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Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
journal homepage: www.elsevier.com/locate/jmatprotec
Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface Rafael Cortell Bataller Departamento de F´ısica Aplicada, Escuela T´ecnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Polit´ecnica de Valencia, 46071 Valencia, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper presents a numerical analysis in connection with the boundary layer flow
Received 4 December 2006
induced in a quiescent fluid by a stretching sheet with velocity uw (x) ∼ x1/3 along with heat
Received in revised form
transfer. The surface temperature is assumed to have a power-law variation. The viscous
13 August 2007
dissipation and thermal radiation are considered in the energy equation. The governing
Accepted 16 September 2007
partial differential equations are converted into ordinary ones by a similarity transformation. The variations of dimensionless surface temperature as well as flow and heat transfer characteristics with the governing parameters are graphed and tabulated. Two cases are
Keywords:
studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the
Laminar boundary layer
sheet with prescribed heat flux (PHF case). Similarity solutions of the aforementioned prob-
Nonlinearly stretching surfaces
lem are given for two values of the surface temperature parameter m, namely, m = 2/3 in
Viscous dissipation
the PST case and m = 1/3 in the PHF case. Moreover, the mechanical characteristics of the
Thermal radiation
corresponding flow are also presented. © 2007 Elsevier B.V. All rights reserved.
1.
Introduction
In contrast to the well-known Blasius flow problem (Cortell, 2005a), which involves the boundary-layer flow passing through a stationary flat plate, Sakiadis (1961) considered the boundary-layer flow on a moving plate in a quiescent ambient fluid. The aforementioned problems are two special cases of more general studies (Ishak et al., 2007; Cortell, 2007) in which flow and heat transfer of a moving sheet in the presence of a co-flowing fluid were analyzed. Unfortunately, in reality, many flow applications in industrial processes are concerned with non-Newtonian fluids. Specifically, these include polymer melts and solutions, heattreated materials travelling between a feed roll and a wind-up roll or materials manufactured by extrusion, glass–fiber and paper production, cooling of metallic sheets or electronic chips, crystal growing and many others. In these cases, the final product of desired characteristics depends
E-mail address: rcortell@fis.upv.es. 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.09.055
on the rate of cooling in the process and the process of stretching. Very recently Sadeghy et al. (2005) studied the boundary layer of an upper-convected Maxwell fluid, and the role played by the fluid’s elasticity on flow characteristics was analyzed. Moreover, several linearly stretching sheet problems related with non-Newtonian fluids have been analyzed by other researchers in several geometries including, sometimes, a porous medium alike (Cortell, 1993a, 1994, 2006a,b; Prasad et al., 2000, 2002; Char, 1994). Stretching sheet problems of practical interest in materials processing were analyzed by Chen (2003). He dealt with a flow over a heated flat surface continuously moving in a parallel free stream of a non-Newtonian fluid, and various interesting aspects of this class of boundarylayer flow were presented. All the aforementioned researches were performed taking into account whether a constant value for the velocity wall or a linearly stretching sheet problem (i.e., uw (x) = ax). Nevertheless, it needs to be underlined that
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
Nomenclature A, D cP Ec Ec f g k k0 k* L m NR qr T u, v x, y
constants specific heat at a constant pressure Eckert number for the case of prescribed surface temperature (PST case) Eckert number for the case of prescribed heat flux (PHF case) dimensionless stream function dimensionless temperature fluid thermal conductivity dimensionless parameter related with thermal radiation mean absorption coefficient reference length surface temperature parameter radiation parameter radiative heat flux temperature across the thermal boundarylayer velocity components along x and y directions, respectively Cartesian coordinates along the plate and normal to it, respectively
Greek symbols ˛ thermal diffusivity dimensionless similarity variable dimensionless temperature dynamic viscosity kinematic viscosity density Prandtl number * Stefan–Boltzmann constant shear stress Subscripts w, ∞ conditions at the surface and in the free stream, respectively Superscript derivative with respect to
the stretching of the sheet may not necessarily be linear. In view of this, the flow influenced by a nonlinearly stretching sheet was investigated by Vajravelu (2001), and power-law or exponentially stretching sheet was studied by Ali (1995) and Elbashbeshy (2001), respectively. Further, momentum, heat and mass transfer over an exponentially stretching surface were considered by Sanjayanand and Khan (2006). They also enclosed the effects of viscous dissipation and work done by deformation in the energy equation. Vajravelu (2001) studied flow and heat transfer in a viscous fluid over a nonlinearly stretching sheet without viscous dissipation, but the heat transfer in this flow is analyzed in the only case when the sheet is held at a constant temperature. In order to obtain more realistic solutions where non-isothermal conditions at the flat sheet are present, in this paper we study flow
177
and heat transfer on a nonlinearly stretching sheet with velocity uw (x) ∼ x1/3 for two different types of thermal boundary conditions on the sheet, that is, prescribed surface temperature (PST case) and prescribed heat flux (PHF case). In this work, we investigate a problem of considerable engineering interest, that is, the boundary layer flow over a nonlinearly stretching sheet when the temperature of the surface or the surface heat flux vary as a power function of the distance x from the leading edge of the surface. Realize that many of these boundary conditions may be difficult to realize in practice, but we are carrying out these analyses with an emphasis on the similarity solutions for heat transfer, which constitute the basis for real engineering problems. To our knowledge, the aforementioned problem has not yet been considered within available literature. On the other hand, the effect of thermal radiation on temperature profiles can be quite significant at high operating temperature (say, engineering processes involving nuclear power plant, gas turbines and many others). However, as shown in this research, the influence on temperature profiles of such an effect becomes important when the radiation parameter NR is low, but it is negligible when NR is much larger. In view of this, Makinde (2005) studied the influence of the thermal radiation on the transient free convection of an absorbing-emitting fluid along a moving vertical porous surface. Another effect which bears great importance on heat transfer is the viscous dissipation. When the viscosity of the fluid and/or the velocity gradient is high, the dissipation term becomes more important. Consequently, the effects of viscous dissipation and thermal radiation also are included in the energy equation. Moreover, sometimes, a lack in available boundary conditions introduces a hardness affixed and prevents us from achieving studies of flow and heat transfer. For that reason, in the boundary-layer flow induced in a quiescent fluid by a nonlinearly stretching sheet, first, we solve the momentum transfer problem, and second, momentum and heat transfer problems via the fourth-order Runge–Kutta scheme along with shooting technique. The fluid is at rest and the motion is created by the surface whose velocity varies nonlinearly with the distance x from a fixed point; also, the sheet is held at a temperature higher than the temperature T∞ of the ambient fluid. This paper aims to find similarity numerical solutions for problem above-mentioned. Section 2 presents the analysis and numerical procedure of the flow and its mechanical characteristics. Heat transfer of a viscous fluid over a nonlinearly stretching sheet taking into account viscous dissipation in the presence of thermal radiation will be analyzed in Sections 3 and 4. The results emphasize the significant influence on the temperature distributions and heat transfer characteristics of the cited effects. To achieve these objectives, it will be shown that, viscous flow and heat transfer problems influenced by a special nonlinearly stretching sheet surface render themselves to a similarity solution.
2.
Basic equations for the flow
Let consider the flow of an incompressible viscous fluid past a flat sheet coinciding with the plane y = 0, the flow being
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
confined to y > 0. Two equal and opposite forces are applied along the x-axis so that the wall is stretched keeping the origin fixed. The steady two-dimensional boundary-layer equations for this fluid, in the usual notation, are ∂u ∂v + = 0, ∂x ∂y u
(1)
∂u ∂2 u ∂u +v = 2, ∂x ∂y ∂y
(2)
where (x, y) denotes the Cartesian coordinates along the sheet and normal to it, u and v are the velocity components of the fluid in the x and y directions, respectively, and is the kinematic viscosity. The boundary conditions to the present problem are uw (x) =
1/3 x , L4/3
v = 0 at y = 0,
(3)
u → 0 as y → ∞
(4)
where L is a characteristic length. Defining new similarity variables: =y
x−1/3 , L2/3
u=
L4/3
x1/3 f (),
v=−
L2/3
x−1/3
2f − f 3 (5)
and substituting into Eq. (2) gives 3f + 2ff − (f ) = 0, 2
f = 1 at = 0,
are in (Cortell, 2005b, 2006c). The numerical solution determines f (0) = −0.677647; ı = 5.58 and f(∞) = 1.301706. Here, ı is defined as the value of the similarity variable at which the dimensionless velocity f () equals 0.01 and then one obtains ı0.01 = 5.58L2/3 x1/3 . On the other hand, based on third equation (5), the transverse velocity component of the edge of the boundary layer v∞ is related on f(∞) as v∞ = −( /L2/3 x1/3 )(2f (∞)/3). This velocity component v∞ determines the amount of fluid which is entrained by the movement of the surface (dragged by the sheet) and one obtains v∞ = −0.867804 /L2/3 x1/3 . Also, the shear stress at the stretched surface is w = −0.677647 /L2 . Further, it is observed from Fig. 1 that the velocity component u decreases in the boundary layer with increase of .
(6)
where a prime denotes differentiation with respect to the independent similarity variable and f is the dimensionless stream function. Realize that with these changes of the variables, Eq. (1) is identically satisfied. The boundary conditions in Eqs. (3) and (4) become f = 0,
Fig. 1 – Plots of the functions f, f and f for problems (6)–(8).
3. Heat transfer analyses and similarity solutions By using usual boundary layer approximations, the equation of the energy for temperature T in the presence of radiation and viscous dissipation is given by
(7) u
f → 0 as → ∞.
(8)
It needs to be mentioned that Eq. (6) was recently derived by Magyari and Wiedmann (2006). They dealt with shear driven flows and, consequently, other boundary conditions were used in Magyari and Wiedmann (2006). The shear stress at the stretched surface is defined as
∂u
w =
∂y
∂T ∂T ∂2 T +v =˛ 2 + ∂x ∂y cP ∂y
∂y
−
1 ∂qr , cP ∂y
(11)
where ˛ is the thermal diffusivity, is the density, cP is the specific heat of the fluid at constant pressure and qr is the radiative heat flux. Using the Rosseland approximation for radiation (Siddheshwar and Mahabaleswar, 2005), the radiative heat flux is simplified as
(9) qr = −
w
∂u 2
4 ∗ ∂T 4 , 3k∗ ∂y
(12)
and, it is obvious from Eqs. (5) and (9) that w =
f (0), L2
(10)
where is the viscosity of the fluid. The solution for problems (6)–(8) is depicted in Fig. 1. It was solved numerically by employing a Runge–Kutta algorithm for high order initial value problems (Cortell, 1993b) and related numerical studies, and the solution procedures
where * and k* are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. It is assumed that the temperature differences within the flow such as that the term T4 may be expressed as a linear function of temperature. Hence, expanding T4 in a Taylor series about T∞ and neglecting higher order terms, one obtains 3 4 T − 3T∞ . T4 ∼ = 4T∞
(13)
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where k is the thermal conductivity. If m = 1/3, it is obvious from Eq. (20) that
In view of Eqs. (12) and (13), Eq. (11) reduces to u
∂T ∂T +v = ∂x ∂y
˛+
3 16 ∗ T∞ 3cP k∗
∂2 T + cP ∂y2
∂u 2 ∂y
.
(14) qw = −
From the above equation, it is seen that the effect of radiation is to enhance the thermal diffusivity. The second term on the right-hand side is the viscous dissipation term. Two kinds of thermal boundary condition at the wall are considered, and they are treated separately in the following sections.
Prescribed surface temperature (PST case)
x m L
at y = 0,
kA (0) L
(22)
It should be appointed that the assumption m = 2/3 reduces Eqs. (17) and (18) to
In this circumstance, the boundary conditions are T = Tw = T∞ + A
(21)
Further, if the thermal radiation’s effects are not considered (i.e., k0 = 1), Eq. (21) reduces to
qw = −
3.1.
kA (0). k0 L
T → T∞ as y → ∞,
+ 23 k0 (f − f ) = −k0 Ec (f ) , 2
(23)
(15) where A is a constant, T∞ is the fluid temperature far away from the surface, Tw is the temperature at the wall and m is the wall temperature parameter. By considering m = 0 and A = Tw − T∞ into Eq. (15), the constant surface temperature case can be obtained. Defining the non dimensional temperature () as () =
T − T∞ Tw − T∞
(16)
and using Eqs. (5), (15) and (16) in Eq. (14), one can find from Eq. (14): +
2k0 2 f − k0 mf = −k0 Ec (f ) , 3
(17)
where
Ec =
2 AL2 cP
and obviously all solutions are then of the similar type.
3.2.
Prescribed heat flux (PHF case)
In PHF case, one may define a dimensionless new temperature variable as g() =
T − T∞ , (D/k) xm+1/3 L2/3−m
(25)
with the following boundary conditions:
at y = 0 :
2 Lm−8/3 , AcP xm−2/3
(24)
∂T
qw = −k
∂y
=D w
x m L
, as y → ∞, T → T∞ ,
(18)
(26)
is the Eckert number, (= /˛) is the Prandtl number, the primes 3 is denote differentiation with respect to , NR = k∗ k/4 ∗ T∞ the radiation parameter and k0 = 3NR /(3NR + 4). It is easy to check that by setting k0 = 1 in Eq. (17), the thermal radiation is neglected. Realize that the x-coordinate cannot be eliminated from Eq. (17) when m = 2/3. So, the temperature profiles always depend on x. In other words, one may look for the availability of local similarity solutions. It should be noted that the assumptions Ec = 0 and k0 = 1, reduces Eq. (17) to that of Magyari and Wiedmann (2006) (Eq. (8a) in their paper). The boundary conditions for () follow from (15) and (16) as
where D is a constant, and m = 0 provides the constant heat flux case. Using Eqs. (5) and (25) into Eq. (14), one can find
Ec =
= 1 at = 0;
→ 0 as → ∞.
(19)
The rate of heat transfer of the surface is derived from Eqs. (12), (15) and (16) as
∂T
qw = −k
∂y
y=0
+ (qr )w = −
m−1/3
kA x (0) k0 L L
,
(20)
g +
2k0 1 fg − k0 m + 3 3
f g = −k0 Ec (f ) , 2
(27)
where,
Ec =
2 Lm−10/3 (D/k)cP xm−1/3
(28)
is the Eckert number, (= /␣) is the Prandtl number and k0 = 3NR /(3NR + 4). Realize that the x-coordinate cannot be eliminated from Eq. (25) when m = 1/3. So, the temperature profiles always depend on x. The boundary conditions for () follow from Eqs. (23) and (24) as g (0) = −1;
→ 0 as → ∞.
(29)
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
If m = 1/3, one can obtain from Eqs. (27) and (28) as g +
2 2 k0 (fg − f g) = −k0 Ec (f ) , 3
(30)
Ec =
2 k DL3 cP
(31)
and obviously all solutions are then of the similar type.
4.
Illustrative results
Without a break, we begin now the development of the procedure for completing the numerical solution for () and g(). There is no analytical solution for the flow problem and, accordingly, one had to use numerical techniques. It is clear that f (0) = −0.677647 in such a problem. Since the flow problem is uncoupled from the thermal problem, changes in the values of , NR and Ec (Ec ) will not affect the fluid velocity. For that reason, both the function f and its derivatives are identical in the complete problem (flow and heat transfer). In other words, the only parameters which affect the solution for the sheet temperature are the Prandtl number , the radiation parameter NR and the Eckert number Ec (Ec ). In view of the above discussions, first, the problem (Eqs. (6)–(8)), which provides the f (0) value, has been solved numerically, and second, with this numerical result, the heat transfer problems will have a suitable treatment. This procedure has already been applied to discuss some flow and heat transfer problems (Cortell, 2005b). (0) < 0 implies that the heat flows from the surface to the ambient fluid (i.e., Tw > T∞ ) and in accordance with Eq. (16) a negative is not realistic. Consequently, for a physically consistent numerical result, the corresponding is everywhere finite and non-negative. Table 1 extracts information including viscous dissipation and thermal radiation for both PST and PHF cases. The
Table 1 – Wall temperature gradient [− (0)] (PST case with m = 2/3) and wall temperature g(0) (PHF case with m = 1/3) for several values of , NR and Ec (Ec )
NR
Ec (Ec )
− (0)
g(0)
2
0.5 1 3 7
0.2
0.53553 0.71646 0.95660 1.06963
1.82027 1.37205 1.04239 0.93923
1
0.2
0.35480 0.71646 0.91586 1.79029
2.72682 1.37205 1.08597 0.59398
1
0 0.2 1 1.5
0.97887 0.91586 0.66393 0.50639
1.02151 1.08597 1.34327 1.50414
0.71 2 3 10
3
values of the wall temperature gradient [− (0)] for the PST case when m = 2/3 and the wall temperature g(0) for the PHF case when m = 1/3 as a function of all the parameters of the thermal boundary layer treated here, have been tabulated (see Table 1). In heat transfer phenomena these two physical parameters are usually analyzed from the corresponding numerical results. As shown in Table 1, the effect of Prandtl number is to increase the wall temperature gradient [− (0)] in PST case, and to decrease the wall temperature g(0) in the PHF case. The effect of increasing Ec (Ec ) is to increase the magnitude of both () and g(), whereas the opposite behaviour is seen for the thermal radiation parameter NR . In other words, it means that the thermal boundary layer thickness is a function of all the above-mentioned parameters. The combined effect of increasing values of and NR is to increase the numerical value of the wall temperature gradient [− (0)]; accordingly, more heat is carried out of the sheet, resulting in a decrease of the thermal boundary layer thickness, and hence increasing the heat transfer rate; however, an opposite behaviour can be seen for Ec (Ec ). It is also clear from Table 1 that, as and NR increase, the wall temperature g(0) in PHF case decreases, while it increases when, for fixed and NR , an augment in the value of Ec (Ec ) occurs. On the other hand, one concludes that dimensionless temperatures () and g() decrease as the radiation parameter NR increases.
Table 2 – Temperatures () and g() when = 2 and NR = 3 for several values of m without viscous dissipation m
()
− (0)
g()
−g ()
0
0 0.2 0.4 0.6 0.8 1.0 2.0 5.0 8.5
1.0 0.87912 0.76229 0.65270 0.55254 0.46303 0.17079 0.00545 0.00016
0.60798 0.59735 0.56830 0.52577 0.47483 0.42002 0.18247 0.00640 0.00010
1.20349 1.01467 0.84823 0.70359 0.57951 0.47431 0.16226 0.00482 0.00006
1.0 0.88797 0.77692 0.67056 0.57165 0.48197 0.18131 0.00578 0.00009
1
0 0.2 0.4 0.6 0.8 1.0 2.0 5.0 8.5
1.0 0.78696 0.61857 0.48582 0.38137 0.29930 0.08911 0.00237 0.00001
1.19416 0.94531 0.74607 0.58751 0.46191 0.36277 0.10782 0.00289 0.00004
0.74157 0.56663 0.43364 0.33249 0.25547 0.19672 0.05492 0.00142 0.00003
1.0 0.76025 0.57801 0.43984 0.33521 0.25596 0.06885 0.00170 0.00003
3
0 0.2 0.4 0.6 0.8 1.0 2.0 5.0 7.0
1.0 0.67812 0.46511 0.32274 0.22655 0.16083 0.03367 0.00067 0.00005
1.96996 1.29791 0.86337 0.58034 0.39440 0.27107 0.04907 0.00083 0.00007
0.48203 0.32035 0.21567 0.14711 0.10166 0.07115 0.01415 0.00028 0.00003
1.0 0.64420 0.41961 0.27657 0.18456 0.12473 0.02114 0.00033 0.00003
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
181
Fig. 2 – Temperature profiles in PST case for two values of without viscous dissipation when NR = 2 and m = 1 with (solid line) and without (broken line) thermal radiation.
Fig. 5 – Plot of (a) () for the problem (Eqs. (32)–(19)) in PST case for several values of NR when = 3 and m = 1; (b) g() for the problem (Eqs. (33)–(29)) in PHF case for several values of NR when = 3 and m = 1. for the PST case, and Fig. 3 – Plot of g() in PHF case for two values of without viscous dissipation when NR = 2 and m = 1 with (solid line) and without (broken line) thermal radiation.
g +
2k0 1 fg − k0 m + 3 3
f g = 0
(33)
for the PHF case. Here, k0 = 3NR /(3NR + 4).
By neglecting the second term on the right-hand side in Eqs. (17) and (27) (i.e., without viscous dissipation), it is possible to obtain similarity solutions for all m. Under this consideration, one may obtain the simpler equations: +
2k0 f − k0 mf = 0 3
(32)
Fig. 4 – Plot of g() in PHF case for two values of without viscous dissipation when NR = 2 and m = −1/3 (curves as in Fig. 3).
Fig. 6 – Plot of (a) () for the problem (Eqs. (32)–(19)) in PST case for several values of m when = 3 and NR = 2. (b) g() for the problem (Eqs. (33)–(29)) in PHF case for several values of m when = 3 and NR = 2.
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
It is seen from Table 2 that for = 2; NR = 3 and for a given position , () and g() decrease as the wall temperature parameter m increases. At this stage, it is worth noting (see Table 2) that, for each m, the values (0) and g(0) are estimated under the assumptions (∞) → 0 and | (∞)| → 0, or g(∞) → 0 and |g (∞)| → 0, respectively. This approach corrects unphysical behaviours of the solution and seems to play an important role in this class of problems (Cortell, 1994, 2005a, 2006b). Figs. 2–4 show that the imposition of the thermal radiation’s effect broadens the temperature distribution, and this is true for both PST and PHF cases. It is also seen from those figures that, as the Prandtl number increases, the dimensionless temperatures () and g() decrease. This yields a diminution of the thermal boundary layer thickness. An analysis of the graphical results shows the effect of NR and m on temperature profiles. The influence of the radiation parameter NR on both () and g() dimensionless temperature profiles is shown in Fig. 5a and b. From these figures one can observe that, as the radiation parameter NR increases, () and g() decrease. This result qualitatively agrees with the fact of the effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid. The influence of the wall temperature parameter m on dimensionless temperature profiles is shown in Fig. 6a–b. It is observed from these Figures that, as the wall temperature parameter m increases, () and g() decrease. The results of PST case are qualitatively similar to those of PHF case, but they differ quantitatively.
5.
Discussions and conclusions
Subject of this article is a research of the flow influenced by a nonlinearly stretching sheet with heat transfer. The stretching of the sheet is assumed to be proportional to the x1/3 quantity, x being the distance from the slit, and the paper highlights the conditions for the existence of similarity solutions. The numerical results obtained in this work have technological applications in fluid flows influenced by stretchable materials. The effects of various physical parameters on heat transfer phenomena in a viscous flow over a nonlinearly stretching sheet have been analysed. The governing momentum and heat transfer equations, which are partial and partially uncoupled, are transformed to ordinary differential equations by exploiting the similarity procedure. The resulting equation system is then solved numerically using a fourth-order Runge–Kutta algorithm along with a shooting technique. Similarity solutions for both PST/PHF cases have been found for all m when the viscous dissipation is neglected, whereas it is taken into account, m = 2/3(PST case) and m = 1/3(PHF case) provide this type of solution, too. It needs to be mentioned that, in general, the combined effect of increasing values of , NR , and m is to decrease the boundary layer thickness in PST and PHF cases. Accordingly, more heat is carried out of the surface, and hence increasing the heat transfer rate; however, an oppo-
site behaviour has been found for the Eckert number Ec (Ec ).
Acknowledgements The insightful reviews and invaluable suggestions from the reviewers are greatly acknowledged.
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Sadeghy, K., Najafi, A.H., Saffaripour, M., 2005. Sakiadis flow of an upper-convected Maxwell fluid. Int. J. Non-Linear Mech. 40, 1220–1228. Sakiadis, B.C., 1961. Boundary-layer behaviour on continuous solid surfaces. AIChE J. 7, 26–28. Sanjayanand, E., Khan, S.K., 2006. On heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet. Int. J. Therm. Sci. 45, 819–828.
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Siddheshwar, P.G., Mahabaleswar, U.S., 2005. Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet. Int. J. Non-Linear Mech. 40, 807–820. Vajravelu, K., 2001. Viscous flow over a nonlinearly stretching sheet. Appl. Math. Comput. 124, 281–288.
journal homepage: www.elsevier.com/locate/jmatprotec
Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface Rafael Cortell Bataller Departamento de F´ısica Aplicada, Escuela T´ecnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Polit´ecnica de Valencia, 46071 Valencia, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper presents a numerical analysis in connection with the boundary layer flow
Received 4 December 2006
induced in a quiescent fluid by a stretching sheet with velocity uw (x) ∼ x1/3 along with heat
Received in revised form
transfer. The surface temperature is assumed to have a power-law variation. The viscous
13 August 2007
dissipation and thermal radiation are considered in the energy equation. The governing
Accepted 16 September 2007
partial differential equations are converted into ordinary ones by a similarity transformation. The variations of dimensionless surface temperature as well as flow and heat transfer characteristics with the governing parameters are graphed and tabulated. Two cases are
Keywords:
studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the
Laminar boundary layer
sheet with prescribed heat flux (PHF case). Similarity solutions of the aforementioned prob-
Nonlinearly stretching surfaces
lem are given for two values of the surface temperature parameter m, namely, m = 2/3 in
Viscous dissipation
the PST case and m = 1/3 in the PHF case. Moreover, the mechanical characteristics of the
Thermal radiation
corresponding flow are also presented. © 2007 Elsevier B.V. All rights reserved.
1.
Introduction
In contrast to the well-known Blasius flow problem (Cortell, 2005a), which involves the boundary-layer flow passing through a stationary flat plate, Sakiadis (1961) considered the boundary-layer flow on a moving plate in a quiescent ambient fluid. The aforementioned problems are two special cases of more general studies (Ishak et al., 2007; Cortell, 2007) in which flow and heat transfer of a moving sheet in the presence of a co-flowing fluid were analyzed. Unfortunately, in reality, many flow applications in industrial processes are concerned with non-Newtonian fluids. Specifically, these include polymer melts and solutions, heattreated materials travelling between a feed roll and a wind-up roll or materials manufactured by extrusion, glass–fiber and paper production, cooling of metallic sheets or electronic chips, crystal growing and many others. In these cases, the final product of desired characteristics depends
E-mail address: rcortell@fis.upv.es. 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.09.055
on the rate of cooling in the process and the process of stretching. Very recently Sadeghy et al. (2005) studied the boundary layer of an upper-convected Maxwell fluid, and the role played by the fluid’s elasticity on flow characteristics was analyzed. Moreover, several linearly stretching sheet problems related with non-Newtonian fluids have been analyzed by other researchers in several geometries including, sometimes, a porous medium alike (Cortell, 1993a, 1994, 2006a,b; Prasad et al., 2000, 2002; Char, 1994). Stretching sheet problems of practical interest in materials processing were analyzed by Chen (2003). He dealt with a flow over a heated flat surface continuously moving in a parallel free stream of a non-Newtonian fluid, and various interesting aspects of this class of boundarylayer flow were presented. All the aforementioned researches were performed taking into account whether a constant value for the velocity wall or a linearly stretching sheet problem (i.e., uw (x) = ax). Nevertheless, it needs to be underlined that
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
Nomenclature A, D cP Ec Ec f g k k0 k* L m NR qr T u, v x, y
constants specific heat at a constant pressure Eckert number for the case of prescribed surface temperature (PST case) Eckert number for the case of prescribed heat flux (PHF case) dimensionless stream function dimensionless temperature fluid thermal conductivity dimensionless parameter related with thermal radiation mean absorption coefficient reference length surface temperature parameter radiation parameter radiative heat flux temperature across the thermal boundarylayer velocity components along x and y directions, respectively Cartesian coordinates along the plate and normal to it, respectively
Greek symbols ˛ thermal diffusivity dimensionless similarity variable dimensionless temperature dynamic viscosity kinematic viscosity density Prandtl number * Stefan–Boltzmann constant shear stress Subscripts w, ∞ conditions at the surface and in the free stream, respectively Superscript derivative with respect to
the stretching of the sheet may not necessarily be linear. In view of this, the flow influenced by a nonlinearly stretching sheet was investigated by Vajravelu (2001), and power-law or exponentially stretching sheet was studied by Ali (1995) and Elbashbeshy (2001), respectively. Further, momentum, heat and mass transfer over an exponentially stretching surface were considered by Sanjayanand and Khan (2006). They also enclosed the effects of viscous dissipation and work done by deformation in the energy equation. Vajravelu (2001) studied flow and heat transfer in a viscous fluid over a nonlinearly stretching sheet without viscous dissipation, but the heat transfer in this flow is analyzed in the only case when the sheet is held at a constant temperature. In order to obtain more realistic solutions where non-isothermal conditions at the flat sheet are present, in this paper we study flow
177
and heat transfer on a nonlinearly stretching sheet with velocity uw (x) ∼ x1/3 for two different types of thermal boundary conditions on the sheet, that is, prescribed surface temperature (PST case) and prescribed heat flux (PHF case). In this work, we investigate a problem of considerable engineering interest, that is, the boundary layer flow over a nonlinearly stretching sheet when the temperature of the surface or the surface heat flux vary as a power function of the distance x from the leading edge of the surface. Realize that many of these boundary conditions may be difficult to realize in practice, but we are carrying out these analyses with an emphasis on the similarity solutions for heat transfer, which constitute the basis for real engineering problems. To our knowledge, the aforementioned problem has not yet been considered within available literature. On the other hand, the effect of thermal radiation on temperature profiles can be quite significant at high operating temperature (say, engineering processes involving nuclear power plant, gas turbines and many others). However, as shown in this research, the influence on temperature profiles of such an effect becomes important when the radiation parameter NR is low, but it is negligible when NR is much larger. In view of this, Makinde (2005) studied the influence of the thermal radiation on the transient free convection of an absorbing-emitting fluid along a moving vertical porous surface. Another effect which bears great importance on heat transfer is the viscous dissipation. When the viscosity of the fluid and/or the velocity gradient is high, the dissipation term becomes more important. Consequently, the effects of viscous dissipation and thermal radiation also are included in the energy equation. Moreover, sometimes, a lack in available boundary conditions introduces a hardness affixed and prevents us from achieving studies of flow and heat transfer. For that reason, in the boundary-layer flow induced in a quiescent fluid by a nonlinearly stretching sheet, first, we solve the momentum transfer problem, and second, momentum and heat transfer problems via the fourth-order Runge–Kutta scheme along with shooting technique. The fluid is at rest and the motion is created by the surface whose velocity varies nonlinearly with the distance x from a fixed point; also, the sheet is held at a temperature higher than the temperature T∞ of the ambient fluid. This paper aims to find similarity numerical solutions for problem above-mentioned. Section 2 presents the analysis and numerical procedure of the flow and its mechanical characteristics. Heat transfer of a viscous fluid over a nonlinearly stretching sheet taking into account viscous dissipation in the presence of thermal radiation will be analyzed in Sections 3 and 4. The results emphasize the significant influence on the temperature distributions and heat transfer characteristics of the cited effects. To achieve these objectives, it will be shown that, viscous flow and heat transfer problems influenced by a special nonlinearly stretching sheet surface render themselves to a similarity solution.
2.
Basic equations for the flow
Let consider the flow of an incompressible viscous fluid past a flat sheet coinciding with the plane y = 0, the flow being
178
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confined to y > 0. Two equal and opposite forces are applied along the x-axis so that the wall is stretched keeping the origin fixed. The steady two-dimensional boundary-layer equations for this fluid, in the usual notation, are ∂u ∂v + = 0, ∂x ∂y u
(1)
∂u ∂2 u ∂u +v = 2, ∂x ∂y ∂y
(2)
where (x, y) denotes the Cartesian coordinates along the sheet and normal to it, u and v are the velocity components of the fluid in the x and y directions, respectively, and is the kinematic viscosity. The boundary conditions to the present problem are uw (x) =
1/3 x , L4/3
v = 0 at y = 0,
(3)
u → 0 as y → ∞
(4)
where L is a characteristic length. Defining new similarity variables: =y
x−1/3 , L2/3
u=
L4/3
x1/3 f (),
v=−
L2/3
x−1/3
2f − f 3 (5)
and substituting into Eq. (2) gives 3f + 2ff − (f ) = 0, 2
f = 1 at = 0,
are in (Cortell, 2005b, 2006c). The numerical solution determines f (0) = −0.677647; ı = 5.58 and f(∞) = 1.301706. Here, ı is defined as the value of the similarity variable at which the dimensionless velocity f () equals 0.01 and then one obtains ı0.01 = 5.58L2/3 x1/3 . On the other hand, based on third equation (5), the transverse velocity component of the edge of the boundary layer v∞ is related on f(∞) as v∞ = −( /L2/3 x1/3 )(2f (∞)/3). This velocity component v∞ determines the amount of fluid which is entrained by the movement of the surface (dragged by the sheet) and one obtains v∞ = −0.867804 /L2/3 x1/3 . Also, the shear stress at the stretched surface is w = −0.677647 /L2 . Further, it is observed from Fig. 1 that the velocity component u decreases in the boundary layer with increase of .
(6)
where a prime denotes differentiation with respect to the independent similarity variable and f is the dimensionless stream function. Realize that with these changes of the variables, Eq. (1) is identically satisfied. The boundary conditions in Eqs. (3) and (4) become f = 0,
Fig. 1 – Plots of the functions f, f and f for problems (6)–(8).
3. Heat transfer analyses and similarity solutions By using usual boundary layer approximations, the equation of the energy for temperature T in the presence of radiation and viscous dissipation is given by
(7) u
f → 0 as → ∞.
(8)
It needs to be mentioned that Eq. (6) was recently derived by Magyari and Wiedmann (2006). They dealt with shear driven flows and, consequently, other boundary conditions were used in Magyari and Wiedmann (2006). The shear stress at the stretched surface is defined as
∂u
w =
∂y
∂T ∂T ∂2 T +v =˛ 2 + ∂x ∂y cP ∂y
∂y
−
1 ∂qr , cP ∂y
(11)
where ˛ is the thermal diffusivity, is the density, cP is the specific heat of the fluid at constant pressure and qr is the radiative heat flux. Using the Rosseland approximation for radiation (Siddheshwar and Mahabaleswar, 2005), the radiative heat flux is simplified as
(9) qr = −
w
∂u 2
4 ∗ ∂T 4 , 3k∗ ∂y
(12)
and, it is obvious from Eqs. (5) and (9) that w =
f (0), L2
(10)
where is the viscosity of the fluid. The solution for problems (6)–(8) is depicted in Fig. 1. It was solved numerically by employing a Runge–Kutta algorithm for high order initial value problems (Cortell, 1993b) and related numerical studies, and the solution procedures
where * and k* are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. It is assumed that the temperature differences within the flow such as that the term T4 may be expressed as a linear function of temperature. Hence, expanding T4 in a Taylor series about T∞ and neglecting higher order terms, one obtains 3 4 T − 3T∞ . T4 ∼ = 4T∞
(13)
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
where k is the thermal conductivity. If m = 1/3, it is obvious from Eq. (20) that
In view of Eqs. (12) and (13), Eq. (11) reduces to u
∂T ∂T +v = ∂x ∂y
˛+
3 16 ∗ T∞ 3cP k∗
∂2 T + cP ∂y2
∂u 2 ∂y
.
(14) qw = −
From the above equation, it is seen that the effect of radiation is to enhance the thermal diffusivity. The second term on the right-hand side is the viscous dissipation term. Two kinds of thermal boundary condition at the wall are considered, and they are treated separately in the following sections.
Prescribed surface temperature (PST case)
x m L
at y = 0,
kA (0) L
(22)
It should be appointed that the assumption m = 2/3 reduces Eqs. (17) and (18) to
In this circumstance, the boundary conditions are T = Tw = T∞ + A
(21)
Further, if the thermal radiation’s effects are not considered (i.e., k0 = 1), Eq. (21) reduces to
qw = −
3.1.
kA (0). k0 L
T → T∞ as y → ∞,
+ 23 k0 (f − f ) = −k0 Ec (f ) , 2
(23)
(15) where A is a constant, T∞ is the fluid temperature far away from the surface, Tw is the temperature at the wall and m is the wall temperature parameter. By considering m = 0 and A = Tw − T∞ into Eq. (15), the constant surface temperature case can be obtained. Defining the non dimensional temperature () as () =
T − T∞ Tw − T∞
(16)
and using Eqs. (5), (15) and (16) in Eq. (14), one can find from Eq. (14): +
2k0 2 f − k0 mf = −k0 Ec (f ) , 3
(17)
where
Ec =
2 AL2 cP
and obviously all solutions are then of the similar type.
3.2.
Prescribed heat flux (PHF case)
In PHF case, one may define a dimensionless new temperature variable as g() =
T − T∞ , (D/k) xm+1/3 L2/3−m
(25)
with the following boundary conditions:
at y = 0 :
2 Lm−8/3 , AcP xm−2/3
(24)
∂T
qw = −k
∂y
=D w
x m L
, as y → ∞, T → T∞ ,
(18)
(26)
is the Eckert number, (= /˛) is the Prandtl number, the primes 3 is denote differentiation with respect to , NR = k∗ k/4 ∗ T∞ the radiation parameter and k0 = 3NR /(3NR + 4). It is easy to check that by setting k0 = 1 in Eq. (17), the thermal radiation is neglected. Realize that the x-coordinate cannot be eliminated from Eq. (17) when m = 2/3. So, the temperature profiles always depend on x. In other words, one may look for the availability of local similarity solutions. It should be noted that the assumptions Ec = 0 and k0 = 1, reduces Eq. (17) to that of Magyari and Wiedmann (2006) (Eq. (8a) in their paper). The boundary conditions for () follow from (15) and (16) as
where D is a constant, and m = 0 provides the constant heat flux case. Using Eqs. (5) and (25) into Eq. (14), one can find
Ec =
= 1 at = 0;
→ 0 as → ∞.
(19)
The rate of heat transfer of the surface is derived from Eqs. (12), (15) and (16) as
∂T
qw = −k
∂y
y=0
+ (qr )w = −
m−1/3
kA x (0) k0 L L
,
(20)
g +
2k0 1 fg − k0 m + 3 3
f g = −k0 Ec (f ) , 2
(27)
where,
Ec =
2 Lm−10/3 (D/k)cP xm−1/3
(28)
is the Eckert number, (= /␣) is the Prandtl number and k0 = 3NR /(3NR + 4). Realize that the x-coordinate cannot be eliminated from Eq. (25) when m = 1/3. So, the temperature profiles always depend on x. The boundary conditions for () follow from Eqs. (23) and (24) as g (0) = −1;
→ 0 as → ∞.
(29)
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
If m = 1/3, one can obtain from Eqs. (27) and (28) as g +
2 2 k0 (fg − f g) = −k0 Ec (f ) , 3
(30)
Ec =
2 k DL3 cP
(31)
and obviously all solutions are then of the similar type.
4.
Illustrative results
Without a break, we begin now the development of the procedure for completing the numerical solution for () and g(). There is no analytical solution for the flow problem and, accordingly, one had to use numerical techniques. It is clear that f (0) = −0.677647 in such a problem. Since the flow problem is uncoupled from the thermal problem, changes in the values of , NR and Ec (Ec ) will not affect the fluid velocity. For that reason, both the function f and its derivatives are identical in the complete problem (flow and heat transfer). In other words, the only parameters which affect the solution for the sheet temperature are the Prandtl number , the radiation parameter NR and the Eckert number Ec (Ec ). In view of the above discussions, first, the problem (Eqs. (6)–(8)), which provides the f (0) value, has been solved numerically, and second, with this numerical result, the heat transfer problems will have a suitable treatment. This procedure has already been applied to discuss some flow and heat transfer problems (Cortell, 2005b). (0) < 0 implies that the heat flows from the surface to the ambient fluid (i.e., Tw > T∞ ) and in accordance with Eq. (16) a negative is not realistic. Consequently, for a physically consistent numerical result, the corresponding is everywhere finite and non-negative. Table 1 extracts information including viscous dissipation and thermal radiation for both PST and PHF cases. The
Table 1 – Wall temperature gradient [− (0)] (PST case with m = 2/3) and wall temperature g(0) (PHF case with m = 1/3) for several values of , NR and Ec (Ec )
NR
Ec (Ec )
− (0)
g(0)
2
0.5 1 3 7
0.2
0.53553 0.71646 0.95660 1.06963
1.82027 1.37205 1.04239 0.93923
1
0.2
0.35480 0.71646 0.91586 1.79029
2.72682 1.37205 1.08597 0.59398
1
0 0.2 1 1.5
0.97887 0.91586 0.66393 0.50639
1.02151 1.08597 1.34327 1.50414
0.71 2 3 10
3
values of the wall temperature gradient [− (0)] for the PST case when m = 2/3 and the wall temperature g(0) for the PHF case when m = 1/3 as a function of all the parameters of the thermal boundary layer treated here, have been tabulated (see Table 1). In heat transfer phenomena these two physical parameters are usually analyzed from the corresponding numerical results. As shown in Table 1, the effect of Prandtl number is to increase the wall temperature gradient [− (0)] in PST case, and to decrease the wall temperature g(0) in the PHF case. The effect of increasing Ec (Ec ) is to increase the magnitude of both () and g(), whereas the opposite behaviour is seen for the thermal radiation parameter NR . In other words, it means that the thermal boundary layer thickness is a function of all the above-mentioned parameters. The combined effect of increasing values of and NR is to increase the numerical value of the wall temperature gradient [− (0)]; accordingly, more heat is carried out of the sheet, resulting in a decrease of the thermal boundary layer thickness, and hence increasing the heat transfer rate; however, an opposite behaviour can be seen for Ec (Ec ). It is also clear from Table 1 that, as and NR increase, the wall temperature g(0) in PHF case decreases, while it increases when, for fixed and NR , an augment in the value of Ec (Ec ) occurs. On the other hand, one concludes that dimensionless temperatures () and g() decrease as the radiation parameter NR increases.
Table 2 – Temperatures () and g() when = 2 and NR = 3 for several values of m without viscous dissipation m
()
− (0)
g()
−g ()
0
0 0.2 0.4 0.6 0.8 1.0 2.0 5.0 8.5
1.0 0.87912 0.76229 0.65270 0.55254 0.46303 0.17079 0.00545 0.00016
0.60798 0.59735 0.56830 0.52577 0.47483 0.42002 0.18247 0.00640 0.00010
1.20349 1.01467 0.84823 0.70359 0.57951 0.47431 0.16226 0.00482 0.00006
1.0 0.88797 0.77692 0.67056 0.57165 0.48197 0.18131 0.00578 0.00009
1
0 0.2 0.4 0.6 0.8 1.0 2.0 5.0 8.5
1.0 0.78696 0.61857 0.48582 0.38137 0.29930 0.08911 0.00237 0.00001
1.19416 0.94531 0.74607 0.58751 0.46191 0.36277 0.10782 0.00289 0.00004
0.74157 0.56663 0.43364 0.33249 0.25547 0.19672 0.05492 0.00142 0.00003
1.0 0.76025 0.57801 0.43984 0.33521 0.25596 0.06885 0.00170 0.00003
3
0 0.2 0.4 0.6 0.8 1.0 2.0 5.0 7.0
1.0 0.67812 0.46511 0.32274 0.22655 0.16083 0.03367 0.00067 0.00005
1.96996 1.29791 0.86337 0.58034 0.39440 0.27107 0.04907 0.00083 0.00007
0.48203 0.32035 0.21567 0.14711 0.10166 0.07115 0.01415 0.00028 0.00003
1.0 0.64420 0.41961 0.27657 0.18456 0.12473 0.02114 0.00033 0.00003
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
181
Fig. 2 – Temperature profiles in PST case for two values of without viscous dissipation when NR = 2 and m = 1 with (solid line) and without (broken line) thermal radiation.
Fig. 5 – Plot of (a) () for the problem (Eqs. (32)–(19)) in PST case for several values of NR when = 3 and m = 1; (b) g() for the problem (Eqs. (33)–(29)) in PHF case for several values of NR when = 3 and m = 1. for the PST case, and Fig. 3 – Plot of g() in PHF case for two values of without viscous dissipation when NR = 2 and m = 1 with (solid line) and without (broken line) thermal radiation.
g +
2k0 1 fg − k0 m + 3 3
f g = 0
(33)
for the PHF case. Here, k0 = 3NR /(3NR + 4).
By neglecting the second term on the right-hand side in Eqs. (17) and (27) (i.e., without viscous dissipation), it is possible to obtain similarity solutions for all m. Under this consideration, one may obtain the simpler equations: +
2k0 f − k0 mf = 0 3
(32)
Fig. 4 – Plot of g() in PHF case for two values of without viscous dissipation when NR = 2 and m = −1/3 (curves as in Fig. 3).
Fig. 6 – Plot of (a) () for the problem (Eqs. (32)–(19)) in PST case for several values of m when = 3 and NR = 2. (b) g() for the problem (Eqs. (33)–(29)) in PHF case for several values of m when = 3 and NR = 2.
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 3 ( 2 0 0 8 ) 176–183
It is seen from Table 2 that for = 2; NR = 3 and for a given position , () and g() decrease as the wall temperature parameter m increases. At this stage, it is worth noting (see Table 2) that, for each m, the values (0) and g(0) are estimated under the assumptions (∞) → 0 and | (∞)| → 0, or g(∞) → 0 and |g (∞)| → 0, respectively. This approach corrects unphysical behaviours of the solution and seems to play an important role in this class of problems (Cortell, 1994, 2005a, 2006b). Figs. 2–4 show that the imposition of the thermal radiation’s effect broadens the temperature distribution, and this is true for both PST and PHF cases. It is also seen from those figures that, as the Prandtl number increases, the dimensionless temperatures () and g() decrease. This yields a diminution of the thermal boundary layer thickness. An analysis of the graphical results shows the effect of NR and m on temperature profiles. The influence of the radiation parameter NR on both () and g() dimensionless temperature profiles is shown in Fig. 5a and b. From these figures one can observe that, as the radiation parameter NR increases, () and g() decrease. This result qualitatively agrees with the fact of the effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid. The influence of the wall temperature parameter m on dimensionless temperature profiles is shown in Fig. 6a–b. It is observed from these Figures that, as the wall temperature parameter m increases, () and g() decrease. The results of PST case are qualitatively similar to those of PHF case, but they differ quantitatively.
5.
Discussions and conclusions
Subject of this article is a research of the flow influenced by a nonlinearly stretching sheet with heat transfer. The stretching of the sheet is assumed to be proportional to the x1/3 quantity, x being the distance from the slit, and the paper highlights the conditions for the existence of similarity solutions. The numerical results obtained in this work have technological applications in fluid flows influenced by stretchable materials. The effects of various physical parameters on heat transfer phenomena in a viscous flow over a nonlinearly stretching sheet have been analysed. The governing momentum and heat transfer equations, which are partial and partially uncoupled, are transformed to ordinary differential equations by exploiting the similarity procedure. The resulting equation system is then solved numerically using a fourth-order Runge–Kutta algorithm along with a shooting technique. Similarity solutions for both PST/PHF cases have been found for all m when the viscous dissipation is neglected, whereas it is taken into account, m = 2/3(PST case) and m = 1/3(PHF case) provide this type of solution, too. It needs to be mentioned that, in general, the combined effect of increasing values of , NR , and m is to decrease the boundary layer thickness in PST and PHF cases. Accordingly, more heat is carried out of the surface, and hence increasing the heat transfer rate; however, an oppo-
site behaviour has been found for the Eckert number Ec (Ec ).
Acknowledgements The insightful reviews and invaluable suggestions from the reviewers are greatly acknowledged.
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