Theor. Comput. Fluid Dyn. (2007) 21: 435–446 DOI 10.1007/s00162-007-0056-z

O R I G I NA L A RT I C L E

Rafael Cortell

Flow and heat transfer in a moving fluid over a moving flat surface

Received: 6 September 2006 / Accepted: 3 July 2007 / Published online: 14 August 2007 © Springer-Verlag 2007

Abstract In this paper, a numerical analysis of the momentum and heat transfer of an incompressible fluid past a parallel moving sheet based on composite reference velocity U is carried out. A single set of equations has been formulated for both momentum and thermal boundary layer problems containing the following parameters: r the ratio of the free stream velocity to the composite reference velocity, σ (Prandtl number) the ratio of the momentum diffusivity of the fluid to its thermal diffusivity, and E c (E ck ) (Eckert number). The present study has been carried out in the domain 0 ≤ r ≤ 1. It is found that the direction of the wall shear changes in such an interval and an increase of the parameter r yields an increase in temperature. Keywords Flat-plate boundary layer · Parallel moving surface · Heat transfer · Viscous dissipation PACS 44.20.+b · 47.27.ek · 47.11.-j 1 Introduction Boundary layer behaviour over a moving continuous solid surface is an important type of flow occurring in several engineering processes. Specifically, these include heat-treated 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 on the rate of cooling in the process and the process of stretching. Since the pioneering work of Sakiadis [1], various aspects of the problem have been investigated by many authors. Crane [2] and Gupta and Gupta [3] have analyzed the stretching problem with constant surface temperature while Soundalgekar [4] investigated the Stokes problem for a viscoelastic fluid. This flow was examined by Siddappa and Khapate [5] for a special class of non-Newtonian fluids known as second-order fluids which are viscoelastic in nature. Danberg and Fansler [6] studied the solution for the boundary-layer flow past a wall that is stretched with a speed proportional to the distance along the wall and Rao et al. [7] studied arbitrary injection/suction at a moving wall in a power-law fluid. Recently, Magyari and Keller [8] studied the stretching problem of an incompressible fluid. On the other hand, the author of this work has studied heat transfer in an incompressible second-order fluid caused by a stretching sheet [9]. He also has analyzed the flow of a fluid of grade three past an infinite porous flat plate subject to suction at the plate [10]. Usually, in the above studies, a similarity transformation for the velocity field is used. This procedure reduces the governing equations into a system of ordinary differential equations, thereby rendering the boundary-layer problem amenable to analysis. However, sometimes, a study of flow and Communicated by R. Grimshaw R. Cortell Departamento de Física Aplicada, Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Valencia, 46071 Valencia, Spain E-mail: [email protected]

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heat transfer it is not possible to carry out due to a lack in available boundary conditions. For that reason, in our analysis, first, we solve the momentum transfer problem and, second, momentum and heat transfer problems. Flow and heat transfer in a moving sheet which is situated in either a quiescent or moving fluid has gained growing interest motivated by its importance in many engineering applications such as cooling of sheets or films, metallic plates and cylinders, conveyor belts just to name a few. The corresponding momentum transfer problem through the moving-wall laminar boundary layer with external stream was studied by Siekman [11] and a highly wrought work for the case of reverse flow was carried out by Klemp and Acrivos [12]. Two sets of boundary value problems in two cases Uw > U∞ and U∞ > Uw were separately analyzed by Abdulhafez [13] and Chappidi and Gunnerson [14]; U∞ is the constant free stream velocity and Uw is the constant sheet velocity. Later, Afzal et al. [15] formulated a single set of equations by using the composite reference velocity U (= U∞ + Uw ) irrespective of whether Uw > U∞ or U∞ > Uw . This single set of equations remains valid for both the forward and reverse motion of the sheet. On the other hand, the case of reverse moving surface is separately considered by Hussaini et al. [16] who treated in detail the existence and non-uniqueness of the solution. They showed that the solution of that boundary value problem depends on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. Further, parallel and reverse flows with heat transfer were treated by Lin and Haung [17] and, very recently, Sparrow and Abraham [18] studied flow and heat transfer of a moving sheet in the presence of a co-flowing fluid. In that work, a relative-velocity model for flow analysis was employed and a non-isothermal condition at the flat surface is considered. In order to obtain several realistic solutions where another non-isothermal conditions at the flat sheet are present, in this paper we study flow and heat transfer on a moving sheet in the presence of a co-flowing fluid for two different types of thermal boundary conditions on the surface, that is, constant surface temperature (CST case) and prescribed power-law surface temperature of second degree (PST case). The surface is held at a temperature Tw (x) higher than the temperature T∞ of the ambient fluid. Another effect which bears great importance on heat transfer is viscous dissipation. When the viscosity of the fluid and/or the velocity gradient is high, the dissipation term becomes important, although it disappears at infinity. Consequently, the effects of viscous dissipation are also included in the energy equation. For many cases, such as polymer processing, which is at very high temperature [19] and flows appeared in glacier physics [20], viscous dissipation plays an important role. On the other hand, numerical results confirm that, in the laminar forced convection in straight micro-channels, both temperature dependence of viscosity and viscous dissipation effects cannot be neglected in a wide range of real operative circumstances. These issues are discussed in Ref. [21]. It is shown that for different fluids the viscous dissipation effect will be different and, for the aforementioned class of flow, a criterion has been provided to discern the limit for which the effect of viscous dissipation can no longer be ignored. The aim of the article is to analyse the effects on both momentum and heat transfer problems of three physical parameters: (a) the ratio r of the fluid velocity U∞ to the composite reference velocity U , (b) the Prandtl number σ and (c) the Eckert number E c (E ck ). In Sect. 2 we shall consider the analysis of the flow when the movement direction of the plate is the same as the direction of the main flow, that is, the positive xdirection; in Sect. 3 we shall examine the thermal problem when the viscous dissipation is included and the influence on the numerical results of these additional effects will also be discussed. Finally, several interesting behaviours of the effects of the aforementioned dimensionless parameters are discussed in Sect. 4.

2 Flow analysis Let us consider the flow of an incompressible viscous fluid past a flat and impermeable sheet y = 0 with a constant velocity Uw . Assuming that the flow in the laminar boundary layer is two-dimensional, the continuity equation and the boundary layer equation may be expressed as ∂v ∂u + = 0, ∂x ∂y ∂u ∂u ∂ 2u u +v =υ 2 ∂x ∂y ∂y

(1) (2)

where (x, y) are the Cartesian coordinates, u and v are the velocity components along x and y-axes and υ is the kinematic viscosity of the fluid.

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Table 1 The velocity gradient at the wall f  (0) in the domain 0 ≤ r ≤ 1 r

f  (0)

0 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.9 1.0

−0.627547 −0.493711 −0.363308 −0.299593 −0.237132 −0.115777 0.0 0.109652 0.212374 0.261075 0.307378 0.393567 0.469602

The boundary conditions of Eqs. (1) and (2) are at y = 0 : u = Uw , v = 0, as y → ∞ : u = U∞ .

(3) (4)

where U∞ and Uw are constant. They represent the free stream and sheet velocities, respectively. The similarity transformation based on the composite reference velocity U (= U∞ + Uw ) following Afzal et al. [13] is given by
U ηf  − f  u = U f (η), η = y . (5) , v=U √ 2νx 2Rex where f is the dimensionless stream function, Rex = Uνx is the local Reynolds number, the prime denotes the differentiation with respect to η and the laminar boundary layer Eqs. (1)–(4) reduces to f  + f f  = 0, f (0) = 0,

f  (0) = 1 − r,

(6) f  (∞) = r

(7a,b,c)

U∞ . (8) U Here, r is a constant and the present study is carried out in the domain 0 ≤ r ≤ 1. Equations (6) and (7), for r = 1 (i.e., Uw = 0), are the classical Blasius flat-plate flow problem and a numerical study in detail of that problem has been carried out by the author of this work [22] and for r = 0 (i.e., U∞ = 0) are the equations for the stretching sheet proposed by Afzal [23,24]. For 0 < r < 1 both the wall and the free stream are moving in positive x- direction. There are dual solutions for 1 < r < 1.548 and no solution for r > 1.548 (see References [15,25]). The cases r = 0 and r = 1 correspond to well-known momentum transfer problems and no comment is needed. A listing of the velocity gradient wall f  (0) values is given in Table 1. These values were obtained by employing a Runge–Kutta algorithm for high-order initial value problems [26] and related numerical studies, ∞ , the fluid motion is faster than the and the solution procedures are in [22,27]. Because of r = UU∞ = U∞U+U w sheet motion if r > 0.5 (i.e., U∞ > Uw ); on the contrary, when the sheet motion is faster than the fluid motion, we have r < 0.5 (i.e., U∞ < Uw ). Further, it is worth mentioning here that throughout the paper we analyze flow and heat transfer in the interval 0 < r < 1. In other words, all of the situations are of the type when both the sheet and the free stream are moving in the positive x-direction. On the other hand, the shear stress at a point on the surface is
  ∂u U  τ0 = µ = µf (0)U . (9) ∂ y y=0 2νx r=

where µ is the viscosity of the fluid.

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Fig. 1 Variation with η of the functions f (solid line); f  (broken line) and f  (dash dot line) for several values of r

The non-dimensional form of the shear stress is τ=

τ0 , ρU 2

(10)

and we obtain from (9) f  (0) τ=√ , 2Rex

(11)

being the skin friction coefficient c f = 2τ . It is observed from Table 1 that the direction of the wall shear depends on whether the free stream velocity (=U∞ ) exceeds the sheet velocity (=Uw )(i.e., U∞ > Uw ; r > 0.5 and f  (0) > 0) or, on the contrary, the sheet velocity exceeds the free stream velocity (i.e., U∞ < Uw ; r < 0.5 and f  (0) < 0). When the free stream velocity exceeds the sheet velocity, it is obvious that the fluid drags the sheet and the shear stress at the surface acts in the positive x direction (plus signs in Table 1), whereas in the case of the faster moving sheet the direction of the wall shear is quite the opposite (minus signs in Table 1). For the case in which only one of the media moves, we also see from this table that the magnitude of the wall shear is smaller for Uw = 0 (i.e., the sheet is at rest and we have the fluid in motion over it; r = 1 and f  (0) = 0.469602) in comparison to the  case of U∞ = 0 (i.e., the fluid is at rest and the motion is created by the sheet; r = 0;and  f  (0) = 0.627547). On the other hand, Fig. 1 shows the variations with η of the functions f, f  and f  for several values of r . Values of f and f  are represented in the vertical left axis and f  (< 0) appears in the vertical right axis. Also, dimensionless variable η is measured on the two horizontal axes from the corresponding origins shown in Fig. 1. From this figure we observe that the gradient f  (η) is positive for r > 1/2 and negative for r < 1/2. The case r = 0.5 corresponds to parallel flow solution (i.e., f  (η) = 0.5). Moreover, irrespective of r, f  (η) → 0 as η → ∞. 3 Heat transfer analyses By using boundary layer approximations and taking into account the viscous dissipation, we find the equation of the energy for temperature T given by   ∂T ν ∂u 2 ∂T ∂2T u (12) +v =α 2 + ∂x ∂y ∂y cP ∂ y where c P and α are the specific heat of the fluid at constant pressure and the thermal diffusivity, respectively. The last term in the right-hand side of Eq. (12) represents the viscous dissipation.

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3.1 Constant surface temperature (CST case) In this circumstance, the boundary conditions are T = Tw at y = 0; T → T∞ as y → ∞.

(13)

Defining the non dimensional temperature θ (η) as θ (η) =

T − T∞ Tw − T∞

(14)

and using Eqs. (5) in (12), we get θ  + σ f θ  + σ E c ( f  )2 = 0,

(15)

where Ec =

U2 c P (Tw − T∞ )

(16)

is the Eckert number, σ (= αν ) is the Prandtl number and primes denote differentiation with respect to η. Further, the constants Tw , T∞ denote the temperature at the wall and at large distance from the wall, respectively. The boundary conditions for θ (η) follow from (13) and (14) as θ = 1 at η = 0; θ → 0 as η → ∞.

(17)

θ  (0) < 0 implies that the heat flows from the surface to the ambient fluid (i.e., Tw > T∞ ) and in accordance with Eq. (14) a negative θ is not realistic. Consequently, for this problem and for a physically consistent numerical result, the corresponding θ is everywhere finite and non negative. The rate of heat transfer of the surface is derived from (14) as
  dT U  −λT = −λT (Tw − T∞ )θ (0) , (18) dy y=0 2νx where λT is the thermal conductivity. As noted in the Introduction from Eq. (15) that when the viscosity of the fluid and/or the velocity gradient is high, the dissipation term (i.e., σ E c ( f  )2 ) becomes important; however it disappears at large η because of f  (η) → 0 as η → ∞(see Fig. 1). Without a break, we begin now the development of the procedure for completing the solution for θ (η). An analytical solution for the flow problem does not exist, and, accordingly, one has to use numerical techniques. It is clear that f  (0) in that problem depends on r (see Table 1). Since the flow problem in uncoupled from the thermal problem given by Eqs. (15)–(17), changes in the values of σ and E c will not affect the fluid velocity. For this reason, when r is given, the function f and its derivatives are identical in the complete problem (flow and heat transfer). In view of the above discussions, we have solved numerically, first, the problem {(6)–(8)}, and we have obtained f  (0) for several values of r and second, with these results, we shall solve numerically via “shooting method” together with a 4th-order Runge–Kutta method the problem {(6), (7), (15), (17)}. Equations (6) and (15) can easily be written as the equivalent first-order system: w1 w2 w3 w4

= w2 = w3 = −w1 w3 = w5

w5 = −σ w1 w5 − σ E c w32

(19)

Here, w1 = f (η) and w4 = θ (η). In accordance with conditions (7a, b) and first condition (17), we obtain for a given r : w1 (0) = 0; w2 (0) = 1 − r ; w4 (0) = 1.

(20)

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Fig. 2 Temperature and temperature gradient profiles in the CST case for two values of r and σ with E c = 0 [r = 0.3 (solid line); r = 0.7 (dash dot line)]

Note that the w3 (0) value is reached from Table 1. Using numerical methods of integration and disregarding temporarily the boundary conditions for f and θ at infinity, one can obtain a family of solutions of (19) for arbitrarily chosen values of w5 (0) = dθ ≤ 0. Tentatively we assume that a special value of θ  (0) yields dη η=0

a solution for which, at a certain η = η∞ , w2 (η∞ ) = r ; w4 (η∞ ) = 0 and satisfies these additional conditions. We estimate w5 (0) and integrate Eq. (19) together with the aforementioned boundary conditions as an initial value problem by the Runge–Kutta method of fourth order with step size η = 0.02.We follow an  iterative procedure which is stopped to give the functions aforementioned boundary   θ (η) and θ (η) when the   conditions are reached and the error in the value of θ (0) becomes less than 10−4 . In this manner, we firstly obtain numerical solutions for the case of E c = 0. Although some of these results are already known from the literature [13] and [17], we plot them in Fig. 2 with a view to analyze the effect of different parameters on temperature profiles. At this stage, it is worth indicating that Figs. 2–4 depict the functions θ (η) and θ  (η) for a set of selected heat transfer analyses. The θ (η) values are measured on the vertical left axis, whereas the θ  (η) values (< 0) are measured on the vertical right axis. The latter has been ranged from zero to −1. Further, the dimensionless variable η is measured onto the two horizontal axes from the corresponding origin as shown graphically. It is seen from Fig. 2 that, for fixed σ , the influence of r is to increase the temperature with its increases whereas the effect of σ is to decrease this temperature. We also observe that when the Prandtl number σ is high, the influence of r on temperature profiles becomes more important. For fixed σ , for a given position η and without viscous dissipation (i.e., E c = 0), when the fluid motion is faster than sheet motion (i.e., r = 0.7), the temperature θ (η) is greater than when the sheet motion is faster than fluid motion (i.e., r = 0.3). Moreover, it can be seen from Table 2 that the aforementioned trend is also met when the viscous dissipation is considered (i.e., E c  = 0). Numerical results are presented in Table 2 with and without viscous dissipation. When this effect is taken into account, an increase of the fluid temperature occurs. 3.2 Prescribed surface temperature (PST case) To seek a local similarity solution of the thermal boundary layer equation, in case of moving parallel plate problem, we contemplate to deal with quadratic power-law thermal boundary condition as follow, T = Tw (= T∞ + Ax 2 ) at y = 0; T → T∞ as y → ∞.

(21)

Using Eqs. (5), (14) and (21) in (12), we get θ  + σ f θ  − 4σ f  θ = −σ E ck ( f  )2 ,

(22)

θ (0) = 1; θ (∞) → 0.

(23)

and

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Fig. 3 Temperature and temperature gradient profiles in the PST case for two values of r and σ with s = −0.2 and E ck = 0; [r = 0.3 (solid line); r = 0.7 (dash dot line)]

Fig. 4 Temperature and temperature gradient profiles in the PST case for two values of r and σ with s = 1 and E ck = 0; [r = 0.3 (solid line); r = 0.7 (dash dot line)]  2 U x

Here, E ck = Ac P represents the appropriate form of the local Eckert number for this particular problem. A close look at Eq. (22) reveals that, for prescribed quadratic power-law surface temperature, the temperature profiles are not similar because the x-coordinate can not be eliminated from this equation. In other words, one may look for the availability of local similarity solutions. Keeping this in view, it was decided to proceed with finding local similarity solutions and the local similarity approach (Kays and Crawford [28]) is applied to solve the governing Eq. (22) with the boundary conditions (23). For a given fluid, the role played by the composite reference velocity U on temperature distributions can be analyzed. That is, by varying the local Eckert number E ck one can study those effects on temperature distributions at any given x-location far enough from the leading edge at which point the local Eckert number can not be defined. So, we do not take into account the upstream history of the flow, except as it influences the similarity variable η. The local surface heat flux can be expressed as  qw = −λT

∂T ∂y

 w


2 

= −λT Ax θ (0)

U . 2νx

(24)

If we neglect the viscous dissipation term into Eq. (12) and taking a general power-law thermal boundary condition as T = Tw (= T∞ + Ax s ) at y = 0; T → T∞ as y → ∞.

(25)

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Table 2 Values of θ and θ  for σ = 0.7 and several values of r with and without viscous dissipation (CST case) σ

Ec

r

η

θ

−θ 

0.7

0.0

0.1

0.7

0.03

0.1

0.7

0.03

0.3

0.7

0.03

0.7

0.7

0.03

0.9

0.0 0.2 0.4 0.6 1.0 2.0 5.0 10.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0 10.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0 10.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0 10.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0 10.0

1.0 0.901677 0.805643 0.713803 0.547941 0.255705 0.015312 0.000070 1.0 0.902440 0.806957 0.715470 0.549841 0.256954 0.015367 0.000040 1.0 0.903332 0.808420 0.716860 0.548760 0.242777 0.006565 0.000027 1.0 0.909677 0.820133 0.732261 0.565400 0.237509 0.001670 0.000059 1.0 0.914939 0.830096 0.745986 0.582835 0.245723 0.000951 0.000012

0.493622 0.487666 0.471082 0.446136 0.380747 0.209600 0.015963 0.000050 0.489284 0.484384 0.468840 0.444822 0.380770 0.210452 0.016051 0.000050 0.484773 0.480384 0.467404 0.447096 0.390569 0.222418 0.009768 0.0 0.452177 0.450381 0.444309 0.433624 0.398077 0.247666 0.003745 0.0 0.425217 0.425115 0.422887 0.417644 0.395190 0.263218 0.002537 0.0

we obtain the simpler equation θ  + σ f θ  − 2σ s f  θ = 0

(26)

where s is now arbitrary. Problem {(26)–(23)} was solved and the numerical solutions for two values of r, σ and s are shown in Fig. 3 (i.e., s = −0.2) and Fig. 4 (i.e., s = 1). It is seen from these figures that when s increases, the thermal boundary layer thickness decreases. On the other hand, the problem (22) and (23) has also been solved and the numerical results are shown in Table 3 for several values of r when σ = 1 and E ck = 0.03. As we can see from  this table the effect of increasing values of r is to decrease the numerical value of wall temperature gradient θ  (0) and, consequently, less heat is carried out of the sheet, resulting in an increase of the thermal boundary layer thickness and hence decreasing the heat transfer rate. Also, a selected set of numerical solutions for PST case are plotted in Figs. 5–9. We can observe from these figures that the effect of increasing values of σ is to decrease the temperature distribution, whereas an opposite behaviour can be seen for the Eckert number E ck . Also, the effect of viscous dissipation becomes more important with the increase of σ . Finally, Figs. 8–9 depict the influences of viscous dissipation on temperature profiles for several values of σ and for both CST/PST cases when the fluid motion is faster than the sheet motion (i.e., r = 0.75; U∞ = 3Uw ). As shown, when viscous dissipation’s effects are taken into account an increase in temperature occurs. These

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Table 3 Values of θ and θ  for σ = 1, E ck = 0.03 and several values of r (PST case) σ

E ck

r

η

θ

−θ 

1.0

0.03

0.1

1.0

0.03

0.3

1.0

0.03

0.7

1.0

0.03

0.9

0.0 0.2 0.4 0.6 1.0 2.0 5.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0 0.0 0.2 0.4 0.6 1.0 2.0 5.0

1.0 0.692078 0.482483 0.339113 0.171719 0.034924 0.000308 1.0 0.712633 0.506872 0.359829 0.180092 0.029847 0.000056 1.0 0.763783 0.571986 0.419925 0.213139 0.027519 0.000052 1.0 0.796921 0.617177 0.464728 0.242491 0.029681 0.000014

1.856336 1.261652 0.860671 0.590773 0.285406 0.054040 0.000496 1.688832 1.210653 0.865858 0.618279 0.314120 0.055650 0.000104 1.298444 1.066580 0.855345 0.669817 0.382747 0.063240 0.000027 1.062373 0.961883 0.832318 0.691604 0.427628 0.072222 0.000007

Fig. 5 Plot of temperature distributions for two values of E ck and r when σ = 0.71[E ck = 0.02 (solid line); E ck = 0.6 (dash dot line)]

commented effects become slightly more significant with the increase of σ . The cases in which r < 0.5 also maintain the aforementioned trends.

4 Discussions and conclusions A numerical analysis has been carried out for momentum and heat transfer of an incompressible fluid past a moving surface based on composite reference velocity U . The problem is reduced to a single third-order non-linear ordinary differential equation for the momentum transfer problem [i.e., Eq. (6)] and to a five firstorder ODEs for the momentum and heat transfer problems [i.e., Eqs. (19) for the CST case]. Solutions for heat transfer are obtained by employing a Runge–Kutta algorithm for high-order initial value problems (see [26]) along with shooting procedure and the influences of the parameters r, s, σ and E c (E ck ) on both temperature

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Fig. 6 Plot of temperature distributions for two values of E ck and r when σ = 5[E ck = 0.02 (solid line); E ck = 0.6 (dash dot line)]

Fig. 7 Temperature and temperature gradient distributions for different values of E ck when σ = 5 and r = 0.7[E ck = 0.02 (solid line); E ck = 0.6 (broken line); E ck = 1.6 (dash dot line)]

Fig. 8 Influence of viscous dissipation on temperature distributions in CST case for various values of σ when r = 0.75 (i.e., U∞ = 3Uw )[E c = 0 (solid line); E c = 0.6 (broken line)]

and temperature-gradient profiles were examined in this analysis. The velocity ratio r ranged from the case in which the fluid was stationary in the presence of a moving sheet (i.e., r = 0) to the case in which a flowing fluid passed over a non-moving surface (i.e., r = 1). From our numerical results and for 0 ≤ r ≤ 1, the following conclusions may be drawn:

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Fig. 9 Influence of viscous dissipation on temperature distributions in PST case for various values of σ when r = 0.75 (i.e., U∞ = 3Uw )[E ck = 0 (solid line); E ck = 0.6 (broken line)]

1. The direction of the wall shear depends on whether U∞ > Uw or U∞ < Uw . 2. For the case in which only one of the media moves, the magnitude of the wall shear is smaller for Uw = 0  (i.e., f  (0) = 0.469602) in comparison to the case of U∞ = 0 (i.e.,  f  (0) = 0.627547). This trend was already predicted by Sakiadis [1] theoretically. His conclusion (later corroborated by Tsou et al. [29] experimentally) dealt with an increase in the wall  [30]). Our numerical result  shear about of 34% (see Ref. 0.627547−0.469602 . for this prediction is easily seen to be 33.63% = 100 × 0.469602 3. The increase of the parameter r leads to the increase of dimensionless surface temperature. 4. For fixed σ and for both CST/PST cases, when the fluid motion is faster than the sheet motion (i.e., r > 0.5; U∞ > Uw ), the temperature θ (η) for a given position η is greater than when the sheet motion is faster than the fluid motion (i.e., r < 0.5; U∞ < Uw ). 5. When the Prandtl number σ is high, the influence of r on temperature profiles becomes more important. 6. The influence of an increase in the Prandtl number σ is to decrease the temperature distribution. 7. The effect of the inclusion of viscous dissipation is to increase the temperature distribution. This effect becomes slightly more relevant when σ becomes larger (see Figs. 8–9). These two figures reveal that, at selected values of r, σ and E c (E ck ), the dimensionless surface temperature θ (η) of the fluid CST case is larger than that of the PST case. Also, the increase in the fluid temperature due to viscous dissipation is observed to be more relevant for a higher value of E c (E ck ), as expected. Acknowledgement The author expresses sincere thanks to the reviewers for their valuable suggestions and comments on the paper.

References 1. Sakiadis, B.C.: Boundary-layer behaviour on continuous solid surfaces.. A.I.Ch.E.Jl. 7, 26–28 (1961) 2. Crane, L.J.: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645–647 (1970) 3. Gupta, P.S., Gupta, A.S.: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55, 744– 746 (1977) 4. Soundalgekar, V.M.: Stokes problem for elastic-viscous fluid. Rheol. Acta 13, 177–179 (1974) 5. Siddappa, B., Khapate, B.S.: Rivlin-Ericksen fluid flow past a stretching sheet. Rev. Roum. Sci. Tech. Mech. (Appl.) 2, 497– 505 (1976) 6. Danberg, J.E., Fansler, K.S.: A non similar moving wall boundary-layer problem. Q. Appl. Math. 34, 305–309 (1976) 7. Rao, J.H., Jeng, D.R., De Witt, K.J.: Momentum and heat transfer in a power-law fluid with arbitrary injection/suction at a moving wall. Int. J. Heat Mass Transf. 42, 2837–2847 (1999) 8. Magyari, E., Keller, B.: Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls. Eur. J. Mech. B Fluids 10, 109–122 (2000) 9. Cortell, R.: Similarity solutions for flow and heat transfer of a viscoelastic fluid over a stretching sheet. Int. J. Nonlinear Mech. 29, 155–161 (1994) 10. Cortell, R.: Numerical solutions for the flow of a fluid of grade three past an infinite porous plate. Int. J. Nonlinear Mech. 28, 623–626 (1993) 11. Siekman, J.: The laminar boundary layer along a flat plate. Z. Flugwiss. 10, 278–281 (1962) 12. Klemp, J.B., Acrivos, A.: The moving-wall boundary layer with reverse flow. J. Fluid Mech. 76, 363–381 (1976)

446

R. Cortell

13. Abdulhafez, T.A.: Skin friction and heat transfer on a continuous flat surface moving in a parallel free stream. Int. J. Heat Mass Transf. 28, 1234–1237 (1985) 14. Chappidi, P.R., Gunnerson, F.S.: Analysis of heat and momentum transport along a moving surface. Int. J. Heat Mass Transf. 32, 1383–1386 (1989) 15. Afzal, N., Baderuddin, A., Elgarvi, A.A.: Momentum and heat transport on a continuous flat surface moving in a parallel stream. Int. J. Heat Mass Transf. 36, 3399–3403 (1993) 16. Hussaini, M.Y., Lakin, W.D., Nachman, A.: On similarity solutions of a boundary-layer problem with an upstream moving wall. SIAM J. Appl. Math. 47, 699–709 (1987) 17. Lin, H.T., Haung, S.F.: Flow and heat transfer of plane surface moving in parallel and reversely to the free stream. Int. J. Heat Mass Transf. 37, 333–336 (1994) 18. Sparrow, E.M., Abraham, J.P.: Universal solutions for the streamwise variation of the temperature of a moving sheet in the presence of a moving fluid. Int. J. Heat Mass Transf. 48, 3047–3056 (2005) 19. Winter, H.H.: Viscous dissipation in shear flows of molten polymers. Adv. Heat Trans. 13, 280 (1977) 20. Hutter, K.: Theoretical Glaciology. D. Reidel, Dordrecht (1983) 21. Celata, G.P., Morini, G.L., Marconi, V., Mc Phail, S.J., Zummo, G.: Using viscous heating to determine the friction factor in micro-channels- An Experimental validation. Exp. Therm. Fluid Sci. 30, 725–731 (2006) 22. Cortell, R.: Numerical solutions of the classical Blasius flat-plate problem. Appl. Math. Comp. 170, 706–710 (2005) 23. Afzal, N.: The cooling of a low heat resistance sheet moving through a fluid. Warme Stoffubertrag 17, 217–219 (1983) 24. Afzal, N.: Heat transfer from a stretching surface. Int. J. Heat Mass Transf. 36, 1128–1131 (1993) 25. Afzal, N.: Momentum transfer on power law stretching plate with free stream pressure gradient. Int. J. Eng. Sci. 41, 1197– 1207 (2003) 26. Cortell, R.: Application of the fourth-order Runge–Kutta method for the solution of high-order general initial value problems. Comput. Struct. 49, 897–900 (1993) 27. Cortell, R.: Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dyn. Res. 37, 231–245 (2005) 28. Kays, W.M., Crawford, M.E.: Convective Heat and Mass Transfer, 2nd edn. McGrawHill, New York (1987) 29. Tsou, F., Sparrow, E., Goldstein, R.: Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf. 10, 219–235 (1967) 30. Sadeghy, K., Sharifi, M.: Local similarity solution for the flow of a “second grade” viscoelastic fluid above a moving plate. Int. J. Nonlinear Mech. 39, 1265–1273 (2004)