- Email: [email protected]
Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition
Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition Abdul Aziz * Department of Mechanical Engineering, School of Engineering & Applied Science Gonzaga University, Spokane, WA 99258, USA
a r t i c l e
i n f o
Article history: Received 19 November 2008 Received in revised form 11 April 2009 Accepted 11 April 2009 Available online 3 May 2009 PACS: 44.20.+b 47.45.Gx Keywords: Flat plate Hydrodynamic and thermal boundary layers Slip flow Constant heat flux
a b s t r a c t In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The boundary layer flow over a flat plate in a uniform stream of fluid has been studied extensively in fluid mechanics. When fluid flows in micro electro mechanical systems or MEMS are encountered, the no slip condition at the solid–fluid interface is abandoned in favor of a slip flow model which represents more accurately the non-equilibrium region near the interface. The first-order slip flow model proposes a relationship between the tangential component of the velocity at the surface, the velocity gradient normal to the surface, the mean free path, and the tangential momentum accommodation coefficient [1]. The effect of slip flow on the hydrodynamic boundary layer over a stationary flat plate has been studied by Martin and Boyd [2] who employed the Maxwell slip condition. The same problem has been studied by Vedantam [3] with three different models for the slip flow. The stationary plate model was extended by Fang and Lee [4] to a moving flat plate
* Tel.: +1 509 323 3540; fax: +1 509 323 5871. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.04.026
574
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
and by Anderson [5] to a stretching surface. For a stationary plate, the general conclusion from these investigations is that as the slip (rarefaction) parameter increases, the slip velocity increases and the wall shear stress decreases. For the moving plate, however, the momentum equation has two solutions and the location of the maximum shear stress which occurs on the plate surface for a stationary plate moves out into the fluid at a certain distance from the plate when the plate is moving. This distance depends on the ratio of the plate velocity and the free stream velocity [4]. The effect of slip flow on the flat plate thermal boundary layer characteristics has received limited attention. In a recent paper, Martin and Boyd [6] considered the thermal boundary layer over an isothermal flat plate in the presence of slip flow. In this analysis they took into account the effect of slip parameter K (a function of x, direction along the plate) in transforming the momentum equation and found that the stream function and consequently the y (direction normal to the plate) component of the velocity (v) also depends on the variation of the similarity variable f with respect to K. They neglected this in their earlier work [2]. With this modification, the similarity equation for the hydrodynamic boundary layer became a partial differential equation instead of the classical Blasius ordinary differential equation. With the introduction of temperature jump condition, the transformed energy equation also became a partial differential equation. They solved both equations numerically using center-difference approximation. With the neglect of temperature jump condition, the heat transfer was found to increase as the slip velocity increased. However, when the temperature jump condition was retained the heat transfer decreased as the slip velocity increased. Other papers that are relevant to the present work are those of Wang [7], Anderson [8]. Abel and Mahesha [9,10]. Wang [7] considered the analysis of viscous flow due to stretching sheet with surface slip and suction, while Anderson [8] provided an exact solution of the Navier–Stokes equations describing the flow past a stretching boundary with partial slip. The papers by Abel and Mahesha [9,10] are important contributions because they included effect of variable thermal conductivity, heat source, radiation, buoyancy, magneto-hydrodynamic effects, and viscoelastic behavior of the fluid. This paper considers the effect of slip flow on the thermal boundary layer over a flat plate with a constant heat flux boundary condition instead of a constant temperature boundary condition used by Martin and Boyd [6]. This situation arises in a MEM condensation application where a fixed heat dissipation (constant heat flux) due to condensation on lower surface of the plate is be removed by the gas flowing over the top surface. As a preliminary study the temperature jump condition will be neglected. As noted in [6], this is a reasonable assumption for the flow of liquids at the micro scale level particularly because of the lack of data on the thermal accommodation coefficient. 2. Momentum and energy equations The continuity and momentum equations for the laminar hydrodynamic boundary layer over a flat plate in a uniform stream of fluid can be reduced to the Blasius differential equation. 00
2f 000 þ ff ¼ 0
ð1Þ
where the dependent variable f(g) and the independent variable g are defined as
wðx; yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 1 mx=U 1 1=2 U g¼y 1 mx f ðgÞ ¼
ð2Þ ð3Þ
In Eq. (2), w(x,y) is the stream function, U1 is the free stream velocity, x is the coordinate along the plate, y is the coordinate normal to the plate, and m is the kinematic viscosity of the fluid. The x component of the velocity, u(x,y) is given by
f 0 ðgÞ ¼
uðx; yÞ U1
ð4Þ
and the y component of the velocity, v(x,y), is given by
v ðx; yÞ ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 mU1=xðgf f Þ 2
ð5Þ
The boundary condition at large normal distance from the plate is
f 0 ð1Þ ¼ 1
ð6Þ
For rarefied flow, the slip condition of the form given Martin and Boyd [2] is used.
uy¼0 ¼
ð2 rÞ ou k oy r
ð7Þ
y¼0
where k is the mean free path and r is the tangential momentum accommodation coefficient. Eq. (7) may be expressed in dimensionless form as
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
f 0 ð0Þ ¼
2r
r
00
00 Knx Re1=2 x f ð0Þ ¼ Kf ð0Þ
575
ð8Þ
where the local Knudsen number Knx and the local Reynolds number Rex are defined as
Knx ¼ k=x U1 x Rex ¼
m
ð9Þ ð10Þ
and
K¼
2r
r
Knx Re1=2 x
ð11Þ
It should be noted that the slip (rarefaction) parameter K is a function of x and as a consequence the solutions of Eq. (1) subject to the boundary conditions (6) and (8) for fixed values of K would be locally similar. 00 For the thermal boundary layer without viscous dissipation, the energy equation for constant q may be written as
oT oT o2 T þv ¼a 2 ox oy oy oT k ðx; 0Þ ¼ q00 oy Tðx; 1Þ ¼ T 1
u
ð12Þ ð13Þ ð14Þ
Following Bejan [11], we introduce the following dimensionless variable for the temperature
h¼
T T1 1=2
q00 k
ð15Þ
mx U1
and use the other variables defined previously. The energy equation reduces to the following ordinary differential equation.
1 h00 þ Prðf h0 f 0 hÞ ¼ 0 2 h0 ð0Þ ¼ 1 hð1Þ ¼ 0
ð16Þ ð17Þ ð18Þ
3. Numerical solutions Hydrodynamic results The numerical solution of Eq. (1) subject to the boundary conditions (6) and (8) was obtained using the symbolic algebra software Maple 11. The dimensionless stream function f as a function of g is shown in Fig. 1 for K = 0 (no slip), 1, 2, 3, 4, and 5. For any location in the boundary layer (i.e. for any value of g), the stream function f increases as the flow becomes more rarefied, that is, as K increases. This is consistent with the results shown in Fig. 2 of Martin and Boyd [6] but perhaps the authors mistakenly use the wording ‘‘f decreases” in the text. The results for the dimensionless x component of the velocity f0 (g) appear in Fig. 2. As the flow becomes more rarefied, the slip velocity f0 (0) increases and so does the x component of the velocity for any value of g. However, the dimensionless y component of the velocity, (gf0 f), decreases as K increases as illustrated in Fig. 3. Some of the results for the stream function and the two components of the velocity read from Martin and Boyd [6] were checked against the present results and found to be close despite the fact that Martin and Boyd [6] took into account the variation of the stream function with parameter K which was neglected in the present analysis. Though precise, this refinement does not appear to have a significant effect on the y component of the velocity. Fig. 4 shows the effect of increasing rarefaction on the slip velocity and the wall shear. As noted earlier, the slip velocity increases as the flow becomes more rarefied. The wall shear, however, decreases with the increase in rarefaction. The shear which is given by the initial slope of the curves in Fig. 2 is a maximum when K = 0 (no slip). These conclusions are consistent with those reported by Martin and Boyd [2]. It is interesting to point the same authors in a later paper [6] found that the maximum shear occurs at K 0.5 and not at K = 0. Similarly, they found that the y component of the velocity exhibited a local peak for K < 1. They attributed these anomalies to the ‘‘loss of self-similarity in the flow”. Thermal results The solution of the thermal Eqs. (16)–(18) was also obtained using Maple 11. Fig. 5 shows the temperature distribution in the thermal boundary layer for Prandtl number of 0.72 for K = 0, 1, 2, 3, 4, 5, and 6. As the flow rarefaction increases, temperatures in the thermal boundary layer are progressively lowered. The effect becomes less pronounced beyond K = 2 when
576
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Fig. 1. Stream function as a function of g for different values of K.
Fig. 2. Effect of flow rarefaction on the dimensionless x component of the velocity, f 0 .
the curves become very close to each other. Because of the constant heat flux boundary condition, the initial slope of each curve is the same and is equal to 1. It can also be seen that the thickness of the thermal boundary decreases as the flow becomes more rarefied. The effect of Prandtl number on the temperature distribution in the thermal boundary layer at K = 1 is illustrated in Fig. 6. As Prandtl number increases, the temperature at every location in the thermal boundary layer decreases. The thickness and the Reynolds of the boundary layer decreases as Prandtl number increases as in the classical case of an isothermal flat plate with no slip.
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A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Fig. 3. Effect of flow rarefaction on the dimensionless y component of the velocity, (gf 0 f ).
1 0.9 0.8 0.7 0.6 0.5 f´´(0)
0.4
f´(0)
0.3 0.2 0.1 0 0
1
2
3
4
K
5
6
7
8
9
10
Fig. 4. Slip velocity and wall shear as a function of K.
Fig. 7 shows the effect of rarefaction on the surface temperature for Prandtl numbers of 0.72, 10, and 100. As Prandtl number increases, the surface temperature decreases. For a given Prandtl number, the surface temperature decreases as the flow becomes more rarefied. Using Eq. (15), the actual surface temperature is
Ts ¼ T1 þ
1=2 q00 mx hð0Þ k U1
ð19Þ
In view of Eqs. (11) and (19), h(0) is a function of the Prandtl number, the local Knudsen number, the local Reynolds number and the tangential momentum accommodation coefficient. To evaluate the local surface temperature at any value of x, the accommodation coefficient must be known. Then the local Knudsen number needs to be calculated to determine the value of K. The values of K and Prandtl number then determine the value of h(0). Finally q00 ,k,m,U1 must be known to determine the value of Ts. Fig. 8 illustrates the variation of the temperature gradient in the thermal boundary layer. At any location in the boundary layer the gradient, h0 , decreases as the flow rarefaction parameter increases. However, beyond K = 2, the curves tend to coalesce into a single curve indicating that the effect of K on the temperature gradient is diminished considerably.
578
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Fig. 5. Temperature distribution in the thermal boundary at Pr = 0.72 for different flow rarefactions.
Fig. 6. Effect of Prandtl number on the temperature distribution in the thermal boundary layer at K = 1.
The hydrodynamic and thermal characteristics of the boundary flow over a flat plate with slip and constant heat flux have been obtained numerically. With the use of the classical similarity variables, the continuity and energy equations can be reduced to a similarity equation that is locally valid. As the fluid becomes more rarefied, the x component of the velocity increases but the y component of the velocity and the wall shear decrease. At any location in the hydrodynamic boundary layer, the stream function increases as the slip increases. The non-similarity arising as a result of the slip affects the otherwise similar energy equation. As the flow rarefaction increases, temperatures in the thermal boundary layer are progressively lowered. However, the effect diminishes as the flow becomes further rarefied. The local surface temperature decreases as Prandtl number and the flow rarefaction increase. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow and fluid properties, and the applied heat flux.
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A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
2.5 Pr = 0.72 Pr = 10 Pr = 100
2 1.5
θ (0) 1 0.5 0
0
1
2
K
3
4
5
Fig. 7. Surface temperature as a function of Prandtl number and the flow rarefaction.
Fig. 8. Temperature gradient variation in the thermal boundary layer for different flow rarefactions.
4. Conclusions The hydrodynamic and thermal characteristics of the boundary flow over a flat plate with slip and constant heat flux have been obtained numerically. With the use of the classical similarity variables, the continuity and energy equations can be reduced to a similarity equation that is locally valid. As the fluid becomes more rarefied, the x component of the velocity increases but the y component of the velocity and the wall shear decrease. At any location in the hydrodynamic boundary layer, the stream function increases as the slip increases. The non-similarity arising as a result of the slip affects the otherwise similar energy equation. As the flow rarefaction increases, temperatures in the thermal boundary layer are progressively lowered. However, the effect diminishes as the flow becomes further rarefied. The local surface temperature decreases as Prandtl number and the flow rarefaction increase. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow and fluid properties, and the applied heat flux.
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A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
References [1] Gad-el-Hak M. In: Gad-el-Hak M, editor. Flow physics in the MEMS handbook. Boca Raton, FL: CRC Press; 2002. Chapter 4. [2] Martin MJ, Boyd ID. Blasius boundary layer with slip flow conditions. In: Bartel TJ, Gallis MA, editors. 22nd Rarefied gas dynamics symposium. Sydney, Australia, July 2000. [3] Vedantam NK. Effects of slip on the flow characteristics of a laminar flat plate boundary layer. In: Proceedings of ASME fluids engineering summer meeting, Miami, FL, July 17–20. 2006. p. 1551–60. [4] Fang T, Lee CF. A moving wall boundary layer flow of a slightly rarefied gas free stream over a moving flat plate. Appl Math Lett 2005;18:487–95. [5] Anderson HI. Slip flow past a stretching surface. Acta Mech 2002;158:121–5. [6] Martin MJ, Boyd ID. Momentum and heat transfer in laminar boundary layer with slip flow. J Thermophys Heat Transfer 2006;20(4):710–9. [7] Wang CY. Analysis of viscous flow due to stretching sheet with surface slip and suction. Nonlinear Anal Real World Appl 2009;10:375–80. [8] Anderson HI. Slip flow past a stretching boundary with partial slip-an exact solution of the Navier–Stokes equations. Chem Eng Sci 2002;57:3745–7. [9] Abel S, Mahesha N. Effects of thermal buoyancy and variable thermal conductivity in a power law fluid past a vertical stretching sheet in the presence of non-uniform heat source. Int J Nonlinear Mech 2009;44:1–12. [10] Abel S, Mahesha N. Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl Math Mod 2008;32. [11] Bejan A. Convection heat transfer. 3rd ed. New York: John Wiley; 2004. p. 84.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition Abdul Aziz * Department of Mechanical Engineering, School of Engineering & Applied Science Gonzaga University, Spokane, WA 99258, USA
a r t i c l e
i n f o
Article history: Received 19 November 2008 Received in revised form 11 April 2009 Accepted 11 April 2009 Available online 3 May 2009 PACS: 44.20.+b 47.45.Gx Keywords: Flat plate Hydrodynamic and thermal boundary layers Slip flow Constant heat flux
a b s t r a c t In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The boundary layer flow over a flat plate in a uniform stream of fluid has been studied extensively in fluid mechanics. When fluid flows in micro electro mechanical systems or MEMS are encountered, the no slip condition at the solid–fluid interface is abandoned in favor of a slip flow model which represents more accurately the non-equilibrium region near the interface. The first-order slip flow model proposes a relationship between the tangential component of the velocity at the surface, the velocity gradient normal to the surface, the mean free path, and the tangential momentum accommodation coefficient [1]. The effect of slip flow on the hydrodynamic boundary layer over a stationary flat plate has been studied by Martin and Boyd [2] who employed the Maxwell slip condition. The same problem has been studied by Vedantam [3] with three different models for the slip flow. The stationary plate model was extended by Fang and Lee [4] to a moving flat plate
* Tel.: +1 509 323 3540; fax: +1 509 323 5871. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.04.026
574
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
and by Anderson [5] to a stretching surface. For a stationary plate, the general conclusion from these investigations is that as the slip (rarefaction) parameter increases, the slip velocity increases and the wall shear stress decreases. For the moving plate, however, the momentum equation has two solutions and the location of the maximum shear stress which occurs on the plate surface for a stationary plate moves out into the fluid at a certain distance from the plate when the plate is moving. This distance depends on the ratio of the plate velocity and the free stream velocity [4]. The effect of slip flow on the flat plate thermal boundary layer characteristics has received limited attention. In a recent paper, Martin and Boyd [6] considered the thermal boundary layer over an isothermal flat plate in the presence of slip flow. In this analysis they took into account the effect of slip parameter K (a function of x, direction along the plate) in transforming the momentum equation and found that the stream function and consequently the y (direction normal to the plate) component of the velocity (v) also depends on the variation of the similarity variable f with respect to K. They neglected this in their earlier work [2]. With this modification, the similarity equation for the hydrodynamic boundary layer became a partial differential equation instead of the classical Blasius ordinary differential equation. With the introduction of temperature jump condition, the transformed energy equation also became a partial differential equation. They solved both equations numerically using center-difference approximation. With the neglect of temperature jump condition, the heat transfer was found to increase as the slip velocity increased. However, when the temperature jump condition was retained the heat transfer decreased as the slip velocity increased. Other papers that are relevant to the present work are those of Wang [7], Anderson [8]. Abel and Mahesha [9,10]. Wang [7] considered the analysis of viscous flow due to stretching sheet with surface slip and suction, while Anderson [8] provided an exact solution of the Navier–Stokes equations describing the flow past a stretching boundary with partial slip. The papers by Abel and Mahesha [9,10] are important contributions because they included effect of variable thermal conductivity, heat source, radiation, buoyancy, magneto-hydrodynamic effects, and viscoelastic behavior of the fluid. This paper considers the effect of slip flow on the thermal boundary layer over a flat plate with a constant heat flux boundary condition instead of a constant temperature boundary condition used by Martin and Boyd [6]. This situation arises in a MEM condensation application where a fixed heat dissipation (constant heat flux) due to condensation on lower surface of the plate is be removed by the gas flowing over the top surface. As a preliminary study the temperature jump condition will be neglected. As noted in [6], this is a reasonable assumption for the flow of liquids at the micro scale level particularly because of the lack of data on the thermal accommodation coefficient. 2. Momentum and energy equations The continuity and momentum equations for the laminar hydrodynamic boundary layer over a flat plate in a uniform stream of fluid can be reduced to the Blasius differential equation. 00
2f 000 þ ff ¼ 0
ð1Þ
where the dependent variable f(g) and the independent variable g are defined as
wðx; yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 1 mx=U 1 1=2 U g¼y 1 mx f ðgÞ ¼
ð2Þ ð3Þ
In Eq. (2), w(x,y) is the stream function, U1 is the free stream velocity, x is the coordinate along the plate, y is the coordinate normal to the plate, and m is the kinematic viscosity of the fluid. The x component of the velocity, u(x,y) is given by
f 0 ðgÞ ¼
uðx; yÞ U1
ð4Þ
and the y component of the velocity, v(x,y), is given by
v ðx; yÞ ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 mU1=xðgf f Þ 2
ð5Þ
The boundary condition at large normal distance from the plate is
f 0 ð1Þ ¼ 1
ð6Þ
For rarefied flow, the slip condition of the form given Martin and Boyd [2] is used.
uy¼0 ¼
ð2 rÞ ou k oy r
ð7Þ
y¼0
where k is the mean free path and r is the tangential momentum accommodation coefficient. Eq. (7) may be expressed in dimensionless form as
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
f 0 ð0Þ ¼
2r
r
00
00 Knx Re1=2 x f ð0Þ ¼ Kf ð0Þ
575
ð8Þ
where the local Knudsen number Knx and the local Reynolds number Rex are defined as
Knx ¼ k=x U1 x Rex ¼
m
ð9Þ ð10Þ
and
K¼
2r
r
Knx Re1=2 x
ð11Þ
It should be noted that the slip (rarefaction) parameter K is a function of x and as a consequence the solutions of Eq. (1) subject to the boundary conditions (6) and (8) for fixed values of K would be locally similar. 00 For the thermal boundary layer without viscous dissipation, the energy equation for constant q may be written as
oT oT o2 T þv ¼a 2 ox oy oy oT k ðx; 0Þ ¼ q00 oy Tðx; 1Þ ¼ T 1
u
ð12Þ ð13Þ ð14Þ
Following Bejan [11], we introduce the following dimensionless variable for the temperature
h¼
T T1 1=2
q00 k
ð15Þ
mx U1
and use the other variables defined previously. The energy equation reduces to the following ordinary differential equation.
1 h00 þ Prðf h0 f 0 hÞ ¼ 0 2 h0 ð0Þ ¼ 1 hð1Þ ¼ 0
ð16Þ ð17Þ ð18Þ
3. Numerical solutions Hydrodynamic results The numerical solution of Eq. (1) subject to the boundary conditions (6) and (8) was obtained using the symbolic algebra software Maple 11. The dimensionless stream function f as a function of g is shown in Fig. 1 for K = 0 (no slip), 1, 2, 3, 4, and 5. For any location in the boundary layer (i.e. for any value of g), the stream function f increases as the flow becomes more rarefied, that is, as K increases. This is consistent with the results shown in Fig. 2 of Martin and Boyd [6] but perhaps the authors mistakenly use the wording ‘‘f decreases” in the text. The results for the dimensionless x component of the velocity f0 (g) appear in Fig. 2. As the flow becomes more rarefied, the slip velocity f0 (0) increases and so does the x component of the velocity for any value of g. However, the dimensionless y component of the velocity, (gf0 f), decreases as K increases as illustrated in Fig. 3. Some of the results for the stream function and the two components of the velocity read from Martin and Boyd [6] were checked against the present results and found to be close despite the fact that Martin and Boyd [6] took into account the variation of the stream function with parameter K which was neglected in the present analysis. Though precise, this refinement does not appear to have a significant effect on the y component of the velocity. Fig. 4 shows the effect of increasing rarefaction on the slip velocity and the wall shear. As noted earlier, the slip velocity increases as the flow becomes more rarefied. The wall shear, however, decreases with the increase in rarefaction. The shear which is given by the initial slope of the curves in Fig. 2 is a maximum when K = 0 (no slip). These conclusions are consistent with those reported by Martin and Boyd [2]. It is interesting to point the same authors in a later paper [6] found that the maximum shear occurs at K 0.5 and not at K = 0. Similarly, they found that the y component of the velocity exhibited a local peak for K < 1. They attributed these anomalies to the ‘‘loss of self-similarity in the flow”. Thermal results The solution of the thermal Eqs. (16)–(18) was also obtained using Maple 11. Fig. 5 shows the temperature distribution in the thermal boundary layer for Prandtl number of 0.72 for K = 0, 1, 2, 3, 4, 5, and 6. As the flow rarefaction increases, temperatures in the thermal boundary layer are progressively lowered. The effect becomes less pronounced beyond K = 2 when
576
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Fig. 1. Stream function as a function of g for different values of K.
Fig. 2. Effect of flow rarefaction on the dimensionless x component of the velocity, f 0 .
the curves become very close to each other. Because of the constant heat flux boundary condition, the initial slope of each curve is the same and is equal to 1. It can also be seen that the thickness of the thermal boundary decreases as the flow becomes more rarefied. The effect of Prandtl number on the temperature distribution in the thermal boundary layer at K = 1 is illustrated in Fig. 6. As Prandtl number increases, the temperature at every location in the thermal boundary layer decreases. The thickness and the Reynolds of the boundary layer decreases as Prandtl number increases as in the classical case of an isothermal flat plate with no slip.
577
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Fig. 3. Effect of flow rarefaction on the dimensionless y component of the velocity, (gf 0 f ).
1 0.9 0.8 0.7 0.6 0.5 f´´(0)
0.4
f´(0)
0.3 0.2 0.1 0 0
1
2
3
4
K
5
6
7
8
9
10
Fig. 4. Slip velocity and wall shear as a function of K.
Fig. 7 shows the effect of rarefaction on the surface temperature for Prandtl numbers of 0.72, 10, and 100. As Prandtl number increases, the surface temperature decreases. For a given Prandtl number, the surface temperature decreases as the flow becomes more rarefied. Using Eq. (15), the actual surface temperature is
Ts ¼ T1 þ
1=2 q00 mx hð0Þ k U1
ð19Þ
In view of Eqs. (11) and (19), h(0) is a function of the Prandtl number, the local Knudsen number, the local Reynolds number and the tangential momentum accommodation coefficient. To evaluate the local surface temperature at any value of x, the accommodation coefficient must be known. Then the local Knudsen number needs to be calculated to determine the value of K. The values of K and Prandtl number then determine the value of h(0). Finally q00 ,k,m,U1 must be known to determine the value of Ts. Fig. 8 illustrates the variation of the temperature gradient in the thermal boundary layer. At any location in the boundary layer the gradient, h0 , decreases as the flow rarefaction parameter increases. However, beyond K = 2, the curves tend to coalesce into a single curve indicating that the effect of K on the temperature gradient is diminished considerably.
578
A. Aziz / Commun Nonlinear Sci Numer Simulat 15 (2010) 573–580
Fig. 5. Temperature distribution in the thermal boundary at Pr = 0.72 for different flow rarefactions.
Fig. 6. Effect of Prandtl number on the temperature distribution in the thermal boundary layer at K = 1.
The hydrodynamic and thermal characteristics of the boundary flow over a flat plate with slip and constant heat flux have been obtained numerically. With the use of the classical similarity variables, the continuity and energy equations can be reduced to a similarity equation that is locally valid. As the fluid becomes more rarefied, the x component of the velocity increases but the y component of the velocity and the wall shear decrease. At any location in the hydrodynamic boundary layer, the stream function increases as the slip increases. The non-similarity arising as a result of the slip affects the otherwise similar energy equation. As the flow rarefaction increases, temperatures in the thermal boundary layer are progressively lowered. However, the effect diminishes as the flow becomes further rarefied. The local surface temperature decreases as Prandtl number and the flow rarefaction increase. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow and fluid properties, and the applied heat flux.
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2.5 Pr = 0.72 Pr = 10 Pr = 100
2 1.5
θ (0) 1 0.5 0
0
1
2
K
3
4
5
Fig. 7. Surface temperature as a function of Prandtl number and the flow rarefaction.
Fig. 8. Temperature gradient variation in the thermal boundary layer for different flow rarefactions.
4. Conclusions The hydrodynamic and thermal characteristics of the boundary flow over a flat plate with slip and constant heat flux have been obtained numerically. With the use of the classical similarity variables, the continuity and energy equations can be reduced to a similarity equation that is locally valid. As the fluid becomes more rarefied, the x component of the velocity increases but the y component of the velocity and the wall shear decrease. At any location in the hydrodynamic boundary layer, the stream function increases as the slip increases. The non-similarity arising as a result of the slip affects the otherwise similar energy equation. As the flow rarefaction increases, temperatures in the thermal boundary layer are progressively lowered. However, the effect diminishes as the flow becomes further rarefied. The local surface temperature decreases as Prandtl number and the flow rarefaction increase. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow and fluid properties, and the applied heat flux.
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