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Unsteady oscillatory stagnation-point flow of a viscoelastic fluid
International Journal of Engineering Science 42 (2004) 625–633 www.elsevier.com/locate/ijengsci
Unsteady oscillatory stagnation-point flow of a viscoelastic fluid F. Labropulu *, M. Chinichian Luther College––Mathematics, University of Regina, Regina, SK, Canada S4S 0A2 Received 30 July 2003; accepted 11 September 2003 (Communicated by L. DEBNATH)
Abstract The unsteady stagnation point flow of the Walters B0 fluid is examined and solutions are obtained. It is assumed that the infinite plate at y ¼ 0 is oscillating and the fluid impinges obliquely on the plate. Ó 2004 Published by Elsevier Ltd. Keywords: Oscillatory; Viscoelastic; Stagnation
1. Introduction Interest in the flows of viscoelastic liquids has increased substantially over the past decades due to the occurrence of these liquids in industrial processes. Among those liquids is the Walters B0 fluid [1]. The equations of motion of non-Newtonian fluids are highly non-linear and one order higher than the Navier–Stokes equations. Due to the complexity of these equations, finding accurate solutions is not easy. In the history of fluid dynamics, considerable attention has been given to the study of 2-D stagnation point flow. Hiemenz [2] derived an exact solution of the steady flow of a Newtonian fluid impinging orthogonally on an infinite flat plate. Stuart [3], Tamada [4] and Dorrepaal [5] independently investigated the solutions of a stagnation point flow when the fluid impinges obliquely on the plate. Beard and Walters [6] used boundary-layer equations to study two-dimensional flow near a stagnation point of a viscoelastic fluid. Dorrepaal et al. [7] investigated the behaviour of a viscoelastic fluid impinging on a flat rigid wall at an arbitrary angle of incidence. *
Corresponding author. Tel.: +1-306-585-5040; fax: +1-306-585-5267. E-mail address: [email protected] (F. Labropulu).
0020-7225/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2003.09.004
626
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
Labropulu et al. [8] studied the oblique flow of a viscoelastic fluid impinging on a porous wall with suction or blowing. Unsteady stagnation point flow of a Newtonian fluid has also been studied extensively. Rott [9] and Glauert [10] studied the stagnation point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane. Srivastava [11] investigated the same problem for a nonNewtonian second grade fluid using the Karman–Pohlhausen method. Labropulu et al. [12] used series methods to solve the unsteady stagnation point flow of a viscoelastic fluid impinging on an oscillating flat plate. Matunobu [13,14] and Kawaguti and Hamano [15] examined the fundamental character of the unsteady flow near a stagnation point for a Newtonian fluid. Takemitsu and Matunobu [16] studied the oblique stagnation point flow for a Newtonian fluid and obtained the general features of a periodic stagnation point flow. In this work, the unsteady stagnation point flow of the Walters B0 fluid is examined and solutions are obtained. We assume that the infinite plate at y ¼ 0 is oscillating with velocity U cos Xt, the fluid occupies the entire upper half plane y > 0 and the fluid impinges obliquely on the plate. 2. Flow equations The two-dimensional flow of a viscous incompressible non-Newtonian Walters B0 fluid, neglecting thermal effects and body forces, is governed by (see [6]): ou ov þ ¼0 ox oy
ð1Þ
ou ou ou 1 op a o o o ou ou 2 2 þu þv þ ¼ mr u ðr uÞ þ u þ v r2 u r2 u r2 v ot ox oy q ox q ot ox oy ox oy 2 2 2 ou o u ov o u ou ov o u 2 þ þ þ 2 2 ox ox oy oy oy ox oxoy ov ov ov 1 op a o o o ov ov 2 2 þu þv þ ¼ mr v ðr vÞ þ u þ v r2 v r2 u r2 v ot ox oy q oy q ot ox oy ox oy ou o2 v ov o2 v ou ov o2 v þ 2 þ þ 2 2 ox ox oy oy oy ox oxoy
ð2Þ
ð3Þ
where u ¼ uðx; y; tÞ, v ¼ vðx; y; tÞ are the velocity components, p ¼ pðx; y; tÞ is the pressure, m ¼ lq is the kinematic viscosity and a is a measure of the viscoelasticity of the fluid. Continuity equation (1) implies the existence of a streamfunction wðx; y; tÞ such that
u¼
ow ; oy
v¼
ow ox
ð4Þ
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
627
Substitution of (4) in Eqs. (2) and (3) and elimination of pressure from the resulting equations using pxy ¼ pyx yields o a o oðw; r2 wÞ a oðw; r4 wÞ ðr2 wÞ þ ðr4 wÞ mr4 w ¼ 0 ot q ot oðx; yÞ q oðx; yÞ
ð5Þ
Having obtained a solution of Eq. (5), the velocity components are given by Eq. (4) and the pressure can be found by integrating Eqs. (2) and (3). The shear stress component s12 is given by 2 3 o w o2 w ow o w o3 w ow o3 w o3 w s12 ¼ l a oy 2 ox2 oy oxoy 2 ox3 ox oy 3 ox2 oy o2 w o2 w o2 w o2 w þ2 2 þ2 oxoy oy 2 ox oxoy
ð6Þ
3. Solutions in the fixed frame of reference Following Takemitsu and Matunobu [16], we assume that w ¼ k½xf ðyÞ þ gðy; tÞ
ð7Þ
We assume that the infinite plate at y ¼ 0 is oscillating with velocity U cos Xt and that the fluid occupies the entire upper half plane y > 0. Furthermore, we assume that the streamfunction far from the wall is given by w ¼ 12 cy 2 þ xy (see [3]). Thus, the boundary conditions are f ð0Þ ¼ f 0 ð0Þ ¼ 0; f 0 ð1Þ ¼ 1;
gð0; tÞ ¼ 0;
gy ð0; tÞ ¼
U iXt e k
gy ð1; tÞ ¼ cy
ð8Þ ð9Þ
where c is a non-dimensional constant characterizing the obliqueness of oncoming flow. It is assumed that only the real part of a complex quantity has its physical meaning. Using Eq. (7) in (5), we obtain ak ðvÞ ff f 0 f ðivÞ ¼ 0 mf ðivÞ þ k ff 000 f 0 f 00 þ q
ð10Þ
and m
3 5 o4 g o3 g a o5 g og ak og 00 og ðivÞ og f þ k f f f þ ¼0 oy 4 otoy 2 q otoy 4 oy 3 oy q oy 5 oy
ð11Þ
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F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
Integrating Eqs. (10) and (11) once with respect to y and using the conditions at infinity, we have ak ðivÞ mf 000 þ k ff 00 f 02 þ 2f 0 f 000 þ f 002 ¼ k ff q
ð12Þ
and 2 4 3 2 o3 g o2 g a o4 g og ak og 0 og 0o g 00 o g 000 og þk f 2 f þf f þ f 4 f ¼0 m 3 oy otoy q otoy 3 oy oy q oy oy 3 oy 2 oy ð13Þ Non-dimensionalizing using rffiffiffi rffiffiffi k m y; s ¼ Xt; f ðyÞ ¼ F ðgÞ; g¼ m k
m gðy; tÞ ¼ Gðg; sÞ; k
b¼
X ; k
U ¼ pffiffiffiffiffi mk
ð14Þ
we get F 000 þ FF 00 F 02 þ We ðFF ðivÞ 2F 0 F 000 þ F 002 Þ ¼ 1 F ð0Þ ¼ 0; F 0 ð0Þ ¼ 0; F 0 ð1Þ ¼ 1
ð15Þ
and 4 3 2 o3 G o2 G oG o2 G o4 G 0 oG 0o G 00 o G 000 oG þ W W þ F F F þ F F b ¼0 F b e e og3 og2 og og4 og3 og2 og osog osog3 Gð0; sÞ ¼ 0;
Gg ð0; sÞ ¼ eis ;
Ggg ð1; sÞ ¼ c ð16Þ
is the Weissenberg number. where We ¼ ak qm System (15) has been solved numerically by many authors [6,17]. Using the shooting method with the finite difference technique described by Ariel [17], we find that F 00 ð0Þ ¼ 1:23259 when We ¼ 0. Numerical values of F 00 ð0Þ for different values of We are shown in Table 1. Fig. 1 shows the profiles of F 0 for various We . We observed that as the elasticity of the fluid increases, the velocity near the wall increases. Fig. 2 depicts the profiles of F for various We . Table 1 Numerical values of F 00 ð0Þ, /00 ð0Þ, /01 ð0Þ, /02 ð0Þ and H00 ð0Þ for different values of We We
F 00 ð0Þ
/00 ð0Þ
/01 ð0Þ
/02 ð0Þ
H00 ð0Þ
0.0 0.1 0.2 0.3
1.23259 1.36954 1.5873 2.11092
)0.811318 )0.86709 )0.947485 )1.10879
)0.49307 )0.547302 )0.633897 )0.842867
0.0945488 0.0658565 0.0221985 )0.0761073
0.607965 0.697589 0.846722 1.1889
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F'(η) 1.2
We=0.2
We=0.3 1.0 0.8
We=0.1 We=0.0
0.6
We=0.0 We=0.1 We=0.2 We=0.3
0.4 0.2 0.0 0
1
2
3
η
4
5
Fig. 1. Variation of F 0 ðgÞ with We .
5
4 We=0.2 3
We=0.3
F(η)
We=0.0 2
We=0.1
We=0.0 We=0.1 We=0.2 We=0.3
1
0 0
1
2
η
3
4
5
Fig. 2. Variation of F ðgÞ with We .
Letting Gðg; sÞ ¼ G0 ðgÞ þ G1 ðgÞeis , then system (16) gives
ðivÞ 00 0 0 0 000 00 00 000 0 G000 þ FG F G þ W FG G þ F G F G F e 0 0 0 0 0 0 0 ¼ 0 G0 ð0Þ ¼ 0;
G00 ð0Þ ¼ 0;
G000 ð1Þ ¼ c
ð17Þ
and
0 ðivÞ 00 0 0 0 000 00 00 000 0 000 G000 F þ FG F G þ W FG G þ F G F G e 1 1 1 1 1 1 1 ib G1 þ We G1 ¼ 0 G1 ð0Þ ¼ 0;
G01 ð0Þ ¼ 1;
G01 ð1Þ ¼ 0
ð18Þ
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F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
Letting G00 ¼ cH0 , then system (17) becomes H000 þ FH00 F 0 H0 þ We FH0000 F 0 H000 þ F 00 H00 F 000 H0 ¼ 0 H0 ð0Þ ¼ 0;
ð19Þ
H00 ð1Þ ¼ 1
This system is solved numerically using a shooting method and it is found that for We ¼ 0, H00 ð0Þ ¼ 0:607965. Since G00 ð0Þ ¼ cH 0 ð0Þ, then for We ¼ 0, G00 ð0Þ ¼ 0:607965c which is in good agreement with the value obtained by Takemitsu and Matunobu [16]. Numerical values of H00 ð0Þ for different values of We are shown in Table 1. Fig. 3 depicts the profiles of H00 for various values of We . Letting /ðgÞ ¼ G01 ðgÞ, then system (18) becomes
/00 þ F /0 F 0 / þ We F /000 F 0 /00 þ F 00 /0 F 000 / ib / þ We /00 ¼ 0 /ð0Þ ¼ 1;
ð20Þ
/ð1Þ ¼ 0
The only parameter in Eq. (20) is the frequency b. Two series solutions valid for small and large values of b have been obtained by Labropulu et al. [12]. They found that for small b, /0 ð0Þ ¼ /00 ð0Þ þ ib/01 ð0Þ þ ðibÞ2 /02 ð0Þ þ
ð21Þ
where the values for /00 ð0Þ; /01 ð0Þ and /02 ð0Þ are given in Table 1 for different values of We .
H'0(η) 1.2 1.1 1.0 0.9 We=0.0 We=0.1 We=0.2 We=0.3
0.8 0.7 0.6 0
1
2
3
η
4
5
Fig. 3. Variation of H00 ðgÞ with We .
6
7
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
631
pffiffiffiffi For large b, they let Y ¼ ag; a ¼ ib and they found that /0 ð0Þ ¼ /00 ð0Þ þ a/01 ð0Þ þ a2 /02 ð0Þ þ a3 /03 ð0Þ þ 1 ð3 4mÞF 00 ð0Þ 2 3 þ 4m ¼ pffiffiffiffiffiffiffiffiffiffiffiffi a þ pffiffiffiffiffiffiffiffiffiffiffiffi a3 8ð1 mÞ a 1m 16 1 m 3 2 002 ð40m 50m þ 28m 33ÞF ð0Þ 5 pffiffiffiffiffiffiffiffiffiffiffiffi a þ 128ð1 mÞ 1 m
ð22Þ
where F 00 ð0Þ is given in Table 1 for different values of We .
4. Solutions in the moving frame of reference We assume that the Cartesian coordinates (x; y) are moving with the plate, the x-axis being along the flat plate and the y-axis normal to the plate. In this case, following Takemitsu and Matunobu [16], we assume the streamfunction is given by w ¼ k ½ xf ðyÞ þ hðy; tÞ
ð23Þ
and the boundary conditions are f ð0Þ ¼ f 0 ð0Þ ¼ 0;
hð0; tÞ ¼ hy ð0; tÞ ¼ 0;
f 0 ð1Þ ¼ 1;
hy ð1; tÞ ¼ cy
U iXt e k
ð24Þ
We note that the flow is oscillating with the velocity U cos Xt at infinity. Using Eq. (23) in (5), equating different powers of x to zero and integrating once with respect to y using the conditions at infinity, we obtain mf 000 þ kðff 00 f 02 Þ þ
ak ðff ðivÞ 2f 0 f 000 þ f 002 Þ ¼ k q
ð25Þ
and 2 4 3 2 o3 h o2 h a o4 h oh ak oh 0 oh 0o h 00 o h 000 oh m 3 þk f 2 f þf f þ f 4f oy otoy q otoy 3 oy oy q oy oy 3 oy 2 oy iX UeiXt ¼ 1þ k
ð26Þ
Non-dimensionalizing using rffiffiffi k g¼ y; m
s ¼ Xt;
rffiffiffi m f ðyÞ ¼ F ðgÞ; k
m hðy; tÞ ¼ Gðg; sÞ; k
b¼
X ; k
U ¼ pffiffiffiffiffi mk
ð27Þ
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F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
we get F 000 þ FF 00 F 02 þ We ðFF ðivÞ 2F 0 F 000 þ F 002 Þ ¼ 1 F ð0Þ ¼ 0;
F 0 ð0Þ ¼ 0;
F 0 ð1Þ ¼ 1
ð28Þ
and 4 3 2 o3 G o2 G oG o2 G o4 G 0 oG 0o G 00 o G 000 oG þ F F F þ F F b ¼ ð1 þ ibÞeis F þ W b W e e og3 og2 og og4 og3 og2 og osog osog3 Gð0; sÞ ¼ 0; Gg ð0; sÞ ¼ 0; Gg ð1; sÞ ¼ cg eis
ð29Þ System (28) has been solved numerically in Section 3. Letting Gðg; sÞ ¼ G0 ðgÞ H ðgÞeis , then system (29) gives
ðivÞ 00 0 0 0 000 00 00 000 0 F þ FG F G þ W FG G þ F G F G G000 e 0 0 0 0 0 0 0 ¼ 0 G0 ð0Þ ¼ 0;
G00 ð0Þ ¼ 0;
G000 ð1Þ ¼ c
ð30Þ
and H 000 þ FH 00 F 0 H 0 þ We FH ðivÞ F 0 H 000 þ F 00 H 00 F 000 H 0 ib H 0 þ We H 000 ¼ 1 ib Hð0Þ ¼ 0;
H 0 ð0Þ ¼ 0;
H 0 ð1Þ ¼ 1
ð31Þ
System (30) has been solved numerically by Labropulu et al. [12]. It can be easily shown that H0 ¼
F 0 þ ibG01 ib 1 ib
ð32Þ
where F 0 and G01 have been found in Section 3 is a solution of system (31) since it satisfies both the equation and the boundary conditions. References [1] K. Walters, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, 1964, p. 507. [2] K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, DinglerÕs Polytech. J. 326 (1911) 321. [3] J.T. Stuart, The viscous flow near a stagnation point when the external flow has uniform vorticity, J. Aerospace Sci. 26 (1959) 124. [4] K.J. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn. 46 (1979) 310. [5] J.M. Dorrepaal, An Exact solution of the Navier–Stokes equation which describes non-orthogonal stagnationpoint flow in two dimensions, J. Fluid Mech. 163 (1986) 141. [6] B.W. Beard, K. Walters, Elastico-viscous boundary layer flows, Proc. Camb. Philos. Soc. 60 (1964) 667.
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[7] J.M. Dorrepaal, O.P. Chandna, F. Labropulu, The flow of a visco-elastic fluid near a point of re-attachment, ZAMP 43 (1992) 708. [8] F. Labropulu, J.M. Dorrepaal, O.P. Chandna, Viscoelastic fluid flow impinging on a wall with suction or blowing, Mech. Res. Commun. 20 (2) (1993) 143. [9] N. Rott, Unsteady viscous flow in the vicinity of a stagnation point, Quart. Appl. Math. 13 (1956) 444. [10] M.B. Glauert, The laminar boundary layer on oscillating plates and cylinders, J. Fluid Mech. 1 (1956) 97. [11] A.C. Srivastava, Unsteady flow of a second-order fluid near a stagnation point, J. Fluid Mech. 24 (Part 1) (1966) 33. [12] F. Labropulu, X. Xu, M. Chinichian, Unsteady stagnation-point flow of a non-Newtonian second grade fluid, Int. J. Math. Math. Sci., in press. [13] Y. Matunobu, Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. I, J. Phys. Soc. Jpn. 42 (1977) 2041. [14] Y. Matunobu, Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. II, J. Phys. Soc. Jpn. 43 (1977) 326. [15] M. Kawaguti, K. Hamano, Two-dimensional model of pulsatile flow through constricted artery, in: Proceedings of Xth International Congress on Angiology, Session 26, No. 4, Tokyo, 1976. [16] N. Takemitsu, Y. Matunobu, Unsteady stagnation-point flow impinging oqliquely on an oscillating flat plate, J. Phys. Soc. Jpn. 47 (4) (1979) 1347. [17] P.D. Ariel, A hybrid method for computing the flow of viscoelastic fluids, Int. J. Numer. Meth. Fluids 14 (1992) 323.
Unsteady oscillatory stagnation-point flow of a viscoelastic fluid F. Labropulu *, M. Chinichian Luther College––Mathematics, University of Regina, Regina, SK, Canada S4S 0A2 Received 30 July 2003; accepted 11 September 2003 (Communicated by L. DEBNATH)
Abstract The unsteady stagnation point flow of the Walters B0 fluid is examined and solutions are obtained. It is assumed that the infinite plate at y ¼ 0 is oscillating and the fluid impinges obliquely on the plate. Ó 2004 Published by Elsevier Ltd. Keywords: Oscillatory; Viscoelastic; Stagnation
1. Introduction Interest in the flows of viscoelastic liquids has increased substantially over the past decades due to the occurrence of these liquids in industrial processes. Among those liquids is the Walters B0 fluid [1]. The equations of motion of non-Newtonian fluids are highly non-linear and one order higher than the Navier–Stokes equations. Due to the complexity of these equations, finding accurate solutions is not easy. In the history of fluid dynamics, considerable attention has been given to the study of 2-D stagnation point flow. Hiemenz [2] derived an exact solution of the steady flow of a Newtonian fluid impinging orthogonally on an infinite flat plate. Stuart [3], Tamada [4] and Dorrepaal [5] independently investigated the solutions of a stagnation point flow when the fluid impinges obliquely on the plate. Beard and Walters [6] used boundary-layer equations to study two-dimensional flow near a stagnation point of a viscoelastic fluid. Dorrepaal et al. [7] investigated the behaviour of a viscoelastic fluid impinging on a flat rigid wall at an arbitrary angle of incidence. *
Corresponding author. Tel.: +1-306-585-5040; fax: +1-306-585-5267. E-mail address: [email protected] (F. Labropulu).
0020-7225/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2003.09.004
626
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
Labropulu et al. [8] studied the oblique flow of a viscoelastic fluid impinging on a porous wall with suction or blowing. Unsteady stagnation point flow of a Newtonian fluid has also been studied extensively. Rott [9] and Glauert [10] studied the stagnation point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane. Srivastava [11] investigated the same problem for a nonNewtonian second grade fluid using the Karman–Pohlhausen method. Labropulu et al. [12] used series methods to solve the unsteady stagnation point flow of a viscoelastic fluid impinging on an oscillating flat plate. Matunobu [13,14] and Kawaguti and Hamano [15] examined the fundamental character of the unsteady flow near a stagnation point for a Newtonian fluid. Takemitsu and Matunobu [16] studied the oblique stagnation point flow for a Newtonian fluid and obtained the general features of a periodic stagnation point flow. In this work, the unsteady stagnation point flow of the Walters B0 fluid is examined and solutions are obtained. We assume that the infinite plate at y ¼ 0 is oscillating with velocity U cos Xt, the fluid occupies the entire upper half plane y > 0 and the fluid impinges obliquely on the plate. 2. Flow equations The two-dimensional flow of a viscous incompressible non-Newtonian Walters B0 fluid, neglecting thermal effects and body forces, is governed by (see [6]): ou ov þ ¼0 ox oy
ð1Þ
ou ou ou 1 op a o o o ou ou 2 2 þu þv þ ¼ mr u ðr uÞ þ u þ v r2 u r2 u r2 v ot ox oy q ox q ot ox oy ox oy 2 2 2 ou o u ov o u ou ov o u 2 þ þ þ 2 2 ox ox oy oy oy ox oxoy ov ov ov 1 op a o o o ov ov 2 2 þu þv þ ¼ mr v ðr vÞ þ u þ v r2 v r2 u r2 v ot ox oy q oy q ot ox oy ox oy ou o2 v ov o2 v ou ov o2 v þ 2 þ þ 2 2 ox ox oy oy oy ox oxoy
ð2Þ
ð3Þ
where u ¼ uðx; y; tÞ, v ¼ vðx; y; tÞ are the velocity components, p ¼ pðx; y; tÞ is the pressure, m ¼ lq is the kinematic viscosity and a is a measure of the viscoelasticity of the fluid. Continuity equation (1) implies the existence of a streamfunction wðx; y; tÞ such that
u¼
ow ; oy
v¼
ow ox
ð4Þ
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
627
Substitution of (4) in Eqs. (2) and (3) and elimination of pressure from the resulting equations using pxy ¼ pyx yields o a o oðw; r2 wÞ a oðw; r4 wÞ ðr2 wÞ þ ðr4 wÞ mr4 w ¼ 0 ot q ot oðx; yÞ q oðx; yÞ
ð5Þ
Having obtained a solution of Eq. (5), the velocity components are given by Eq. (4) and the pressure can be found by integrating Eqs. (2) and (3). The shear stress component s12 is given by 2 3 o w o2 w ow o w o3 w ow o3 w o3 w s12 ¼ l a oy 2 ox2 oy oxoy 2 ox3 ox oy 3 ox2 oy o2 w o2 w o2 w o2 w þ2 2 þ2 oxoy oy 2 ox oxoy
ð6Þ
3. Solutions in the fixed frame of reference Following Takemitsu and Matunobu [16], we assume that w ¼ k½xf ðyÞ þ gðy; tÞ
ð7Þ
We assume that the infinite plate at y ¼ 0 is oscillating with velocity U cos Xt and that the fluid occupies the entire upper half plane y > 0. Furthermore, we assume that the streamfunction far from the wall is given by w ¼ 12 cy 2 þ xy (see [3]). Thus, the boundary conditions are f ð0Þ ¼ f 0 ð0Þ ¼ 0; f 0 ð1Þ ¼ 1;
gð0; tÞ ¼ 0;
gy ð0; tÞ ¼
U iXt e k
gy ð1; tÞ ¼ cy
ð8Þ ð9Þ
where c is a non-dimensional constant characterizing the obliqueness of oncoming flow. It is assumed that only the real part of a complex quantity has its physical meaning. Using Eq. (7) in (5), we obtain ak ðvÞ ff f 0 f ðivÞ ¼ 0 mf ðivÞ þ k ff 000 f 0 f 00 þ q
ð10Þ
and m
3 5 o4 g o3 g a o5 g og ak og 00 og ðivÞ og f þ k f f f þ ¼0 oy 4 otoy 2 q otoy 4 oy 3 oy q oy 5 oy
ð11Þ
628
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
Integrating Eqs. (10) and (11) once with respect to y and using the conditions at infinity, we have ak ðivÞ mf 000 þ k ff 00 f 02 þ 2f 0 f 000 þ f 002 ¼ k ff q
ð12Þ
and 2 4 3 2 o3 g o2 g a o4 g og ak og 0 og 0o g 00 o g 000 og þk f 2 f þf f þ f 4 f ¼0 m 3 oy otoy q otoy 3 oy oy q oy oy 3 oy 2 oy ð13Þ Non-dimensionalizing using rffiffiffi rffiffiffi k m y; s ¼ Xt; f ðyÞ ¼ F ðgÞ; g¼ m k
m gðy; tÞ ¼ Gðg; sÞ; k
b¼
X ; k
U ¼ pffiffiffiffiffi mk
ð14Þ
we get F 000 þ FF 00 F 02 þ We ðFF ðivÞ 2F 0 F 000 þ F 002 Þ ¼ 1 F ð0Þ ¼ 0; F 0 ð0Þ ¼ 0; F 0 ð1Þ ¼ 1
ð15Þ
and 4 3 2 o3 G o2 G oG o2 G o4 G 0 oG 0o G 00 o G 000 oG þ W W þ F F F þ F F b ¼0 F b e e og3 og2 og og4 og3 og2 og osog osog3 Gð0; sÞ ¼ 0;
Gg ð0; sÞ ¼ eis ;
Ggg ð1; sÞ ¼ c ð16Þ
is the Weissenberg number. where We ¼ ak qm System (15) has been solved numerically by many authors [6,17]. Using the shooting method with the finite difference technique described by Ariel [17], we find that F 00 ð0Þ ¼ 1:23259 when We ¼ 0. Numerical values of F 00 ð0Þ for different values of We are shown in Table 1. Fig. 1 shows the profiles of F 0 for various We . We observed that as the elasticity of the fluid increases, the velocity near the wall increases. Fig. 2 depicts the profiles of F for various We . Table 1 Numerical values of F 00 ð0Þ, /00 ð0Þ, /01 ð0Þ, /02 ð0Þ and H00 ð0Þ for different values of We We
F 00 ð0Þ
/00 ð0Þ
/01 ð0Þ
/02 ð0Þ
H00 ð0Þ
0.0 0.1 0.2 0.3
1.23259 1.36954 1.5873 2.11092
)0.811318 )0.86709 )0.947485 )1.10879
)0.49307 )0.547302 )0.633897 )0.842867
0.0945488 0.0658565 0.0221985 )0.0761073
0.607965 0.697589 0.846722 1.1889
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
629
F'(η) 1.2
We=0.2
We=0.3 1.0 0.8
We=0.1 We=0.0
0.6
We=0.0 We=0.1 We=0.2 We=0.3
0.4 0.2 0.0 0
1
2
3
η
4
5
Fig. 1. Variation of F 0 ðgÞ with We .
5
4 We=0.2 3
We=0.3
F(η)
We=0.0 2
We=0.1
We=0.0 We=0.1 We=0.2 We=0.3
1
0 0
1
2
η
3
4
5
Fig. 2. Variation of F ðgÞ with We .
Letting Gðg; sÞ ¼ G0 ðgÞ þ G1 ðgÞeis , then system (16) gives
ðivÞ 00 0 0 0 000 00 00 000 0 G000 þ FG F G þ W FG G þ F G F G F e 0 0 0 0 0 0 0 ¼ 0 G0 ð0Þ ¼ 0;
G00 ð0Þ ¼ 0;
G000 ð1Þ ¼ c
ð17Þ
and
0 ðivÞ 00 0 0 0 000 00 00 000 0 000 G000 F þ FG F G þ W FG G þ F G F G e 1 1 1 1 1 1 1 ib G1 þ We G1 ¼ 0 G1 ð0Þ ¼ 0;
G01 ð0Þ ¼ 1;
G01 ð1Þ ¼ 0
ð18Þ
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F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
Letting G00 ¼ cH0 , then system (17) becomes H000 þ FH00 F 0 H0 þ We FH0000 F 0 H000 þ F 00 H00 F 000 H0 ¼ 0 H0 ð0Þ ¼ 0;
ð19Þ
H00 ð1Þ ¼ 1
This system is solved numerically using a shooting method and it is found that for We ¼ 0, H00 ð0Þ ¼ 0:607965. Since G00 ð0Þ ¼ cH 0 ð0Þ, then for We ¼ 0, G00 ð0Þ ¼ 0:607965c which is in good agreement with the value obtained by Takemitsu and Matunobu [16]. Numerical values of H00 ð0Þ for different values of We are shown in Table 1. Fig. 3 depicts the profiles of H00 for various values of We . Letting /ðgÞ ¼ G01 ðgÞ, then system (18) becomes
/00 þ F /0 F 0 / þ We F /000 F 0 /00 þ F 00 /0 F 000 / ib / þ We /00 ¼ 0 /ð0Þ ¼ 1;
ð20Þ
/ð1Þ ¼ 0
The only parameter in Eq. (20) is the frequency b. Two series solutions valid for small and large values of b have been obtained by Labropulu et al. [12]. They found that for small b, /0 ð0Þ ¼ /00 ð0Þ þ ib/01 ð0Þ þ ðibÞ2 /02 ð0Þ þ
ð21Þ
where the values for /00 ð0Þ; /01 ð0Þ and /02 ð0Þ are given in Table 1 for different values of We .
H'0(η) 1.2 1.1 1.0 0.9 We=0.0 We=0.1 We=0.2 We=0.3
0.8 0.7 0.6 0
1
2
3
η
4
5
Fig. 3. Variation of H00 ðgÞ with We .
6
7
F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
631
pffiffiffiffi For large b, they let Y ¼ ag; a ¼ ib and they found that /0 ð0Þ ¼ /00 ð0Þ þ a/01 ð0Þ þ a2 /02 ð0Þ þ a3 /03 ð0Þ þ 1 ð3 4mÞF 00 ð0Þ 2 3 þ 4m ¼ pffiffiffiffiffiffiffiffiffiffiffiffi a þ pffiffiffiffiffiffiffiffiffiffiffiffi a3 8ð1 mÞ a 1m 16 1 m 3 2 002 ð40m 50m þ 28m 33ÞF ð0Þ 5 pffiffiffiffiffiffiffiffiffiffiffiffi a þ 128ð1 mÞ 1 m
ð22Þ
where F 00 ð0Þ is given in Table 1 for different values of We .
4. Solutions in the moving frame of reference We assume that the Cartesian coordinates (x; y) are moving with the plate, the x-axis being along the flat plate and the y-axis normal to the plate. In this case, following Takemitsu and Matunobu [16], we assume the streamfunction is given by w ¼ k ½ xf ðyÞ þ hðy; tÞ
ð23Þ
and the boundary conditions are f ð0Þ ¼ f 0 ð0Þ ¼ 0;
hð0; tÞ ¼ hy ð0; tÞ ¼ 0;
f 0 ð1Þ ¼ 1;
hy ð1; tÞ ¼ cy
U iXt e k
ð24Þ
We note that the flow is oscillating with the velocity U cos Xt at infinity. Using Eq. (23) in (5), equating different powers of x to zero and integrating once with respect to y using the conditions at infinity, we obtain mf 000 þ kðff 00 f 02 Þ þ
ak ðff ðivÞ 2f 0 f 000 þ f 002 Þ ¼ k q
ð25Þ
and 2 4 3 2 o3 h o2 h a o4 h oh ak oh 0 oh 0o h 00 o h 000 oh m 3 þk f 2 f þf f þ f 4f oy otoy q otoy 3 oy oy q oy oy 3 oy 2 oy iX UeiXt ¼ 1þ k
ð26Þ
Non-dimensionalizing using rffiffiffi k g¼ y; m
s ¼ Xt;
rffiffiffi m f ðyÞ ¼ F ðgÞ; k
m hðy; tÞ ¼ Gðg; sÞ; k
b¼
X ; k
U ¼ pffiffiffiffiffi mk
ð27Þ
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F. Labropulu, M. Chinichian / International Journal of Engineering Science 42 (2004) 625–633
we get F 000 þ FF 00 F 02 þ We ðFF ðivÞ 2F 0 F 000 þ F 002 Þ ¼ 1 F ð0Þ ¼ 0;
F 0 ð0Þ ¼ 0;
F 0 ð1Þ ¼ 1
ð28Þ
and 4 3 2 o3 G o2 G oG o2 G o4 G 0 oG 0o G 00 o G 000 oG þ F F F þ F F b ¼ ð1 þ ibÞeis F þ W b W e e og3 og2 og og4 og3 og2 og osog osog3 Gð0; sÞ ¼ 0; Gg ð0; sÞ ¼ 0; Gg ð1; sÞ ¼ cg eis
ð29Þ System (28) has been solved numerically in Section 3. Letting Gðg; sÞ ¼ G0 ðgÞ H ðgÞeis , then system (29) gives
ðivÞ 00 0 0 0 000 00 00 000 0 F þ FG F G þ W FG G þ F G F G G000 e 0 0 0 0 0 0 0 ¼ 0 G0 ð0Þ ¼ 0;
G00 ð0Þ ¼ 0;
G000 ð1Þ ¼ c
ð30Þ
and H 000 þ FH 00 F 0 H 0 þ We FH ðivÞ F 0 H 000 þ F 00 H 00 F 000 H 0 ib H 0 þ We H 000 ¼ 1 ib Hð0Þ ¼ 0;
H 0 ð0Þ ¼ 0;
H 0 ð1Þ ¼ 1
ð31Þ
System (30) has been solved numerically by Labropulu et al. [12]. It can be easily shown that H0 ¼
F 0 þ ibG01 ib 1 ib
ð32Þ
where F 0 and G01 have been found in Section 3 is a solution of system (31) since it satisfies both the equation and the boundary conditions. References [1] K. Walters, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, 1964, p. 507. [2] K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, DinglerÕs Polytech. J. 326 (1911) 321. [3] J.T. Stuart, The viscous flow near a stagnation point when the external flow has uniform vorticity, J. Aerospace Sci. 26 (1959) 124. [4] K.J. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn. 46 (1979) 310. [5] J.M. Dorrepaal, An Exact solution of the Navier–Stokes equation which describes non-orthogonal stagnationpoint flow in two dimensions, J. Fluid Mech. 163 (1986) 141. [6] B.W. Beard, K. Walters, Elastico-viscous boundary layer flows, Proc. Camb. Philos. Soc. 60 (1964) 667.
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[7] J.M. Dorrepaal, O.P. Chandna, F. Labropulu, The flow of a visco-elastic fluid near a point of re-attachment, ZAMP 43 (1992) 708. [8] F. Labropulu, J.M. Dorrepaal, O.P. Chandna, Viscoelastic fluid flow impinging on a wall with suction or blowing, Mech. Res. Commun. 20 (2) (1993) 143. [9] N. Rott, Unsteady viscous flow in the vicinity of a stagnation point, Quart. Appl. Math. 13 (1956) 444. [10] M.B. Glauert, The laminar boundary layer on oscillating plates and cylinders, J. Fluid Mech. 1 (1956) 97. [11] A.C. Srivastava, Unsteady flow of a second-order fluid near a stagnation point, J. Fluid Mech. 24 (Part 1) (1966) 33. [12] F. Labropulu, X. Xu, M. Chinichian, Unsteady stagnation-point flow of a non-Newtonian second grade fluid, Int. J. Math. Math. Sci., in press. [13] Y. Matunobu, Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. I, J. Phys. Soc. Jpn. 42 (1977) 2041. [14] Y. Matunobu, Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. II, J. Phys. Soc. Jpn. 43 (1977) 326. [15] M. Kawaguti, K. Hamano, Two-dimensional model of pulsatile flow through constricted artery, in: Proceedings of Xth International Congress on Angiology, Session 26, No. 4, Tokyo, 1976. [16] N. Takemitsu, Y. Matunobu, Unsteady stagnation-point flow impinging oqliquely on an oscillating flat plate, J. Phys. Soc. Jpn. 47 (4) (1979) 1347. [17] P.D. Ariel, A hybrid method for computing the flow of viscoelastic fluids, Int. J. Numer. Meth. Fluids 14 (1992) 323.