh1 h1 ¼ 2kf00 2 f1 ð0Þ ¼ f10 ð0Þ ¼ 0; h1 ð0Þ ¼ 0; f10 ð1Þ ¼ 0;
125
ð22Þ h1 ð1Þ ¼ 0
1 1 f2000 þ gf200 2f20 ¼ gf100 f000 f1 f0 f100 þ 2f00 f10 f10 2kh1 2 2 1 1 h002 þ gh02 2h2 ¼ gh01 þ f00 h1 f0 h01 h1 þ 2kf10 2 2 0 f2 ð0Þ ¼ f2 ð0Þ ¼ 0; h2 ð0Þ ¼ 0; f20 ð1Þ ¼ 0; h2 ð1Þ ¼ 0
ð23Þ
The closed form solution of Eq. (21) is given by (15), and thus, 1 f000 ð0Þ ¼ pffiffiffi ¼ 0:5642 p
ð24Þ
The analytical solution of Eq. (22) is given by 1 2 1 2 1 1 1 2 0 g2 =4 1 g erfc2 ðg=2Þ f1 ðgÞ ¼ 1 þ g erfcðg=2Þ pffiffiffi ge 2 3p 2 2 2 p 3 2 1 1 4 2 2 2 g þ pffiffiffi eg =4 pffiffiffi geg 4 erfcðg=2Þ þ eg =2 pffiffiffi p 2 p p 4 3 p 2 2 h1 ðgÞ ¼ kg2 erfcðg=2Þ pffiffiffi kgeg =4 p where f100 ð0Þ
1 ¼ pffiffiffi p
7 4 þ 4 3p
¼ 0:7479;
ð25Þ
ð26Þ
2 h01 ð0Þ ¼ pffiffiffi k ¼ 1:1284k p
Eq. (23) has been solved numerically. Thus, the skin friction coefficients can be expressed as 1 7 4 1=2 1=2 x 1=2 þ þ Cf Rex ¼ pffiffiffi n n þ h:o:t: 4 3p p 1 2 Cfy Rex1=2 ¼ pffiffiffi 0 pffiffiffi kn1=2 þ h:o:t: p p
ð27Þ
ð28Þ
ð29Þ
for n 1 or s 1. Using the approximate polynomial relationship 1 1 n ¼ s s2 þ s3 þ h:o:t: 2 6
ð30Þ
the skin friction coefficients can be expressed in terms of s.
4. Results and discussion Eqs. (7) and (8) under the boundary conditions (9) are solved numerically for 0 6 n 6 1 and some values of the parameter k using the Keller-box method described by Cebeci and Bradshaw (1984). In order to validate our results, we have compared the values of the reduced skin frictions f 00 ðn; 0Þ and h0 ðn; 0Þ when
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R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
n ¼ 1 (final steady state flow) with those of Wang (1988). The results are found to be in excellent agreement. The comparison is shown in Table 1. The evolution of the similarity velocity profiles f 0 ðgÞ and hðgÞ in the x and y directions, respectively, at the final steady state flow (n ¼ 1) are shown for some values of k in Figs. 2 and 3, respectively. From Fig. 2 we notice that for zero and small values of k, the velocities decay monotonically exponentially, while for large values of k, the decay is oscillatory. This behaviour has also been observed by Wang (1988). Figs. 4–7 show the variation of the skin friction coefficients Cfx Re1=2 and Cfy Rex1=2 with n and s for some x values of k by solving Eqs. (7) and (8) numerically. The steady state solution (n ¼ 1) obtained by solving Eqs. (18) and (19) are also included in Figs. 4 and 6. We notice that there is a very good agreement between the results when we solved the full unsteady boundary layer equations and the steady state equations. It is also noticed that the transition from unsteady to steady flows take place smoothly. Further, in Figs. 5 and Table 1 Some values of f 00 ð0Þ and h0 ð0Þ for n ¼ 1 and different values of k k
f 00 ð0Þ Wang (1988)
f 00 ð0Þ present
h0 ð0Þ Wang (1988)
h0 ð0Þ present
0 0.5 1 2
)1 )1.1384 )1.3250 )1.6523
)1 )1.1384 )1.3250 )1.6523
0 )0.5128 )0.8371 )1.2873
0 )0.5128 )0.8371 )1.2873
1.2
f ′(η) 1 0.8 0.6 0.4 0.2 0 λ = 0, 0.5, 1, 2, 5
-0.2 0
1
2
3
4
5
η
6
Fig. 2. Similarity velocity profile in x direction for n ¼ 1.
0.1
h (η)
λ = 0.5, 1, 2, 5
0 -0.1 -0.2 -0.3 -0.4 -0.5 0
1
2
3
4
5
η
6
Fig. 3. Similarity velocity profile in y direction for n ¼ 1.
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
127
0
Cf x Rex1/2 -1 -2 -3 λ = 0, 0.5, 1, 2, 5 -4 Numerical solution, Eqs. (7) &(8) Steadystate solution, Eqs. (18)& (19)
-5 -6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ξ
Fig. 4. Variation with n of the skin friction coefficient in x direction for some values of k.
0 -0.5
Cf x Rex1/2-1 -1.5 -2 -2.5 -3
λ = 0, 0.5,1, 2, 5
-3.5
Numerical solution, Eqs. (7) & (8) Small τ solution, Eqs. (28) & (29)
-4 -4.5 -5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
τ
5
Fig. 5. Variation with s of the skin friction coefficient in x direction for some values of k.
0.5 Numerical solution, Eqs. (7)& (8) Steady state solution, Eqs. (18) &(19)
Cf y Rex1/2
0
λ = 0.5
-0.5
1 -1 2 -1.5 -2 -2.5
5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ξ
1
Fig. 6. Variation with n of the skin friction coefficient in y direction for some values of k.
7, the small s solution given by Eqs. (28) and (29) is shown, and we can see that for relatively small values of k, the numerical and analytical solutions are in excellent agreement, but for relatively large values of k, the results are in good agreement only for very small s. However, this agreement can be improved by taking more terms in Eqs. (28) and (29). In addition, it is worth mentioning that as the value of k (the fluid rotation) increases, the values of jCfx Rex1=2 j and jCfy Rex1=2 j also increase, which is in agreement with the steady state flow case studied by Wang (1988). We have also found that the behaviour of the skin friction coefficient is oscillatory as the fluid rotation increases (large values of k 1).
128
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128 0
Cf y Rex1/2 λ = 0.5
-0.5
1 -1 2 -1.5
Numerical solution, Eqs. (7) & (8) Small τ solution, Eqs. (28) & (29)
-2
5
-2.5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
τ
5
Fig. 7. Variation with s of the skin friction coefficient in y direction for some values of k.
5. Conclusions In the present paper, the unsteady flow due to the impulsive starting from rest of a stretching surface in a viscous and incompressible rotating fluid is studied. The problem is formulated in such a way that at time t ¼ 0 (unsteady flow) it is represented by the Rayleigh type of equations and for t ! 1 (steady state flow) it is represented by the Wang (1988) type of equations. The partial differential equations governing the flow have been solved numerically using the very efficient implicit finite difference method, known as the Kellerbox method. The flow characteristics for small and large values of time, described by the parameter k have been examined in detail. Also, analytical behavior of the solution for large time (n ¼ 1), and its approach to the steady state (Wang, 1988) solution has been studied. For this steady state flow, we have compared the values of the reduced skin frictions f 00 ðn; 0Þ and h0 ðn; 0Þ when n ¼ 1 (final steady state flow) with those of Wang (1988). The results are found to be in excellent agreement. It is also shown that there is a smooth transition from the short time solution (n ¼ 0) to the large time solutions (n ¼ 1). Thus, the complete transient history from an impulsive start to steady state flow has been analyzed.
References Brady, J.F., Acrivos, A., 1981. J. Fluid Mech. 112, 127. Cebeci, T., Bradshaw, P., 1984. Physical and Computational Aspects of Convective Heat Transfer. Springer, New York. Chen, C.-H., 2000. Heat Mass Transfer 36, 79. Crane, L.J., 1970. J. Appl. Math. Phys. (ZAMP) 21, 645. Magyari, E., Keller, H.B., 2000. Eur. J. Mech. B––Fluids 19, 109. Mahapatra, T.R., Gupta, A.S., 2002. Heat Mass Transfer 38, 517. Pop, I., Ingham, D.B., 2001. Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon, Oxford. Pop, I., Na, T.Y., 1996. Mech. Res. Comm. 23, 413. Seshadri, R., Sreeshylan, N., Nath, G., 2002. Int. J. Heat Mass Transfer 45, 1345. Wang, C.Y., Du, G., Miklavcic, M., Chang, C.C., 1997. SIAM J. Appl. Math. 57, 1. Wang, C.Y., 1984. Phys. Fluids 27, 1915. Wang, C.Y., 1988. J. Appl. Math. Phys. (ZAMP) 39, 177. Wang, C.Y., 1991. Ann. Rev. Fluid Mech. 23, 159. Williams, J.C., Rhyne, T.H., 1980. SIAM J. Appl. Math. 38, 215.
Mechanics Research Communications 31 (2004) 121–128 www.elsevier.com/locate/mechrescom
Unsteady boundary layer flow due to a stretching surface in a rotating fluid R. Nazar a, N. Amin a, I. Pop b
b,*
a Department of Mathematics, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia Department of Applied Mathematics, Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania
Received 24 February 2003; received in revised form 2 July 2003
Abstract The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Stretching surface; Rotating fluid; Unsteady viscous flow
1. Introduction The description of flow and heat transfer in the boundary layer induced by a stretching surface has many important applications in manufacturing processes in industry such as the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film and condensation processes, the cooling and/or drying of paper and textiles, and glass filer production, to name just a few of these applications. In particular, in the extrusion of a polymer in a melt-spinning process, the extrusion from the die is generally drawn and simultaneously stretched into a sheet, which is then solidified throughout quenching or gradual cooling by direct contact with water. In all these cases, a study of the flow field and heat transfer can be of a significant importance because the quality of the final product depends to a large extent on the skin friction coefficient and the surface heat transfer rate. Recent papers by Wang et al. (1997), Magyari and Keller (2000), Chen (2000) and Mahapatra and Gupta (2002), and the book by Pop and Ingham (2001) show considerable research activities in this area.
*
Corresponding author. Tel.: +40-264-594315; fax: +40-264-591906. E-mail address: [email protected] (I. Pop).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.09.004
122
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
The steady state flow due to two-dimensional stretching of a surface in a quiescent fluid, has been first studied by Crane (1970). Axisymmetric and three-dimensional cases were studied by Brady and Acrivos (1981), and Wang (1984). The two-dimensional stretching of a sheet is particularly interesting because Crane (1970) has shown that this problem possesses a closed form solution. The steady state flow due to a stretching surface in a rotating fluid, has also been considered by Wang (1988). It is worth mentioning that these problems belong to an important class of exact solutions of the Navier–Stokes equations described by Wang (1991). However, relatively little work has been done on the unsteady boundary layer flow due to stretching of a surface in a quiescent fluid. To our best knowledge, there are only two papers, namely, by Pop and Na (1996) and Wang et al. (1997), which deal with unsteady boundary layer flow due to impulsive starting from rest of a stretching sheet in a viscous fluid. Practical situations cited above of boundary layer flows due to stretching surfaces call for a complete analysis of the fluid dynamics that would be a three-dimensional time-dependent flow. The aim of this paper is therefore, to extend the problem of the stretching of a surface in a steady rotating fluid considered by Wang (1988) to the case when the induced rotating fluid is due to a suddenly stretched surface. When there is an impulsive change in the velocity field, the inviscid flow is developed instantaneously, but the flow in the viscous layer near the wall is developed slowly and becomes fully developed steady flow after some times. For small time, the flow is dominated by the viscous forces, the pressure gradient and the convective acceleration. The problem is formulated in such a manner that for t ¼ 0 (initial unsteady flow) and t ! 1 (final steady state flow), the governing equations reduce to those of Rayleigh equations and Wang (1988) equations, respectively. Thus, it is convenient to choose a time scale n so that the region of time integration may become finite. Such transformations have been found by Williams and Rhyne (1980), and have been used recently by Seshadri et al. (2002) for the problem of unsteady mixed convection flow along a heated vertical plate.
2. Formulation Consider the two-dimensional stretching of a surface in a rotating fluid as shown in Fig. 1. At time t ¼ 0, the surface at z ¼ 0 is impulsively stretched in the x direction in a rotating fluid. Due to the Coriolis force, the fluid motion is three-dimensional. Let (u; v; w) be the velocity components in the direction of the Cartesian axes (x; y; z), respectively, with the axes rotating at an angular velocity X in the z direction. The unsteady Navier–Stokes equations governing the flow are ou ov ow þ þ ¼0 ox oy oz
ð1Þ
z Ω
y x
Fig. 1. Physical model and coordinate system.
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
123
ou ou ou ou 1 op þ u þ v þ w 2Xv ¼ þ mr2 u ot ox oy oz q ox
ð2Þ
ov ov ov ov 1 op þ u þ v þ w 2Xu ¼ þ mr2 v ot ox oy oz q oy
ð3Þ
ow ow ow ow 1 op þu þv þw ¼ þ mr2 w ot ox oy oz q oz
ð4Þ
where p is the pressure, q is the density, m is the kinematic viscosity and r2 denotes the three-dimensional Laplacean. Let the surface be impulsively stretched in the x direction such that the initial and boundary conditions are t<0:u¼v¼w¼0
for any x; y; z
t P 0 : u ¼ ax; v ¼ w ¼ 0 at z ¼ 0 u ! 0; v ! 0; w ! 0 as z ! 1
ð5Þ
where a (>0) has the dimension of [t1 ] and represents the stretching rate. We now introduce the following similarity variables g ¼ ða=mÞ1=2 n1=2 z; w ¼ ðamÞ
1=2 1=2
n
u ¼ axf 0 ðn; gÞ;
v ¼ axhðn; gÞ; s
f ðn; gÞ;
n¼1e ;
s ¼ at
ð6Þ
Then, Eqs. (2)–(4) become 1 of 0 f 000 þ ð1 nÞgf 00 þ nðff 00 f 02 þ 2khÞ ¼ nð1 nÞ 2 on
ð7Þ
1 oh h00 þ ð1 nÞgh0 þ nðfh0 f 0 h 2kf 0 Þ ¼ nð1 nÞ 2 on
ð8Þ
where k ¼ X=a. The boundary conditions (5) become f ðn; 0Þ ¼ 0;
f 0 ðn; 0Þ ¼ 1;
hðn; 0Þ ¼ 0
f 0 ðn; 1Þ ¼ 0;
hðn; 1Þ ¼ 0
ð9Þ
The wall shear stresses sxw and syw in the x and y directions are related to the non-dimensional skin friction coefficient in x and y directions, Cfx and Cfy , respectively, according to sxw m ou x Cf ¼ ¼ 2 2 qðaxÞ ðaxÞ oz z¼0 ð10Þ y sw m ov Cfy ¼ ¼ 2 2 qðaxÞ ðaxÞ oz z¼0 Using variables (6), we obtain ¼ n1=2 f 00 ðn; 0Þ; Cfx Re1=2 x
Cfy Re1=2 ¼ n1=2 h0 ðn; 0Þ x
where Rex ¼ ðaxÞx=m is the local Reynolds number.
ð11Þ
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R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
3. Solution A numerical solution of Eqs. (7) and (8) subjected to the boundary conditions (9) is obtained for 0 6 n 6 1 using the Keller-box method described in the book by Cebeci and Bradshaw (1984). 3.1. Initial unsteady solution at n ¼ 0 For n ¼ 0 (initial unsteady flow), corresponding to s ¼ 0, we have from (7) and (8) 1 f 000 þ gf 00 ¼ 0 2
ð12Þ
1 h00 þ gh0 ¼ 0 2
ð13Þ
subject to f ð0Þ ¼ 0;
f 0 ð0Þ ¼ 1;
hð0Þ ¼ 0;
f 0 ð1Þ ¼ 0;
hð1Þ ¼ 0
ð14Þ
The solution of these equations is 2 2 f ðgÞ ¼ g erfcðg=2Þ þ pffiffiffi ð1 eg =4 Þ p
ð15Þ
hðgÞ ¼ 0
ð16Þ
where erfcðzÞ is the complementary error function defined as Z 1 2 2 erfcðzÞ ¼ pffiffiffi es ds p z
ð17Þ
3.2. Steady state solution at n ¼ 1 For n ¼ 1 (final steady flow), corresponding to s ! 1, Eqs. (7) and (8) give f 000 þ ff 00 f 02 þ 2kh ¼ 0
ð18Þ
h00 þ fh0 f 0 h 2kf 0 ¼ 0
ð19Þ
subject to (14). 3.3. Solution for small n (or s) On the other hand, approximate solutions of Eqs. (7) and (8) subjected to the boundary conditions (9), which is valid in the region n 1, equivalent to small time s 1 solution, can be expressed as f ðn; gÞ ¼ f0 ðgÞ þ f1 ðgÞn þ f2 ðgÞn2 þ h:o:t: hðn; gÞ ¼ h0 ðgÞ þ h1 ðgÞn þ h2 ðgÞn2 þ h:o:t:
ð20Þ
where 1 f0000 þ gf000 ¼ 0 2 f0 ð0Þ ¼ 0; f00 ð0Þ ¼ 1;
ð21Þ f00 ð1Þ ¼ 0
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
1 1 f1000 þ gf100 f10 ¼ gf000 f0 f000 þ f002 2 2 1 0 00 h1 þ gh1 h1 ¼ 2kf00 2 f1 ð0Þ ¼ f10 ð0Þ ¼ 0; h1 ð0Þ ¼ 0; f10 ð1Þ ¼ 0;
125
ð22Þ h1 ð1Þ ¼ 0
1 1 f2000 þ gf200 2f20 ¼ gf100 f000 f1 f0 f100 þ 2f00 f10 f10 2kh1 2 2 1 1 h002 þ gh02 2h2 ¼ gh01 þ f00 h1 f0 h01 h1 þ 2kf10 2 2 0 f2 ð0Þ ¼ f2 ð0Þ ¼ 0; h2 ð0Þ ¼ 0; f20 ð1Þ ¼ 0; h2 ð1Þ ¼ 0
ð23Þ
The closed form solution of Eq. (21) is given by (15), and thus, 1 f000 ð0Þ ¼ pffiffiffi ¼ 0:5642 p
ð24Þ
The analytical solution of Eq. (22) is given by 1 2 1 2 1 1 1 2 0 g2 =4 1 g erfc2 ðg=2Þ f1 ðgÞ ¼ 1 þ g erfcðg=2Þ pffiffiffi ge 2 3p 2 2 2 p 3 2 1 1 4 2 2 2 g þ pffiffiffi eg =4 pffiffiffi geg 4 erfcðg=2Þ þ eg =2 pffiffiffi p 2 p p 4 3 p 2 2 h1 ðgÞ ¼ kg2 erfcðg=2Þ pffiffiffi kgeg =4 p where f100 ð0Þ
1 ¼ pffiffiffi p
7 4 þ 4 3p
¼ 0:7479;
ð25Þ
ð26Þ
2 h01 ð0Þ ¼ pffiffiffi k ¼ 1:1284k p
Eq. (23) has been solved numerically. Thus, the skin friction coefficients can be expressed as 1 7 4 1=2 1=2 x 1=2 þ þ Cf Rex ¼ pffiffiffi n n þ h:o:t: 4 3p p 1 2 Cfy Rex1=2 ¼ pffiffiffi 0 pffiffiffi kn1=2 þ h:o:t: p p
ð27Þ
ð28Þ
ð29Þ
for n 1 or s 1. Using the approximate polynomial relationship 1 1 n ¼ s s2 þ s3 þ h:o:t: 2 6
ð30Þ
the skin friction coefficients can be expressed in terms of s.
4. Results and discussion Eqs. (7) and (8) under the boundary conditions (9) are solved numerically for 0 6 n 6 1 and some values of the parameter k using the Keller-box method described by Cebeci and Bradshaw (1984). In order to validate our results, we have compared the values of the reduced skin frictions f 00 ðn; 0Þ and h0 ðn; 0Þ when
126
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
n ¼ 1 (final steady state flow) with those of Wang (1988). The results are found to be in excellent agreement. The comparison is shown in Table 1. The evolution of the similarity velocity profiles f 0 ðgÞ and hðgÞ in the x and y directions, respectively, at the final steady state flow (n ¼ 1) are shown for some values of k in Figs. 2 and 3, respectively. From Fig. 2 we notice that for zero and small values of k, the velocities decay monotonically exponentially, while for large values of k, the decay is oscillatory. This behaviour has also been observed by Wang (1988). Figs. 4–7 show the variation of the skin friction coefficients Cfx Re1=2 and Cfy Rex1=2 with n and s for some x values of k by solving Eqs. (7) and (8) numerically. The steady state solution (n ¼ 1) obtained by solving Eqs. (18) and (19) are also included in Figs. 4 and 6. We notice that there is a very good agreement between the results when we solved the full unsteady boundary layer equations and the steady state equations. It is also noticed that the transition from unsteady to steady flows take place smoothly. Further, in Figs. 5 and Table 1 Some values of f 00 ð0Þ and h0 ð0Þ for n ¼ 1 and different values of k k
f 00 ð0Þ Wang (1988)
f 00 ð0Þ present
h0 ð0Þ Wang (1988)
h0 ð0Þ present
0 0.5 1 2
)1 )1.1384 )1.3250 )1.6523
)1 )1.1384 )1.3250 )1.6523
0 )0.5128 )0.8371 )1.2873
0 )0.5128 )0.8371 )1.2873
1.2
f ′(η) 1 0.8 0.6 0.4 0.2 0 λ = 0, 0.5, 1, 2, 5
-0.2 0
1
2
3
4
5
η
6
Fig. 2. Similarity velocity profile in x direction for n ¼ 1.
0.1
h (η)
λ = 0.5, 1, 2, 5
0 -0.1 -0.2 -0.3 -0.4 -0.5 0
1
2
3
4
5
η
6
Fig. 3. Similarity velocity profile in y direction for n ¼ 1.
R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128
127
0
Cf x Rex1/2 -1 -2 -3 λ = 0, 0.5, 1, 2, 5 -4 Numerical solution, Eqs. (7) &(8) Steadystate solution, Eqs. (18)& (19)
-5 -6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ξ
Fig. 4. Variation with n of the skin friction coefficient in x direction for some values of k.
0 -0.5
Cf x Rex1/2-1 -1.5 -2 -2.5 -3
λ = 0, 0.5,1, 2, 5
-3.5
Numerical solution, Eqs. (7) & (8) Small τ solution, Eqs. (28) & (29)
-4 -4.5 -5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
τ
5
Fig. 5. Variation with s of the skin friction coefficient in x direction for some values of k.
0.5 Numerical solution, Eqs. (7)& (8) Steady state solution, Eqs. (18) &(19)
Cf y Rex1/2
0
λ = 0.5
-0.5
1 -1 2 -1.5 -2 -2.5
5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ξ
1
Fig. 6. Variation with n of the skin friction coefficient in y direction for some values of k.
7, the small s solution given by Eqs. (28) and (29) is shown, and we can see that for relatively small values of k, the numerical and analytical solutions are in excellent agreement, but for relatively large values of k, the results are in good agreement only for very small s. However, this agreement can be improved by taking more terms in Eqs. (28) and (29). In addition, it is worth mentioning that as the value of k (the fluid rotation) increases, the values of jCfx Rex1=2 j and jCfy Rex1=2 j also increase, which is in agreement with the steady state flow case studied by Wang (1988). We have also found that the behaviour of the skin friction coefficient is oscillatory as the fluid rotation increases (large values of k 1).
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R. Nazar et al. / Mechanics Research Communications 31 (2004) 121–128 0
Cf y Rex1/2 λ = 0.5
-0.5
1 -1 2 -1.5
Numerical solution, Eqs. (7) & (8) Small τ solution, Eqs. (28) & (29)
-2
5
-2.5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
τ
5
Fig. 7. Variation with s of the skin friction coefficient in y direction for some values of k.
5. Conclusions In the present paper, the unsteady flow due to the impulsive starting from rest of a stretching surface in a viscous and incompressible rotating fluid is studied. The problem is formulated in such a way that at time t ¼ 0 (unsteady flow) it is represented by the Rayleigh type of equations and for t ! 1 (steady state flow) it is represented by the Wang (1988) type of equations. The partial differential equations governing the flow have been solved numerically using the very efficient implicit finite difference method, known as the Kellerbox method. The flow characteristics for small and large values of time, described by the parameter k have been examined in detail. Also, analytical behavior of the solution for large time (n ¼ 1), and its approach to the steady state (Wang, 1988) solution has been studied. For this steady state flow, we have compared the values of the reduced skin frictions f 00 ðn; 0Þ and h0 ðn; 0Þ when n ¼ 1 (final steady state flow) with those of Wang (1988). The results are found to be in excellent agreement. It is also shown that there is a smooth transition from the short time solution (n ¼ 0) to the large time solutions (n ¼ 1). Thus, the complete transient history from an impulsive start to steady state flow has been analyzed.
References Brady, J.F., Acrivos, A., 1981. J. Fluid Mech. 112, 127. Cebeci, T., Bradshaw, P., 1984. Physical and Computational Aspects of Convective Heat Transfer. Springer, New York. Chen, C.-H., 2000. Heat Mass Transfer 36, 79. Crane, L.J., 1970. J. Appl. Math. Phys. (ZAMP) 21, 645. Magyari, E., Keller, H.B., 2000. Eur. J. Mech. B––Fluids 19, 109. Mahapatra, T.R., Gupta, A.S., 2002. Heat Mass Transfer 38, 517. Pop, I., Ingham, D.B., 2001. Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon, Oxford. Pop, I., Na, T.Y., 1996. Mech. Res. Comm. 23, 413. Seshadri, R., Sreeshylan, N., Nath, G., 2002. Int. J. Heat Mass Transfer 45, 1345. Wang, C.Y., Du, G., Miklavcic, M., Chang, C.C., 1997. SIAM J. Appl. Math. 57, 1. Wang, C.Y., 1984. Phys. Fluids 27, 1915. Wang, C.Y., 1988. J. Appl. Math. Phys. (ZAMP) 39, 177. Wang, C.Y., 1991. Ann. Rev. Fluid Mech. 23, 159. Williams, J.C., Rhyne, T.H., 1980. SIAM J. Appl. Math. 38, 215.