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Effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching surface
Nuclear Engineering and Design 241 (2011) 2660–2665
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching surface Swati Mukhopadhyay ∗ Department of Mathematics, The University of Burdwan, Burdwan 713104, W.B., India
a r t i c l e
i n f o
Article history: Received 13 August 2010 Received in revised form 19 April 2011 Accepted 5 May 2011
a b s t r a c t This paper investigates the unsteady mixed convective boundary layer flow and heat transfer over a porous stretching vertical surface in presence of slip. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically. With increasing values of the unsteadiness parameter, fluid velocity and the temperature are found to decrease in both the presence and absence of slip at the boundary. Fluid velocity decreases due to increasing values of the velocity slip parameter resulting an increase in the temperature field. Skin-friction decreases with the velocity slip parameter whereas it increases with unsteadiness parameter. The rate of heat transfer decreases with the velocity slip parameter while increases with unsteadiness parameter. Same feature is also noticed for thermal slip parameter. Effects of increasing mixed convection parameter on the velocity boundary layer is to increase the velocity field and the temperature decreases in this case. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The study of hydrodynamic flow and heat transfer over a stretching sheet has gained considerable attention due to its vast applications in industry and its importance to several technological processes. The production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. The tangential velocity imparted by the sheet induces motion in the surrounding fluid that alters the convection cooling of the sheet. Knowledge of the flow properties of the fluid is desirable because the quality of the resulting sheeting material, as well as the cost of production, is affected by the speed of collection and mass transfer rate. In recent years, a great deal of interest has been generated in the area of boundary layer mixed convection flow on a vertical stretching surface in view of its numerous and ever increasing industrial and technical applications which include aerodynamic extrusion of plastic sheets, cooling of metallic sheets in a cooling bath, crystal growing, etc. In the study of horizontal heated or cooled surfaces, the effect of buoyancy force is neglected. However, for vertical or inclined surfaces, the buoyancy force modifies the flow field and hence the heat transfer rate [1]. The importance of this phenomenon is increasing day by day due to the enhanced concern in science and technology about buoyancy induced motions in the atmosphere, the bodies in water and quasi-solid bodies such as earth. Buoyancy plays an important role where the temperature
∗ Tel.: +91 342 2557741; fax: +91 342 2530452. E-mail address: swati [email protected] 0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.05.007
differences between land and air give rise to a complicated flow in enclosures such as ventilated and heated rooms [2]. The buoyancy force arising due to the heating of a stretching surface, under some circumstances, may alter significantly the flow and thermal fields and thereby the heat transfer behaviour in the manufacturing process [3]. Many studies were reported considering the effect of buoyancy forces on the boundary layer [4–6]. All of the above mentioned studies were restricted in the steady state conditions. The transient or unsteady aspects become interesting in certain practical problems where the motion of the stretched surface may start impulsively from rest. Elbashbeshy and Bazid [2], Sharidan et al. [7], Mukhopadhyay [8] presented similarity solutions for unsteady flow and heat transfer over a stretching surface. The non-adherence of the fluid to a solid boundary, also known as velocity slip, is a phenomenon that has been observed under certain circumstances [9]. Beavers and Joseph [10] proposed a slip flow condition at the boundary. Of late, there has been a revival of interest in the flow problems with partial slip [11]. Wang [12] undertook the study of the flow of a Newtonian fluid past a stretching sheet with partial slip and purportedly gave an exact solution. Polymer melts often exhibit macroscopic wall slip, which in general is governed by a non-linear and monotone relationship between the slip velocity and traction [13]. In the light of this, the purpose of this article is to study the heat transfer of a viscous fluid over an unsteady porous stretching sheet when there are velocity slip and thermal slip on the wall. The slip condition is taken into account in terms of the shear stress. Similarity solutions are obtained and the reduced ordinary differential equations are solved numerically using shooting method. The
S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
the kinematic viscosity, is the coefficient of fluid viscosity, is the fluid density, T is the temperature, is the thermal diffusivity of the fluid, ˇ is the volumetric coefficient of thermal expansion, g is the gravity field and T∞ is the temperature at infinity.
Nomenclature F f f f M Pr p,q S (>0) T Tw T∞ u, z
2661
non-dimensional stream function first order derivative with respect to second order derivative with respect to third order derivative with respect to unsteadiness parameter Prandtl number variables suction parameter temperature of the fluid temperature of the wall of the surface free-stream temperature components of velocity in x and y directions variable
2.1. Boundary conditions The appropriate boundary conditions for the problem are given by u = U(x, t) + N1 = w (x, t),
∂u , ∂y
T = Tw (x, t) + D1
∂T ∂y
at y = 0,
(4)
u → 0,
T → T∞ as y → ∞. (5) √ Here N1 = N 1 −√ ˛t is the velocity slip factor which changes with time, D1 = D 1 − ˛t is the thermal slip factor, also changes with time, N,D are respectively the initial values of velocity and thermal slip factors. The essential slip factors N1 and D1 have dimension (velocity)−1 and length, respectively. The noslip case considered by Andersson et al. [14] is recovered for N1 = D1 = 0. Tw (x,t) = T∞ + T0 (cx/)(1 − ␣t)−2 is the wall temperature where T0 is a reference temperature such that 0 ≤ T0 ≤ Tw . w (x, t) = −(0 /(1 − ˛t)1/2 ) where 0 (>0) is the velocity of suction. The expressions for U(x,t), w (x,t), Tw (x,t) are valid for time t < ␣−1 .
Greek symbols ˇ volumetric coefficient of thermal expansion thermal slip parameter ı similarity variable velocity slip parameter coefficient of thermal diffusivity mixed convection parameter coefficient of viscosity kinematic viscosity stream function density of the fluid
non-dimensional temperature first order derivative with respect to
second order derivative with respect to
2.2. Method of solution We now introduce the following relations for u, and as
analysis of the results obtained shows that the flow field is influenced appreciably by the presence of unsteadiness and velocity slip parameter. To reveal the tendency of the solutions, representative results are presented for the velocity, temperature as well as the skin friction and rate of heat transfer. Comparisons with previously published works are performed and excellent agreement is obtained.
u=
∂ , ∂y
∂ ∂u + = 0, ∂x ∂y
(1)
∂u ∂u ∂u ∂2 u +u + = 2 + gˇ(T − T∞ ), ∂t ∂x ∂y ∂y
(2)
∂T ∂T ∂T ∂2 T +u + = 2, ∂t ∂x ∂y ∂y
(3)
=
c y, (1 − ˛t)
T = T∞ + T0
cx
and
=
T − T∞ , Tw − T∞
(6)
=
c xf (), (1 − ˛t)
(1 − ˛t)−2 ()
(7) (8)
and with the help of the above relations, the governing equations finally reduce to M
2
f + f
+ f − f = f + , 2
(9)
M 1 + 2M + f − f =
, 2 Pr
(10)
where M = ˛/c is the unsteadiness parameter, = (gˇT0 /c) = (Grx /Re2x ) is the mixed convection parameter, Grx = gˇ(Tw − T∞ )x3 /2 is the Grashof number, Rex = Ux/ is the local Reynold’s number based on the stretching velocity U. The boundary conditions (4) and (5) then become f = 1 + f ,
when the viscous dissipation term in the energy equation is neglected (as the fluid velocity is low). Here u and are the components of velocity respectively in the x and y directions, = / is
∂ ∂x
where is the stream function. Introducing the similarity variable and the dimensionless variables f and as
2. Equations of motion We consider mixed convective laminar boundary-layer flow and heat transfer of viscous incompressible fluid over a porous and unsteady stretching sheet emerging out of a slit at origin (x = 0, y = 0 and moving with non-uniform velocity U(x,t) = cx/(1 − ˛t) where c,˛ are positive constants with dimensions (time)−1 , c is the initial stretching rate and c/(1 − ˛t) is the effective stretching rate which is increasing with time. In case of polymer extrusion, the material properties of the extruded sheet may vary with time. Here, the stretching surface is subjected to such amount of tension so that the structure of the porous material does not change. The governing equations of such type of flow are, in the usual notations,
=−
f = S,
= 1 + ı
at = 0
(11)
and f → 0,
→ 0 as → ∞,
(12) √ where Pr = / is the Prandtl number, S = (0 / c)(> 0) is the √ parameter, = N c is the velocity slip parameter, ı = suction D c/ is the thermal slip parameter. The slip parameters ,ı both are dimensionless.
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S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
Table 1 −1/2 [= − (0)] for several values of Prandtl number (Pr) with M = 0, Values of Nux Rex S = 0, = 0, = 0 and ı = 0. Pr
Grubka and Bobba [15]
Chen [5]
Present study
0.01 0.72 1.00 3.00
0.0294 1.0885 1.3333 2.5097
0.02942 1.08853 1.33334 2.50972
0.02944 1.08855 1.33334 2.50971
3. Numerical method for solution The above Eqs. (9) and (10) along with boundary conditions are solved by converting them to an initial value problem. We set f = z, p =
z = p,
= q,
(13)
M p + Mz + z 2 − fp − , 2
q = Pr
M 2
q + 2M + z − fq
(14)
with the boundary conditions f (0) = S,
f (0) = 1 + m,
(0) = n.
(0) = 1 + ın,
f (0) = m, (15)
Here, the values of m and n are a priori unknown and will be determined as a part of the numerical solution. In order to integrate (13) and (14) as an initial value problem we require a value for p(0) i.e. f (0) and q(0) i.e. (0) but no such values are given in the boundary. The suitable guess values for f (0) and (0) are chosen and then integration is carried out. We compare the calculated values for f and at = 10 (say) with the given boundary condition f (10) = 0 and (10) = 0 and adjust the estimated values, f (0) and (0), to give a better approximation for the solution. We take the series of values for f (0) and (0), and apply the fourth order classical Runge–Kutta method with step-size h = 0.01. The above procedure is repeated until we get the results up to the desired degree of accuracy, 10−5 .
Fig. 1. (a) Velocity profiles for variable unsteadiness parameter M with no-slip boundary condition. (b) Velocity profiles for variable unsteadiness parameter M with slip boundary condition.
4. Results and discussions In order to validate the method used in this study and to judge the accuracy of the present analysis, comparison with available results of Grubka and Bobba [15] and Chen [5] for local Nusselt −1/2 number [Nux Rex = − (0)] for steady forced convection flow on a linearly stretching surface in the absence of suction and slip are made (Table 1) and found in excellent agreement. In order to analyse the results, numerical computation has been carried out using the method described in the previous section for various values of the unsteadiness parameter (M), velocity slip parameter (), thermal slip parameter (ı), mixed convection parameter (), suction parameter (S) and Prandtl number (Pr). For illustrations of the results, numerical values are plotted. In the following, the Prandtl number is kept fixed to 0.7 and the influence of the four other dimensionless parameters is considered. First, we present the result for variation of the unsteadiness parameter M in the absence and also presence of slip. In Fig. 1(a), velocity profiles are shown for different values of M when there is noslip at the boundary while in Fig. 1(b) effect of unsteadiness parameter in presence of slip at the boundary is exhibited. From both figures, it is seen that the velocity f () along the sheet decreases with the increase of unsteadiness parameter M and this implies an accompanying reduction of the thickness of the momentum boundary layer. Fig. 2(a) and (b) represent the temperature profiles for the same parameter values as in Fig. 1(a) and (b),
Fig. 2. (a) Temperature profiles for variable unsteadiness parameter M with no-slip boundary condition. (b) Temperature profiles for variable unsteadiness parameter M with slip boundary condition.
S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
Fig. 3. (a) Velocity profiles for variable values of mixed convection parameter . (b) Temperature profiles for variable values of mixed convection parameter .
respectively. For all values of M considered, is found to decrease monotonically with the distance from the sheet. It is noteworthy that the impact of M on the temperature profiles is more pronounced than on the velocity profiles in Fig. 1(a) and (b). The rate of heat transfer increases with increasing unsteadiness parameter M. Same results are obtained in the absence of slip ( = 0, ı = 0), see also Andersson et al. [14] and Tsai et al. [16]. It is important to note that the rate of cooling is much faster for higher values of unsteadiness parameter whereas it may take longer time for cooling during steady flows. Effects of mixed convection parameter on velocity and temperature profiles are clearly exhibited in Fig. 3(a) and (b), respectively. Fluid velocity increases with increasing values of mixed convection parameter but the temperature decreases in this case. Here = 0 gives the result of forced convection case. Physically > 0 means heating of the fluid or cooling of the surface (assisting flow). Also, an increase in the value of can lead to an increase in the temperature difference Tw − T∞ . This leads to an enhancement of the velocity. An increase in the value of mixed convection parameter results in a decrease in the thermal boundary layer thickness and the surface heat transfer rate increases. Fig. 4(a) and (b) display the effects of suction parameter (S) on velocity and temperature profiles. Fluid velocity and the temperature of the sheet both are found to decrease with increasing values of S. The physical explanation for such behaviour is as follows. In this case, the heated fluid is pushed towards the wall where the buoyancy forces can act to retard the fluid due to high influence of viscosity. This effect acts to decrease the wall shear stress. Thermal boundary layer thickness reduces in case of suction. Actually, the effect of suction is to make the velocity and temperature distribution more uniform within the boundary layer. Imposition of fluid suction at the surface has a tendency to reduce both the hydrodynamic and thermal thickness of the boundary layer where
2663
Fig. 4. (a) Velocity profiles for variable values of suction parameter S. (b) Temperature profiles for variable values of suction parameter S.
Fig. 5. (a) Velocity profiles for variable velocity slip parameter . (b) Shear stress profiles for variable velocity slip parameter .
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S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
Fig. 8. Variation of skin-friction coefficient with velocity slip parameter for two values of unsteadiness parameter M.
Fig. 6. (a) Temperature profiles for variable velocity slip parameter . (b) Profiles of temperature gradient for variable velocity slip parameter .
Fig. 7. (a) Temperature profiles for variable thermal slip parameter ı. (b) Profiles of temperature gradient for variable thermal slip parameter ı.
viscous effects dominate. This has the effect of reducing both the fluid velocity and temperature. Next, we present the effects of velocity and thermal slip parameters on velocity and temperature profiles. Fig. 5(a) is the graphical representation of horizontal velocity profiles for different values of velocity slip parameter . With the increasing values of velocity slip parameter the fluid velocity decreases. When slip occurs (for nonzero values of ), the flow velocity near the sheet is no longer equal to the sheet stretching velocity, i.e. a velocity slip exists. With the increase in such a slip velocity increases. Furthermore, increasing the value of will decrease the flow velocity because under the slip condition the pulling of the stretching sheet can be only partly transmitted to the fluid. It is readily seen that has a substantial effect on the solutions. Fig. 5(b) exhibits the shear stress for variable
Fig. 9. (a) Variation of rate of heat transfer with velocity slip parameter for two values of un steadiness parameter M. (b) Variation of rate of heat transfer with thermal slip parameter ı for two values of unsteadiness parameter M.
S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
velocity slip parameter . It is very clear that magnitude of shear stress decreases with increasing slip parameter. Fig. 6(a) and (b) are the graphical representations of temperature and temperature gradient for various values of velocity slip parameter. The temperature increases monotonically with the velocity slip parameter whereas an interesting nature is noticed for temperature gradient. Up to = 1.8, the magnitude of temperature gradient decreases with slip parameter but after this point it increases. Temperature profile () and temperature gradient () for the different values of the thermal slip parameter ı are shown in Fig. 7(a) and (b). As the thermal slip parameter increases, less heat is transferred from the sheet to the fluid; hence, the temperature () decreases [Fig. 7(a)] and the magnitude of temperature gradient decreases in this case [Fig. 7(b)]. Fig. 8 shows the nature of skin-friction with the velocity slip parameter for two values of unsteadiness parameter M. It decreases with the velocity slip parameter while increases with the unsteadiness parameter M, i.e. the slip condition reduces the momentum transfer from the sheet to the fluid. Fig. 9(a) represents the rate of heat transfer for the same parametric values as in Fig. 8. In this case the rate of heat transfer decreases with the velocity slip parameter while increases with the unsteadiness parameter M. Fig. 9(b) shows that for a thermal slip condition at the sheet much less heat is transferred to the fluid, the fluid temperature increases much slower than that with a no-slip thermal condition. As the effect of thermal slip parameter ı on velocity profiles is not significant so it is not presented here. 5. Conclusion The present study gives the similarity solutions for unsteady mixed convective boundary layer flow and heat transfer over a porous stretching surface in presence of both velocity and thermal slip conditions. The results pertaining to the present study indicate that the flow and temperature field are significantly influenced by the unsteadiness parameter, buoyancy force, suction parameter, velocity and thermal slip parameters. Horizontal velocity increases but the temperature decreases with the increase in mixed convection parameter. Increasing the flow slip parameter causes the decrease in the flow velocity, while with the increase in the ther-
2665
mal slip parameter the heat transfer from the sheet to the fluid becomes slower. Acknowledgement The author is thankful to the honourable reviewers for their constructive suggestions which helped a lot to improve the quality of the paper. References Kumari, M., Nath, G., 2004. Radiation effect on mixed convection from a horizontal surface in a porous medium. Mech. Res. Commun. 31, 483–491. Elbashbeshy, E.M.A., Bazid, M.A.A., 2004. Heat transfer over an unsteady stretching surface. Heat Mass Transfer 41, 1–4. Abo-Eldahab, E.M., El Aziz, M.A., 2004. Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. Int. J. Therm. Sci. 43, 709–719. Lin, H.T., Wu, K.Y., Koh, H.L., 1993. Mixed convection from an isothermal horizontal plate moving in parallel or reversibly to a free stream. Int. J. Heat Mass Transfer 36, 3547–3554. Chen, C.H., 1998. Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transfer 33, 471–476. Ali, M., 2006. The effect of variable viscosity on mixed convection heat transfer along a vertical moving surface. Int. J. Therm. Sci. 45, 60–69. Sharidan, S., Mahmood, T., Pop, I., 2006. Similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet. Int. J. Appl. Mech. Eng. 11 (3), 647–654. Mukhopadhyay, S., 2009. Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in porous medium. Int. J. Heat Mass Transfer 52, 3261–3265. Yoshimura, A., Prudhomme, R.K., 1998. Wall slip corrections for Couette and parallel disc viscometers. J. Rheol. 32, 53–67. Beavers, G.S., Joseph, D.D., 1967. Boundary condition at a naturally permeable wall. J. Fluid Mech. 30, 197–207. Ariel, P.D., 2008. Two dimensional stagnation point flow of an elastico-viscous fluid with partial slip. Z. Angew. Math. Mech. 88, 320–324. Wang, C.Y., 2002. Flow due to a stretching boundary with partial slip – an exact solution of the Navier–Stokes equations. Chem. Eng. Sci. 57, 3745–3747. Abbas, Z.Z., Wang, Y., Hayat, T., Oberlack, M., 2009. Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface. Int. J. Numer. Meth. Fluids 59, 443–458. Andersson, H.I., Aarseth, J.B., Dandapat, B.S., 2000. Heat transfer in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transfer 43, 69–74. Grubka, L.J., Bobba, K.M., 1985. Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J. Heat Transfer 107, 248–250. Tsai, R., Huang, K.H., Huang, J.S., 2008. Flow and heat transfer over an unsteady stretching surface with a non-uniform heat source. Int. Commun. Heat Mass Transfer 35, 1340–1343.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching surface Swati Mukhopadhyay ∗ Department of Mathematics, The University of Burdwan, Burdwan 713104, W.B., India
a r t i c l e
i n f o
Article history: Received 13 August 2010 Received in revised form 19 April 2011 Accepted 5 May 2011
a b s t r a c t This paper investigates the unsteady mixed convective boundary layer flow and heat transfer over a porous stretching vertical surface in presence of slip. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically. With increasing values of the unsteadiness parameter, fluid velocity and the temperature are found to decrease in both the presence and absence of slip at the boundary. Fluid velocity decreases due to increasing values of the velocity slip parameter resulting an increase in the temperature field. Skin-friction decreases with the velocity slip parameter whereas it increases with unsteadiness parameter. The rate of heat transfer decreases with the velocity slip parameter while increases with unsteadiness parameter. Same feature is also noticed for thermal slip parameter. Effects of increasing mixed convection parameter on the velocity boundary layer is to increase the velocity field and the temperature decreases in this case. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The study of hydrodynamic flow and heat transfer over a stretching sheet has gained considerable attention due to its vast applications in industry and its importance to several technological processes. The production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. The tangential velocity imparted by the sheet induces motion in the surrounding fluid that alters the convection cooling of the sheet. Knowledge of the flow properties of the fluid is desirable because the quality of the resulting sheeting material, as well as the cost of production, is affected by the speed of collection and mass transfer rate. In recent years, a great deal of interest has been generated in the area of boundary layer mixed convection flow on a vertical stretching surface in view of its numerous and ever increasing industrial and technical applications which include aerodynamic extrusion of plastic sheets, cooling of metallic sheets in a cooling bath, crystal growing, etc. In the study of horizontal heated or cooled surfaces, the effect of buoyancy force is neglected. However, for vertical or inclined surfaces, the buoyancy force modifies the flow field and hence the heat transfer rate [1]. The importance of this phenomenon is increasing day by day due to the enhanced concern in science and technology about buoyancy induced motions in the atmosphere, the bodies in water and quasi-solid bodies such as earth. Buoyancy plays an important role where the temperature
∗ Tel.: +91 342 2557741; fax: +91 342 2530452. E-mail address: swati [email protected] 0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2011.05.007
differences between land and air give rise to a complicated flow in enclosures such as ventilated and heated rooms [2]. The buoyancy force arising due to the heating of a stretching surface, under some circumstances, may alter significantly the flow and thermal fields and thereby the heat transfer behaviour in the manufacturing process [3]. Many studies were reported considering the effect of buoyancy forces on the boundary layer [4–6]. All of the above mentioned studies were restricted in the steady state conditions. The transient or unsteady aspects become interesting in certain practical problems where the motion of the stretched surface may start impulsively from rest. Elbashbeshy and Bazid [2], Sharidan et al. [7], Mukhopadhyay [8] presented similarity solutions for unsteady flow and heat transfer over a stretching surface. The non-adherence of the fluid to a solid boundary, also known as velocity slip, is a phenomenon that has been observed under certain circumstances [9]. Beavers and Joseph [10] proposed a slip flow condition at the boundary. Of late, there has been a revival of interest in the flow problems with partial slip [11]. Wang [12] undertook the study of the flow of a Newtonian fluid past a stretching sheet with partial slip and purportedly gave an exact solution. Polymer melts often exhibit macroscopic wall slip, which in general is governed by a non-linear and monotone relationship between the slip velocity and traction [13]. In the light of this, the purpose of this article is to study the heat transfer of a viscous fluid over an unsteady porous stretching sheet when there are velocity slip and thermal slip on the wall. The slip condition is taken into account in terms of the shear stress. Similarity solutions are obtained and the reduced ordinary differential equations are solved numerically using shooting method. The
S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
the kinematic viscosity, is the coefficient of fluid viscosity, is the fluid density, T is the temperature, is the thermal diffusivity of the fluid, ˇ is the volumetric coefficient of thermal expansion, g is the gravity field and T∞ is the temperature at infinity.
Nomenclature F f f f M Pr p,q S (>0) T Tw T∞ u, z
2661
non-dimensional stream function first order derivative with respect to second order derivative with respect to third order derivative with respect to unsteadiness parameter Prandtl number variables suction parameter temperature of the fluid temperature of the wall of the surface free-stream temperature components of velocity in x and y directions variable
2.1. Boundary conditions The appropriate boundary conditions for the problem are given by u = U(x, t) + N1 = w (x, t),
∂u , ∂y
T = Tw (x, t) + D1
∂T ∂y
at y = 0,
(4)
u → 0,
T → T∞ as y → ∞. (5) √ Here N1 = N 1 −√ ˛t is the velocity slip factor which changes with time, D1 = D 1 − ˛t is the thermal slip factor, also changes with time, N,D are respectively the initial values of velocity and thermal slip factors. The essential slip factors N1 and D1 have dimension (velocity)−1 and length, respectively. The noslip case considered by Andersson et al. [14] is recovered for N1 = D1 = 0. Tw (x,t) = T∞ + T0 (cx/)(1 − ␣t)−2 is the wall temperature where T0 is a reference temperature such that 0 ≤ T0 ≤ Tw . w (x, t) = −(0 /(1 − ˛t)1/2 ) where 0 (>0) is the velocity of suction. The expressions for U(x,t), w (x,t), Tw (x,t) are valid for time t < ␣−1 .
Greek symbols ˇ volumetric coefficient of thermal expansion thermal slip parameter ı similarity variable velocity slip parameter coefficient of thermal diffusivity mixed convection parameter coefficient of viscosity kinematic viscosity stream function density of the fluid
non-dimensional temperature first order derivative with respect to
second order derivative with respect to
2.2. Method of solution We now introduce the following relations for u, and as
analysis of the results obtained shows that the flow field is influenced appreciably by the presence of unsteadiness and velocity slip parameter. To reveal the tendency of the solutions, representative results are presented for the velocity, temperature as well as the skin friction and rate of heat transfer. Comparisons with previously published works are performed and excellent agreement is obtained.
u=
∂ , ∂y
∂ ∂u + = 0, ∂x ∂y
(1)
∂u ∂u ∂u ∂2 u +u + = 2 + gˇ(T − T∞ ), ∂t ∂x ∂y ∂y
(2)
∂T ∂T ∂T ∂2 T +u + = 2, ∂t ∂x ∂y ∂y
(3)
=
c y, (1 − ˛t)
T = T∞ + T0
cx
and
=
T − T∞ , Tw − T∞
(6)
=
c xf (), (1 − ˛t)
(1 − ˛t)−2 ()
(7) (8)
and with the help of the above relations, the governing equations finally reduce to M
2
f + f
+ f − f = f + , 2
(9)
M 1 + 2M + f − f =
, 2 Pr
(10)
where M = ˛/c is the unsteadiness parameter, = (gˇT0 /c) = (Grx /Re2x ) is the mixed convection parameter, Grx = gˇ(Tw − T∞ )x3 /2 is the Grashof number, Rex = Ux/ is the local Reynold’s number based on the stretching velocity U. The boundary conditions (4) and (5) then become f = 1 + f ,
when the viscous dissipation term in the energy equation is neglected (as the fluid velocity is low). Here u and are the components of velocity respectively in the x and y directions, = / is
∂ ∂x
where is the stream function. Introducing the similarity variable and the dimensionless variables f and as
2. Equations of motion We consider mixed convective laminar boundary-layer flow and heat transfer of viscous incompressible fluid over a porous and unsteady stretching sheet emerging out of a slit at origin (x = 0, y = 0 and moving with non-uniform velocity U(x,t) = cx/(1 − ˛t) where c,˛ are positive constants with dimensions (time)−1 , c is the initial stretching rate and c/(1 − ˛t) is the effective stretching rate which is increasing with time. In case of polymer extrusion, the material properties of the extruded sheet may vary with time. Here, the stretching surface is subjected to such amount of tension so that the structure of the porous material does not change. The governing equations of such type of flow are, in the usual notations,
=−
f = S,
= 1 + ı
at = 0
(11)
and f → 0,
→ 0 as → ∞,
(12) √ where Pr = / is the Prandtl number, S = (0 / c)(> 0) is the √ parameter, = N c is the velocity slip parameter, ı = suction D c/ is the thermal slip parameter. The slip parameters ,ı both are dimensionless.
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S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
Table 1 −1/2 [= − (0)] for several values of Prandtl number (Pr) with M = 0, Values of Nux Rex S = 0, = 0, = 0 and ı = 0. Pr
Grubka and Bobba [15]
Chen [5]
Present study
0.01 0.72 1.00 3.00
0.0294 1.0885 1.3333 2.5097
0.02942 1.08853 1.33334 2.50972
0.02944 1.08855 1.33334 2.50971
3. Numerical method for solution The above Eqs. (9) and (10) along with boundary conditions are solved by converting them to an initial value problem. We set f = z, p =
z = p,
= q,
(13)
M p + Mz + z 2 − fp − , 2
q = Pr
M 2
q + 2M + z − fq
(14)
with the boundary conditions f (0) = S,
f (0) = 1 + m,
(0) = n.
(0) = 1 + ın,
f (0) = m, (15)
Here, the values of m and n are a priori unknown and will be determined as a part of the numerical solution. In order to integrate (13) and (14) as an initial value problem we require a value for p(0) i.e. f (0) and q(0) i.e. (0) but no such values are given in the boundary. The suitable guess values for f (0) and (0) are chosen and then integration is carried out. We compare the calculated values for f and at = 10 (say) with the given boundary condition f (10) = 0 and (10) = 0 and adjust the estimated values, f (0) and (0), to give a better approximation for the solution. We take the series of values for f (0) and (0), and apply the fourth order classical Runge–Kutta method with step-size h = 0.01. The above procedure is repeated until we get the results up to the desired degree of accuracy, 10−5 .
Fig. 1. (a) Velocity profiles for variable unsteadiness parameter M with no-slip boundary condition. (b) Velocity profiles for variable unsteadiness parameter M with slip boundary condition.
4. Results and discussions In order to validate the method used in this study and to judge the accuracy of the present analysis, comparison with available results of Grubka and Bobba [15] and Chen [5] for local Nusselt −1/2 number [Nux Rex = − (0)] for steady forced convection flow on a linearly stretching surface in the absence of suction and slip are made (Table 1) and found in excellent agreement. In order to analyse the results, numerical computation has been carried out using the method described in the previous section for various values of the unsteadiness parameter (M), velocity slip parameter (), thermal slip parameter (ı), mixed convection parameter (), suction parameter (S) and Prandtl number (Pr). For illustrations of the results, numerical values are plotted. In the following, the Prandtl number is kept fixed to 0.7 and the influence of the four other dimensionless parameters is considered. First, we present the result for variation of the unsteadiness parameter M in the absence and also presence of slip. In Fig. 1(a), velocity profiles are shown for different values of M when there is noslip at the boundary while in Fig. 1(b) effect of unsteadiness parameter in presence of slip at the boundary is exhibited. From both figures, it is seen that the velocity f () along the sheet decreases with the increase of unsteadiness parameter M and this implies an accompanying reduction of the thickness of the momentum boundary layer. Fig. 2(a) and (b) represent the temperature profiles for the same parameter values as in Fig. 1(a) and (b),
Fig. 2. (a) Temperature profiles for variable unsteadiness parameter M with no-slip boundary condition. (b) Temperature profiles for variable unsteadiness parameter M with slip boundary condition.
S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
Fig. 3. (a) Velocity profiles for variable values of mixed convection parameter . (b) Temperature profiles for variable values of mixed convection parameter .
respectively. For all values of M considered, is found to decrease monotonically with the distance from the sheet. It is noteworthy that the impact of M on the temperature profiles is more pronounced than on the velocity profiles in Fig. 1(a) and (b). The rate of heat transfer increases with increasing unsteadiness parameter M. Same results are obtained in the absence of slip ( = 0, ı = 0), see also Andersson et al. [14] and Tsai et al. [16]. It is important to note that the rate of cooling is much faster for higher values of unsteadiness parameter whereas it may take longer time for cooling during steady flows. Effects of mixed convection parameter on velocity and temperature profiles are clearly exhibited in Fig. 3(a) and (b), respectively. Fluid velocity increases with increasing values of mixed convection parameter but the temperature decreases in this case. Here = 0 gives the result of forced convection case. Physically > 0 means heating of the fluid or cooling of the surface (assisting flow). Also, an increase in the value of can lead to an increase in the temperature difference Tw − T∞ . This leads to an enhancement of the velocity. An increase in the value of mixed convection parameter results in a decrease in the thermal boundary layer thickness and the surface heat transfer rate increases. Fig. 4(a) and (b) display the effects of suction parameter (S) on velocity and temperature profiles. Fluid velocity and the temperature of the sheet both are found to decrease with increasing values of S. The physical explanation for such behaviour is as follows. In this case, the heated fluid is pushed towards the wall where the buoyancy forces can act to retard the fluid due to high influence of viscosity. This effect acts to decrease the wall shear stress. Thermal boundary layer thickness reduces in case of suction. Actually, the effect of suction is to make the velocity and temperature distribution more uniform within the boundary layer. Imposition of fluid suction at the surface has a tendency to reduce both the hydrodynamic and thermal thickness of the boundary layer where
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Fig. 4. (a) Velocity profiles for variable values of suction parameter S. (b) Temperature profiles for variable values of suction parameter S.
Fig. 5. (a) Velocity profiles for variable velocity slip parameter . (b) Shear stress profiles for variable velocity slip parameter .
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S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
Fig. 8. Variation of skin-friction coefficient with velocity slip parameter for two values of unsteadiness parameter M.
Fig. 6. (a) Temperature profiles for variable velocity slip parameter . (b) Profiles of temperature gradient for variable velocity slip parameter .
Fig. 7. (a) Temperature profiles for variable thermal slip parameter ı. (b) Profiles of temperature gradient for variable thermal slip parameter ı.
viscous effects dominate. This has the effect of reducing both the fluid velocity and temperature. Next, we present the effects of velocity and thermal slip parameters on velocity and temperature profiles. Fig. 5(a) is the graphical representation of horizontal velocity profiles for different values of velocity slip parameter . With the increasing values of velocity slip parameter the fluid velocity decreases. When slip occurs (for nonzero values of ), the flow velocity near the sheet is no longer equal to the sheet stretching velocity, i.e. a velocity slip exists. With the increase in such a slip velocity increases. Furthermore, increasing the value of will decrease the flow velocity because under the slip condition the pulling of the stretching sheet can be only partly transmitted to the fluid. It is readily seen that has a substantial effect on the solutions. Fig. 5(b) exhibits the shear stress for variable
Fig. 9. (a) Variation of rate of heat transfer with velocity slip parameter for two values of un steadiness parameter M. (b) Variation of rate of heat transfer with thermal slip parameter ı for two values of unsteadiness parameter M.
S. Mukhopadhyay / Nuclear Engineering and Design 241 (2011) 2660–2665
velocity slip parameter . It is very clear that magnitude of shear stress decreases with increasing slip parameter. Fig. 6(a) and (b) are the graphical representations of temperature and temperature gradient for various values of velocity slip parameter. The temperature increases monotonically with the velocity slip parameter whereas an interesting nature is noticed for temperature gradient. Up to = 1.8, the magnitude of temperature gradient decreases with slip parameter but after this point it increases. Temperature profile () and temperature gradient () for the different values of the thermal slip parameter ı are shown in Fig. 7(a) and (b). As the thermal slip parameter increases, less heat is transferred from the sheet to the fluid; hence, the temperature () decreases [Fig. 7(a)] and the magnitude of temperature gradient decreases in this case [Fig. 7(b)]. Fig. 8 shows the nature of skin-friction with the velocity slip parameter for two values of unsteadiness parameter M. It decreases with the velocity slip parameter while increases with the unsteadiness parameter M, i.e. the slip condition reduces the momentum transfer from the sheet to the fluid. Fig. 9(a) represents the rate of heat transfer for the same parametric values as in Fig. 8. In this case the rate of heat transfer decreases with the velocity slip parameter while increases with the unsteadiness parameter M. Fig. 9(b) shows that for a thermal slip condition at the sheet much less heat is transferred to the fluid, the fluid temperature increases much slower than that with a no-slip thermal condition. As the effect of thermal slip parameter ı on velocity profiles is not significant so it is not presented here. 5. Conclusion The present study gives the similarity solutions for unsteady mixed convective boundary layer flow and heat transfer over a porous stretching surface in presence of both velocity and thermal slip conditions. The results pertaining to the present study indicate that the flow and temperature field are significantly influenced by the unsteadiness parameter, buoyancy force, suction parameter, velocity and thermal slip parameters. Horizontal velocity increases but the temperature decreases with the increase in mixed convection parameter. Increasing the flow slip parameter causes the decrease in the flow velocity, while with the increase in the ther-
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mal slip parameter the heat transfer from the sheet to the fluid becomes slower. Acknowledgement The author is thankful to the honourable reviewers for their constructive suggestions which helped a lot to improve the quality of the paper. References Kumari, M., Nath, G., 2004. Radiation effect on mixed convection from a horizontal surface in a porous medium. Mech. Res. Commun. 31, 483–491. Elbashbeshy, E.M.A., Bazid, M.A.A., 2004. Heat transfer over an unsteady stretching surface. Heat Mass Transfer 41, 1–4. Abo-Eldahab, E.M., El Aziz, M.A., 2004. Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. Int. J. Therm. Sci. 43, 709–719. Lin, H.T., Wu, K.Y., Koh, H.L., 1993. Mixed convection from an isothermal horizontal plate moving in parallel or reversibly to a free stream. Int. J. Heat Mass Transfer 36, 3547–3554. Chen, C.H., 1998. Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transfer 33, 471–476. Ali, M., 2006. The effect of variable viscosity on mixed convection heat transfer along a vertical moving surface. Int. J. Therm. Sci. 45, 60–69. Sharidan, S., Mahmood, T., Pop, I., 2006. Similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet. Int. J. Appl. Mech. Eng. 11 (3), 647–654. Mukhopadhyay, S., 2009. Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in porous medium. Int. J. Heat Mass Transfer 52, 3261–3265. Yoshimura, A., Prudhomme, R.K., 1998. Wall slip corrections for Couette and parallel disc viscometers. J. Rheol. 32, 53–67. Beavers, G.S., Joseph, D.D., 1967. Boundary condition at a naturally permeable wall. J. Fluid Mech. 30, 197–207. Ariel, P.D., 2008. Two dimensional stagnation point flow of an elastico-viscous fluid with partial slip. Z. Angew. Math. Mech. 88, 320–324. Wang, C.Y., 2002. Flow due to a stretching boundary with partial slip – an exact solution of the Navier–Stokes equations. Chem. Eng. Sci. 57, 3745–3747. Abbas, Z.Z., Wang, Y., Hayat, T., Oberlack, M., 2009. Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface. Int. J. Numer. Meth. Fluids 59, 443–458. Andersson, H.I., Aarseth, J.B., Dandapat, B.S., 2000. Heat transfer in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transfer 43, 69–74. Grubka, L.J., Bobba, K.M., 1985. Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J. Heat Transfer 107, 248–250. Tsai, R., Huang, K.H., Huang, J.S., 2008. Flow and heat transfer over an unsteady stretching surface with a non-uniform heat source. Int. Commun. Heat Mass Transfer 35, 1340–1343.