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Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate
International Journal of Engineering Science 42 (2004) 1891–1896 www.elsevier.com/locate/ijengsci
Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate A. Pantokratoras
*
School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece Received 17 February 2004; received in revised form 13 May 2004; accepted 5 July 2004
Abstract The steady laminar boundary layer flow along an isothermal, continuously moving (with constant velocity) plate is studied taking into account the variation of fluid viscosity with temperature. The results are obtained with the direct numerical solution of the boundary layer equations. The study concerns the wall heat transfer, the wall shear stress and velocity and temperature profiles across the boundary layer. When the variation of viscosity with temperature is strong, the results of the present work are completely different from those existing in the literature. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Moving plate; Temperature-dependent viscosity; Forced convection
1. Introduction The problem of forced convection along an isothermal moving plate is a classical problem of fluid mechanics that has been solved for the first time in 1961 by Sakiadis [8]. Thereafter, many solutions have been obtained for different aspects of this class of boundary layer problems. Solutions have been appeared including mass transfer, varying plate velocity, varying plate temperature, fluid injection and fluid suction at the plate. The work by Pop et al. [7] belongs to the above *
Tel.: +30 541 079618. E-mail address: [email protected]
0020-7225/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.07.005
1892
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
class of problems, including the variation of fluid viscosity with temperature. The authors obtained similarity solutions considering that viscosity varies as an inverse function of temperature. However, the Prandtl number, which is a function of viscosity, has been considered constant across the boundary layer and this assumption leads to unrealistic results when the variation of viscosity with temperature is strong. The objective of the present paper is to obtain results considering both viscosity and Prandtl number variable across the boundary layer. As will be shown later the differences of the two methods are very large.
2. The mathematical model Consider the flow along a moving plate placed in a calm environment with u and v denoting respectively the velocity components in the x and y direction, where x is the coordinate along the plate and y is the coordinate perpendicular to x. For steady, two-dimensional flow the boundary layer equations including variable viscosity are ou ov þ ¼0 ox oy
continuity equation :
momentum equation :
energy equation :
u
ou ov 1 o ou u þv ¼ l ox oy q1 oy oy
oT oT o2 T þv ¼a 2 ox oy oy
ð1Þ ð2Þ
ð3Þ
where T is the fluid temperature, l is the dynamic viscosity, a is the thermal diffusivity, and q1 is the ambient fluid density. The boundary conditions are as follows: at y ¼ 0 u ¼ U ; v ¼ 0; T ¼ T w
ð4Þ
as y ! 1 u ¼ 0; T ¼ T 1
ð5Þ
where Tw is the plate temperature, T1 is the ambient fluid temperature and U is the velocity of the moving plate. The viscosity is assumed to be an inverse linear function of temperature given by the following equation [7]: 1 1 ¼ ½1 þ cðT w T 1 Þ l l1
ð6Þ
where l1 is the ambient fluid dynamic viscosity and c is a thermal property of the fluid. Eq. (6) can be rewritten as follows: 1 ¼ aðT T r Þ l
ð7Þ
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
1893
where a = c/l1 and Tr = T1 1/c are constants and their values depend on the reference state and the thermal property of the fluid. Eqs. (1)–(3) form a parabolic system and were solved directly, without any transformation, by a method described by Patankar [6]. The finite difference method is used with primitive variables x, y and a space marching procedure is used in x direction with an expanding grid. A detailed description of the solution procedure may be found in [5] where all fluid properties (viscosity, thermal diffusivity and density) have been considered as functions of temperature.
3. Results and discussion The most important quantities for this problem are the wall heat transfer and the wall shear stress defined as x oT Re1=2 ð8Þ h0 ð0Þ ¼ Tw T1 oy y¼0 #r 1 lw 1=2 ou f ð0Þ ¼ Re #r q1 U 2 oy y¼0 00
ð9Þ
where h is the dimensionless temperature (T T1)/(Tw T1) and f is the dimensionless stream function for which the following equation is valid: u ð10Þ f0 ¼ U The Reynolds number is defined as Ux Re ¼ ð11Þ m1 and hr is a constant defined by hr ¼
Tr T1 1 ¼ cðT w T 1 Þ Tw T1
ð12Þ
In Eqs. (8)–(10) the prime represents differentiation with respect to similarity variable g defined as y g ¼ Re1=2 ð13Þ x It should be mentioned here that when hr ! 1 the fluid viscosity becomes equal to ambient viscosity and we have the classical Sakiadis problem. In order to test the accuracy of the present method, results were compared with those available in the literature. The wall heat transfer h 0 (0) and the wall shear stress f00 (0) for the present problem with constant viscosity and Pr = 0.7 are 0.4446 and 0.3507 respectively. The corresponding quantities calculated by the present method are 0.4438 and 0.3500. The comparison is satisfactory and this happens for other Pr numbers. In contrast to the above direct solution of Eqs. (1)–(3), [7] transformed these equations into the following similarity equations:
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A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
f 000
# #r 00 1 ff #0 f 00 ¼ 0 # #r 2#r
ð14Þ
1 00 1 0 ð15Þ # þ f# ¼ 0 Pr 2 It should be mentioned here that in the transformed energy Eq. (15) the Prandtl number has been assumed constant across the boundary layer. Pop et al. have calculated this Prandtl number at ambient temperature from the following equation: m1 ð16Þ Pr1 ¼ a However, the Prandtl number is a function of viscosity and as viscosity varies across the boundary layer, the Prandtl number varies, too. As will be shown below this assumption leads to unrealistic results for small values of hr. In Tables 1 and 2 the wall shear stress and the wall heat transfer are given for ambient Prandtl numbers 0.7 and 10 respectively. In these tables the results by Pop et al. [7] have been also included for comparison. In the last column of each table the Prandtl numbers at the plate (Prw) are included. From Table 1 it is seen that the wall shear stress, calculated by the two methods, are in agreement. For wall heat transfer things are different. For large values of the parameter jhrj the results are in agreement but as jhrj decreases the results of the two methods diverge. The same observation is valid for Pr1 = 10 but now the divergence appears also in the wall shear stress. It is advocated here that the results of Pop et al. for low jhrj are unrealistic. The explanation is drawn from Fig.1 where the temperature profiles are shown for hr = 0.001 and ambient Pr numbers 0.7 and 10. It is seen that the real temperature profiles, calculated with variable Pr number, are much wider than those given by Pop et al. calculated with constant ambient Pr number. The error is Table 1 Values of f00 (0) and h 0 (0) for Pr1 = 0.7 f00 (0)
h 0 (0)
hr
Present work
Pop et al.
Difference %
Present work
Pop et al.
Difference %
Prw
10 8 6 4 2 1 0.1 0.01 0.001 2 4 6 8 10
0.4712 0.4775 0.4879 0.5078 0.5627 0.6561 1.5021 4.4952 14.1528 0.2795 0.3706 0.3969 0.4094 0.4168
0.4710 0.4774 0.4877 0.5078 0.5629 0.6565 1.5062 4.4857 14.0654 0.2783 0.3699 0.3963 0.4089 0.4163
<1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1
0.3452 0.3441 0.3422 0.3384 0.3278 0.3089 0.1735 0.0711 0.0310 0.3782 0.3632 0.3587 0.3565 0.3552
0.3504 0.3493 0.3476 0.3442 0.3349 0.3189 0.2191 0.1545 0.1341 0.3807 0.3667 0.3625 0.3605 0.3593
1.5 1.5 1.5 2 2 3 26 117 333 <1 <1 1 1 1
0.64 0.62 0.60 0.56 0.47 0.35 0.06 0.007 0.0007 1.40 0.93 0.84 0.80 0.78
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
1895
Table 2 Values of f00 (0) and h 0 (0) for Pr1 = 10 f00 (0)
h 0 (0)
hr
Present work
Pop et al.
Difference %
Present work
Pop et al.
Difference %
Prw
10 8 6 4 2 1 0.1 0.01 0.001 2 4 6 8 10
0.4805 0.4888 0.5032 0.5314 0.6109 0.7522 1.8446 4.6095 14.1323 0.2457 0.3497 0.3824 0.3983 0.4078
0.5067 0.5158 0.5310 0.5608 0.6451 0.7955 1.8734 2.7939 4.8035 0.2571 0.3680 0.4027 0.4196 0.4296
5 5 5 5 5 6 2 39 66 5 5 5 5 5
1.6811 1.6793 1.6771 1.6731 1.6594 1.6366 1.4141 0.7751 0.2723 1.7145 1.6990 1.6937 1.6913 1.6905
1.6816 1.6731 1.6707 1.6659 1.6521 1.6331 1.5762 1.4576 1.4576 1.7129 1.6962 1.6908 1.6882 1.6867
<1 <1 <1 <1 <1 <1 11 88 435 <1 <1 <1 <1 <1
9.10 8.89 8.57 8.00 6.67 5.00 0.91 0.10 0.01 20.00 13.33 12.00 11.43 11.11
Fig. 1. Temperature distribution for hr = 0.001 and ambient Pr numbers 0.7 and 10: solid line, present work: dashed line, [7].
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A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
introduced by considering that the ambient Pr number is valid in the entire boundary layer but this is an approximation that is valid only for large values of jhrj (small variation of viscosity and small variation between ambient and wall Prandtl number). From the above tables we see that for hr = 0.001 the Prandtl number at the plate is 1000 times smaller than the ambient one and this means that the Pr number inside the boundary layer is much smaller than the ambient one. It is well known in the boundary layer theory that large Pr numbers correspond to ‘‘narrow’’ temperature profiles and small Pr numbers to wider temperature profiles. This is the reason for the difference in the temperature profiles and the wall heat transfer between the two methods. The finding of the present work may be valuable to a class of similar problems. For example in the works of Kafoussias and Williams [4], Hady et al. [1], Hossain et al. [3] and Hossain and Munir [2] the Prandtl number has been also considered constant in the transformed energy equation. This assumption may give unrealistic results if the above works are applied to fluids with strong relationship between viscosity and temperature.
References [1] F.M. Hady, A.Y. Bakier, R.S.R. Gorla, Mixed convection boundary layer flow on a continuous flat plate with variable viscosity, Heat and Mass Transfer 31 (1996) 169–172. [2] M.A. Hossain, M.S. Munir, Mixed convection flow from a vertical flat plate with temperature dependent viscosity, International Journal of Thermal Science 39 (2000) 173–183. [3] M.A. Hossain, M.S. Munir, M.Z. Hafiz, H.S. Takhar, Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux, Heat and Mass Transfer 36 (2000) 333– 341. [4] N.G. Kafoussias, E.W. Williams, The effect of temperature-dependent viscosity on the free-forced convective laminar boundary layer flow past a vertical isothermal flat plate, Acta Mechanica 110 (1995) 123–137. [5] A. Pantokratoras, Laminar free-convection over a vertical isothermal plate with uniform blowing or suction in water with variable physical properties, International Journal of Heat and Mass Transfer 45 (2002) 963–977. [6] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Company, New York, 1980. [7] I. Pop, R.S.R. Gorla, M. Rashidi, The effect of variable viscosity on flow and heat transfer to a continuous moving flat plate, International Journal of Engineering Science 30 (1992) 1–6. [8] B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces: the boundary layer on a continuous flat surface, AIChE Journal 7 (1961) 221–225.
Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate A. Pantokratoras
*
School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece Received 17 February 2004; received in revised form 13 May 2004; accepted 5 July 2004
Abstract The steady laminar boundary layer flow along an isothermal, continuously moving (with constant velocity) plate is studied taking into account the variation of fluid viscosity with temperature. The results are obtained with the direct numerical solution of the boundary layer equations. The study concerns the wall heat transfer, the wall shear stress and velocity and temperature profiles across the boundary layer. When the variation of viscosity with temperature is strong, the results of the present work are completely different from those existing in the literature. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Moving plate; Temperature-dependent viscosity; Forced convection
1. Introduction The problem of forced convection along an isothermal moving plate is a classical problem of fluid mechanics that has been solved for the first time in 1961 by Sakiadis [8]. Thereafter, many solutions have been obtained for different aspects of this class of boundary layer problems. Solutions have been appeared including mass transfer, varying plate velocity, varying plate temperature, fluid injection and fluid suction at the plate. The work by Pop et al. [7] belongs to the above *
Tel.: +30 541 079618. E-mail address: [email protected]
0020-7225/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.07.005
1892
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
class of problems, including the variation of fluid viscosity with temperature. The authors obtained similarity solutions considering that viscosity varies as an inverse function of temperature. However, the Prandtl number, which is a function of viscosity, has been considered constant across the boundary layer and this assumption leads to unrealistic results when the variation of viscosity with temperature is strong. The objective of the present paper is to obtain results considering both viscosity and Prandtl number variable across the boundary layer. As will be shown later the differences of the two methods are very large.
2. The mathematical model Consider the flow along a moving plate placed in a calm environment with u and v denoting respectively the velocity components in the x and y direction, where x is the coordinate along the plate and y is the coordinate perpendicular to x. For steady, two-dimensional flow the boundary layer equations including variable viscosity are ou ov þ ¼0 ox oy
continuity equation :
momentum equation :
energy equation :
u
ou ov 1 o ou u þv ¼ l ox oy q1 oy oy
oT oT o2 T þv ¼a 2 ox oy oy
ð1Þ ð2Þ
ð3Þ
where T is the fluid temperature, l is the dynamic viscosity, a is the thermal diffusivity, and q1 is the ambient fluid density. The boundary conditions are as follows: at y ¼ 0 u ¼ U ; v ¼ 0; T ¼ T w
ð4Þ
as y ! 1 u ¼ 0; T ¼ T 1
ð5Þ
where Tw is the plate temperature, T1 is the ambient fluid temperature and U is the velocity of the moving plate. The viscosity is assumed to be an inverse linear function of temperature given by the following equation [7]: 1 1 ¼ ½1 þ cðT w T 1 Þ l l1
ð6Þ
where l1 is the ambient fluid dynamic viscosity and c is a thermal property of the fluid. Eq. (6) can be rewritten as follows: 1 ¼ aðT T r Þ l
ð7Þ
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
1893
where a = c/l1 and Tr = T1 1/c are constants and their values depend on the reference state and the thermal property of the fluid. Eqs. (1)–(3) form a parabolic system and were solved directly, without any transformation, by a method described by Patankar [6]. The finite difference method is used with primitive variables x, y and a space marching procedure is used in x direction with an expanding grid. A detailed description of the solution procedure may be found in [5] where all fluid properties (viscosity, thermal diffusivity and density) have been considered as functions of temperature.
3. Results and discussion The most important quantities for this problem are the wall heat transfer and the wall shear stress defined as x oT Re1=2 ð8Þ h0 ð0Þ ¼ Tw T1 oy y¼0 #r 1 lw 1=2 ou f ð0Þ ¼ Re #r q1 U 2 oy y¼0 00
ð9Þ
where h is the dimensionless temperature (T T1)/(Tw T1) and f is the dimensionless stream function for which the following equation is valid: u ð10Þ f0 ¼ U The Reynolds number is defined as Ux Re ¼ ð11Þ m1 and hr is a constant defined by hr ¼
Tr T1 1 ¼ cðT w T 1 Þ Tw T1
ð12Þ
In Eqs. (8)–(10) the prime represents differentiation with respect to similarity variable g defined as y g ¼ Re1=2 ð13Þ x It should be mentioned here that when hr ! 1 the fluid viscosity becomes equal to ambient viscosity and we have the classical Sakiadis problem. In order to test the accuracy of the present method, results were compared with those available in the literature. The wall heat transfer h 0 (0) and the wall shear stress f00 (0) for the present problem with constant viscosity and Pr = 0.7 are 0.4446 and 0.3507 respectively. The corresponding quantities calculated by the present method are 0.4438 and 0.3500. The comparison is satisfactory and this happens for other Pr numbers. In contrast to the above direct solution of Eqs. (1)–(3), [7] transformed these equations into the following similarity equations:
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A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
f 000
# #r 00 1 ff #0 f 00 ¼ 0 # #r 2#r
ð14Þ
1 00 1 0 ð15Þ # þ f# ¼ 0 Pr 2 It should be mentioned here that in the transformed energy Eq. (15) the Prandtl number has been assumed constant across the boundary layer. Pop et al. have calculated this Prandtl number at ambient temperature from the following equation: m1 ð16Þ Pr1 ¼ a However, the Prandtl number is a function of viscosity and as viscosity varies across the boundary layer, the Prandtl number varies, too. As will be shown below this assumption leads to unrealistic results for small values of hr. In Tables 1 and 2 the wall shear stress and the wall heat transfer are given for ambient Prandtl numbers 0.7 and 10 respectively. In these tables the results by Pop et al. [7] have been also included for comparison. In the last column of each table the Prandtl numbers at the plate (Prw) are included. From Table 1 it is seen that the wall shear stress, calculated by the two methods, are in agreement. For wall heat transfer things are different. For large values of the parameter jhrj the results are in agreement but as jhrj decreases the results of the two methods diverge. The same observation is valid for Pr1 = 10 but now the divergence appears also in the wall shear stress. It is advocated here that the results of Pop et al. for low jhrj are unrealistic. The explanation is drawn from Fig.1 where the temperature profiles are shown for hr = 0.001 and ambient Pr numbers 0.7 and 10. It is seen that the real temperature profiles, calculated with variable Pr number, are much wider than those given by Pop et al. calculated with constant ambient Pr number. The error is Table 1 Values of f00 (0) and h 0 (0) for Pr1 = 0.7 f00 (0)
h 0 (0)
hr
Present work
Pop et al.
Difference %
Present work
Pop et al.
Difference %
Prw
10 8 6 4 2 1 0.1 0.01 0.001 2 4 6 8 10
0.4712 0.4775 0.4879 0.5078 0.5627 0.6561 1.5021 4.4952 14.1528 0.2795 0.3706 0.3969 0.4094 0.4168
0.4710 0.4774 0.4877 0.5078 0.5629 0.6565 1.5062 4.4857 14.0654 0.2783 0.3699 0.3963 0.4089 0.4163
<1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1 <1
0.3452 0.3441 0.3422 0.3384 0.3278 0.3089 0.1735 0.0711 0.0310 0.3782 0.3632 0.3587 0.3565 0.3552
0.3504 0.3493 0.3476 0.3442 0.3349 0.3189 0.2191 0.1545 0.1341 0.3807 0.3667 0.3625 0.3605 0.3593
1.5 1.5 1.5 2 2 3 26 117 333 <1 <1 1 1 1
0.64 0.62 0.60 0.56 0.47 0.35 0.06 0.007 0.0007 1.40 0.93 0.84 0.80 0.78
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
1895
Table 2 Values of f00 (0) and h 0 (0) for Pr1 = 10 f00 (0)
h 0 (0)
hr
Present work
Pop et al.
Difference %
Present work
Pop et al.
Difference %
Prw
10 8 6 4 2 1 0.1 0.01 0.001 2 4 6 8 10
0.4805 0.4888 0.5032 0.5314 0.6109 0.7522 1.8446 4.6095 14.1323 0.2457 0.3497 0.3824 0.3983 0.4078
0.5067 0.5158 0.5310 0.5608 0.6451 0.7955 1.8734 2.7939 4.8035 0.2571 0.3680 0.4027 0.4196 0.4296
5 5 5 5 5 6 2 39 66 5 5 5 5 5
1.6811 1.6793 1.6771 1.6731 1.6594 1.6366 1.4141 0.7751 0.2723 1.7145 1.6990 1.6937 1.6913 1.6905
1.6816 1.6731 1.6707 1.6659 1.6521 1.6331 1.5762 1.4576 1.4576 1.7129 1.6962 1.6908 1.6882 1.6867
<1 <1 <1 <1 <1 <1 11 88 435 <1 <1 <1 <1 <1
9.10 8.89 8.57 8.00 6.67 5.00 0.91 0.10 0.01 20.00 13.33 12.00 11.43 11.11
Fig. 1. Temperature distribution for hr = 0.001 and ambient Pr numbers 0.7 and 10: solid line, present work: dashed line, [7].
1896
A. Pantokratoras / International Journal of Engineering Science 42 (2004) 1891–1896
introduced by considering that the ambient Pr number is valid in the entire boundary layer but this is an approximation that is valid only for large values of jhrj (small variation of viscosity and small variation between ambient and wall Prandtl number). From the above tables we see that for hr = 0.001 the Prandtl number at the plate is 1000 times smaller than the ambient one and this means that the Pr number inside the boundary layer is much smaller than the ambient one. It is well known in the boundary layer theory that large Pr numbers correspond to ‘‘narrow’’ temperature profiles and small Pr numbers to wider temperature profiles. This is the reason for the difference in the temperature profiles and the wall heat transfer between the two methods. The finding of the present work may be valuable to a class of similar problems. For example in the works of Kafoussias and Williams [4], Hady et al. [1], Hossain et al. [3] and Hossain and Munir [2] the Prandtl number has been also considered constant in the transformed energy equation. This assumption may give unrealistic results if the above works are applied to fluids with strong relationship between viscosity and temperature.
References [1] F.M. Hady, A.Y. Bakier, R.S.R. Gorla, Mixed convection boundary layer flow on a continuous flat plate with variable viscosity, Heat and Mass Transfer 31 (1996) 169–172. [2] M.A. Hossain, M.S. Munir, Mixed convection flow from a vertical flat plate with temperature dependent viscosity, International Journal of Thermal Science 39 (2000) 173–183. [3] M.A. Hossain, M.S. Munir, M.Z. Hafiz, H.S. Takhar, Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux, Heat and Mass Transfer 36 (2000) 333– 341. [4] N.G. Kafoussias, E.W. Williams, The effect of temperature-dependent viscosity on the free-forced convective laminar boundary layer flow past a vertical isothermal flat plate, Acta Mechanica 110 (1995) 123–137. [5] A. Pantokratoras, Laminar free-convection over a vertical isothermal plate with uniform blowing or suction in water with variable physical properties, International Journal of Heat and Mass Transfer 45 (2002) 963–977. [6] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Company, New York, 1980. [7] I. Pop, R.S.R. Gorla, M. Rashidi, The effect of variable viscosity on flow and heat transfer to a continuous moving flat plate, International Journal of Engineering Science 30 (1992) 1–6. [8] B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces: the boundary layer on a continuous flat surface, AIChE Journal 7 (1961) 221–225.