sink and thermal radiation on heat transfer over an unsteady stretching permeable surface

Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

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Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface Dulal Pal ⇑ Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal 731235, India

a r t i c l e

i n f o

Article history: Received 15 May 2010 Received in revised form 7 August 2010 Accepted 17 August 2010 Available online 6 September 2010 Keywords: Boundary layer flow Stretching surface Heat transfer Thermal radiation Similarity transformation

a b s t r a c t The present paper is concerned with the study of flow and heat transfer characteristics in the unsteady laminar boundary layer flow of an incompressible viscous fluid over continuously stretching permeable surface in the presence of a non-uniform heat source/sink and thermal radiation. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Similarity transformations are used to convert the governing time-dependent nonlinear boundary layer equations for momentum and thermal energy are reduced to a system of nonlinear ordinary differential equations containing Prandtl number, non-uniform heat source/sink parameter, thermal radiation and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge–Kutta–Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the unsteadiness parameter, thermal radiation, suction/injection parameter, non-uniform heat source/sink parameter on flow and heat transfer characteristics as well as on the local Nusselt number are shown graphically. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The problem of flow and heat transfer induced by continuous stretching heated surfaces placed in a porous medium has received considerable attention in recent years because it is an important type of flow occurring in many engineering disciplines. A class of flow problems with obvious relevance to polymer extrusion, in which the extrudate emerges from a narrow slit. For instance, in a melt-spinning process, the extrudate from the die is generally drawn and simultaneously stretched into a filament or sheet, which is thereafter solidified through rapid quenching or gradual cooling by direct contact with water or chilled metal rolls. In fact, stretching will bring in an unidirectional orientation to the extrudate, thereby improving the quality of the final product considerably which greatly depends on the flow and heat transfer mechanism. Similar situations prevail during the manufacture of plastic and rubber sheets where it is often necessary to blow a gaseous medium through the not-yet solidified material, and where the stretching force may be varying with time. Glass blowing, continuous casting, and spinning of fibers also involve the flow due to a stretching surface. The quality of the resulting sheeting material, as well as the cost of production, may be affected by the speed of collection and the heat transfer rate. The problem of flow due to a stretching sheet has been later extended to many flow situations. Crane [1] was the first to examine the problem of steady two-dimensional boundary layer flow of an incompressible and viscous fluid caused by a stretching sheet whose velocity varies linearly with the distance from a fixed point on the sheet. McCormack and Crane [2] have provided comprehensive discussion on boundary layer flow caused by stretching of an elastic flat sheet moving in its own plane with a ⇑ Tel./fax: +91 3463 261029. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.08.023

D. Pal / Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

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Nomenclature a, b, c A*, B* cp Cf f(g) K* Nux Pr q000 qr qw Rex T Tw T1 t Uw u v x, y

empirical constants coefficients of space and temperature-dependent heat source/sink fluid heat capacity local skin friction coefficient dimensionless stream function mean absorption coefficient local Nusselt number Prandtl number, lcp/j non-uniform heat source/sink radiative heat flux local heat flux (Wm2) local Reynolds number, Uwx/m fluid temperature (K) wall temperature (K) free stream temperature (K) time (s) stretching surface velocity (m/s) x-component of fluid velocity (m/s) y-component of fluid velocity (m/s) vertical or tangential distance; normal distance (m)

Greek letters a unsteadiness parameter, c/a g similarity variable j is the thermal conductivity (Wm1K) l fluid dynamic viscosity (kg m1s1) m fluid kinematic viscosity, l/q (m2s1) q fluid density (kg m3) r* Stephen-Boltzmann constant h(g) dimensionless temperature, (T  T1)/(Tw  T1) sw wall shear stress f suction/injection parameter

velocity varying linearly with distance. Dutta et al. [3] determined the temperature distribution in the flow over a stretching surface subject to uniform heat flux. Chen and Char [4] investigated the heat transfer characteristics over a continuous stretching sheet with variable surface temperature. Ali [5] has investigated flow and heat transfer characteristics on a stretching surface using power-law velocity and temperature distributions. Grubka and Bobba [6] studied the heat transfer over a stretching surface with the non-isothermal wall that is varying as a power-law with the distance. Ishak et al. [7] analyzed mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet. Ali and Magyari [8] studied the unsteady fluid and heat flow induced by a submerged stretching surface while its steady motion is slowed down gradually. A new dimension is added to the study of flow and heat transfer in a viscous fluid over a stretching surface by considering the effect of thermal radiation. The radiative effects have important applications in physics and engineering particularly in space technology and high temperature processes. But very little is known about the effects of radiation on the boundary layer. Thermal radiation effect might play a significant role in controlling heat transfer process in polymer processing industry. The quality of the final product depends to a great extent on the heat controlling factors and the knowledge of radiative heat transfer in the system can perhaps lead to a desired product with a sought characteristic. However, there are no attempts in literature to consider the effect of thermal radiation on the flow and heat transfer in a viscous fluid over an unsteady stretching surface. Pal and Malashetty [9] have presented similarity solutions of the boundary layer equations to analyze the effects of thermal radiation on stagnation point flow over a stretching sheet with internal heat generation or absorption. The effect of radiation on heat transfer problems have been studied by Hossain and Takhar [10], Takhar et al. [11], Hossain et al. [12]. Mukhopadhyay and Layek [13] studied free convective flow and radiative heat transfer of viscous incompressible fluid having variable viscosity over a stretching porous vertical plate. Recently, Pal [14] investigated the effect of thermal radiation on heat and mass transfer in two-dimensional stagnation point flow of an incompressible viscous fluid over a stretching sheet in the presence of buoyancy force. Several authors [15–18] investigated the heat transfer problem in a stretching sheet with a linear, power-law or exponentially surface velocity and an uniform or different surface temperature condition. Abo-Eldahab and El-Aziz [19] extended the problem to involve a space-dependent exponentially decaying with internal heat generation or absorption. Heat source/sink

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effects are crucial in controlling the heat transfer. Many of the authors have studied heat transfer by considering an uniform heat source/sink i.e. a temperature-dependent heat source/sink (see [20]). Abo-Eldahab and El-Aziz [19] included the effect of non-uniform heat source/sink (i.e. space and temperature-dependent heat source/sink) on the heat transfer with suction/ blowing. Abel et al. [21] and Bataller [22] presented the effects of non-uniform heat source on viscoelastic fluid flow and heat transfer over a stretching sheet. Recently, Pal and Mondal [23] examined the effect of non-uniform heat source/sink and variable viscosity on MHD non-Darcy mixed convection heat transfer over a stretching sheet embedded in a porous medium in presence of Ohmic dissipation. The physical situation described in all the above studies is related to the process of uniform stretching sheet case. It is important to include unsteadiness into the governing equations of any problem for the development of a more physically realistic characterization of the flow configuration. Very little attention has been given to the unsteady flows over a stretching surface. Wang [24] was the first to study unsteady boundary layer flow of a finite liquid film by restricting the motion to a specified family of time dependence and reduced the boundary layer equations to nonlinear ordinary differential equations using a dimensionless unsteady parameters and a special type of similarity transformation. Later, Andersson et al. [25] studied the unsteady stretching flow in the case of power-law fluid film whereas Andersson et al. [26] extended Wang’s unsteady thin film stretching problem to the case of heat transfer. Chen [27] investigated on heat transfer in a power-law fluid film over an unsteady stretching sheet. Wang [28] and Wang and Pop [29] discussed the viscous and power-law fluids film on an unsteady stretching surface. Elbashbeshy and Bazid [30] studied similarity solution for unsteady momentum and heat transfer flow whose motion is caused solely by the linear stretching of a horizontal stretching surface. Ishak et al. [31] presented the solution of an unsteady mixed convection boundary layer flow and heat transfer due to a stretching vertical surface. Recently, Ishak et al. [32] presented the heat transfer characteristics caused by an unsteady stretching permeable surface with prescribed wall temperature. Sharidan et al. [33] presented a similarity analysis to investigate the unsteady boundary layer over a stretching sheet for distributions of the stretching velocity, surface temperature and surface heat flux. Wang [34] analyzed viscous flow due to stretching sheet with surface slip and suction. Since no attempt has been made to analyze the combined effects of thermal radiation and non-uniform heat source/sink on unsteady boundary layer flow of an incompressible viscous fluid and heat transfer over a permeable vertical surface in presence of suction/injection, therefore, this problem is examined in the present paper. The stream function is defined differently (compared to uniform stretching sheet case) in arriving at the nonlinear ordinary differential equations. The conservation equations of mass, momentum and energy were transformed into a two-point boundary value problem. These nonlinear equations along with the appropriate boundary conditions are then solved by employing a numerical shooting technique with Runge–Kutta–Fehlberg integration scheme to study the effect of unsteadiness on heat transfer in the laminar flow in a porous medium past a semi-infinite stretching sheet. The results of these studies are of great importance, for example in the prediction of skin friction as well as heat transfer rate over a stretching sheet which would find applications in technological and manufacturing industries such as polymer extrusion. Comparisons with previously published works are performed and excellent agreement between the results is obtained. 2. Mathematical formulations Consider an unsteady two-dimensional laminar boundary layer flow over a continuous moving stretching permeable surface in a quiescent incompressible viscous fluid which issues from a thin slot. The x-axis is taken along the stretching surface in the direction of the motion with the slot as the origin, and the y-axis is perpendicular (see Fig. 1) to the sheet in the outward direction towards the fluid of ambient temperature T1. The flow is assumed to be confined in a region y > 0. In order to get the effect of temperature difference between the surface and the ambient fluid, we consider the non-uniform heat

Fig. 1. Schematic diagram of the problem under consideration.

D. Pal / Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

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source/sink in the flow. We assume that the velocity is proportional to its distance from the slit. Under these assumption along with the boundary layer approximations and neglecting the viscous dissipation, the governing basic boundary layer equations for momentum and heat transfer in the presence of thermal radiation take the following form:

@u @ v þ ¼ 0; @x @y   @u @u @u @2u þu þv ¼l 2; q @t @x @y @y

ð1Þ ð2Þ

@T @T @T j @ 2 T 1 @qr q000  : þu þv ¼ þ @t @x @y qcp @y2 qcp @y qcp

ð3Þ

The associated boundary conditions to the problem are

u ¼ U w ðx; tÞ; u ! 0;

v ¼ V w;

T ! T 1;

T ¼ T w ðx; tÞ;

at y ¼ 0;

as y ! 1;

ð4Þ ð5Þ

where x and y represent coordinate axes along the continuous surface in the direction of motion and perpendicular to it, respectively. u and v are the velocity components along x and y directions, respectively and t is the time. T is the temperature inside the boundary layer, cp is the specific heat at constant pressure, j is the thermal conductivity, l is the fluid viscosity, m = l/q is the kinematics viscosity of the fluid and q is density of fluid. Tw(x, t) temperature of the stretching surface, T1 is the temperature far away from the stretching surface with Tw > T1. The term Vw = (mUw/x)1/2f(0) represents the mass transfer at the surface with Vw > 0 for injection and Vw < 0 for suction. The non-uniform heat source/sink, q000 , is modeled as

q000 ¼

jU w ðx; tÞ  ½A ðT w  T 1 Þf 0 þ ðT  T 1 ÞB ; xm

where A* and B* are the coefficient of space and temperature-dependent heat source/sink, respectively. It is to be noted that the case A* > 0, B* > 0 corresponds to internal heat generation and that A* < 0, B* < 0 corresponds to internal heat absorption. The flow is caused by the stretching of the sheet which moves in its own plane with the surface velocity Uw(x, t) = ax/ (1  ct), where a (stretching rate) and c are positive constants having dimension time 1 (with ct < 1, c P 0). It is noted that the stretching rate a/(1  ct) increases with time since a > 0. The surface temperature of the sheet varies with the distance x from the slot and time t in the form

T w ðx; tÞ ¼ T 1 þ

bx ; 1  ct

ð6Þ

where b is constant with b P 0. It should be noted that when t = 0 (initial motion), Eqs. (1)–(3) describe the case of steady flow over a stretching sheet. The particular form of Uw(x, t) and Tw(x, t) presented in this paper has been chosen in order to devise a similarity transformation (Ishak et al. [32]), which transform the governing partial differential equations (1)–(3) into a set of highly nonlinear ordinary differential equations. The mathematical analysis of the problem is simplified by introducing the following dimensionless functions f and h, and the similarity variable g (see [26,32])

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a y; mð1  ctÞ  ma 12 Wðx; y; tÞ ¼ xf ðgÞ; 1  ct bx Tðx; y; tÞ ¼ T 1 þ hðgÞ; 1  ct

g¼

ð7Þ ð8Þ T  T1 hðgÞ ¼ ; Tw  T1

ð9Þ

where W(x, y, t) is the stream function defined by

u¼

@W ¼ axð1  ctÞ1 f 0 ðgÞ; @y

ð10Þ

and

v¼

1 @W ¼ ½mað1  ctÞ1 2 f ðgÞ; @x

ð11Þ

which automatically satisfies the continuity equation (1). It must be noted that expression (7)–(9) on which the analysis is based are valid only for t < c1 (since ct < 1). By using Rosseland approximation for radiation, the radiative heat flux qr is given by

qr ¼ 

4r @T 4 ; 3K  @y

ð12Þ

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where r* and K* are respectively the Stephan–Boltzman constant and the mean absorption coefficient. We assume that the temperature differences within the flow are such that the term T4 may be expressed as a linear function of temperature. This is accomplished by expanding T4 in a Taylor series about a free stream temperature T1 as follows:

T 4 ¼ T 41 þ 4T 31 ðT  T 1 Þ þ 6T 21 ðT  T 1 Þ2 þ   

ð13Þ

Neglecting higher-order terms [35,36] in the above Eq. (13) beyond the first degree in (T  T1), we get

T 4 ffi 4T 31 T  3T 41 :

ð14Þ

Thus employing Eq. (14) in Eq. (12), we get

16T 31 r @T : 3K  @y

qr ¼ 

ð15Þ

Substituting Eqs. (7)–(11) into (2)–(5), we obtain

  1 2 00 f 000 þ ff  f 0  a f 0 þ gf 00 ¼ 0; 2

ð16Þ

  1 ð1 þ NrÞPr1 h00  f 0 h þ f h0  a h þ gh0 þ Pr1 ðA f 0 þ B hÞ ¼ 0; 2

ð17Þ

 3

lc

Pr ¼ jp is the Prandtl number, Nr ¼ 163Kr Tj1 is the thermal radiation parameter, a ¼ ac is a parameter that measures the unsteadiness. The boundary conditions (4) and (5) take the form

f ð0Þ ¼ f0 ; 0

f ð1Þ ¼ 0;

f 0 ð0Þ ¼ 1;

hð0Þ ¼ 1;

ð18Þ

hð1Þ ¼ 0;

ð19Þ

where f(0) = f0 with f0 < 0 and f0 > 0 corresponding to injection and suction, respectively. In Eqs. (16)–(18), prime denotes derivative with respect to g. It is worth mentioning that the Eq. (16) reduces to Ishak et al. [32] as Nr, A*, B* ? 0. The physical quantity of interest in this problem are the skin friction coefficient Cf and the local Nusselt number, Nux, which are defined as

Cf ¼

sw xqw ; ; Nux ¼ jðT w  T 1 Þ qU 2w =2

ð20Þ

where the wall shear stress sw and the surface heat flux qw are given by [37]



sw ¼ l

 @u ; @y y¼0

!   @T 4r @T 4 qw ¼ j   @y y¼0 3K @y

y¼0

"

¼

jþ

16r T 31 3K 

!#  @T @y y¼0

ð21Þ

with l and j being dynamic viscosity and thermal conductivity, respectively. Using Eq. (21), quantity (20) can be expressed as 1

C f Re2x ¼ f 00 ð0Þ;

Nux

.pffiffiffiffiffiffiffi Rex ¼ ð1 þ NrÞh0 ð0Þ;

ð22Þ

where Rex is the local Reynolds number based on the surface velocity. It is to be noted that the present problem reduces to steady-state flow for a = 0 in absence of thermal radiation and nonuniform heat source/sink (i.e. Nr = A* = B* = 0), then the closed-form solutions for flow and thermal fields in terms of Kummer’s functions are respectively given by [32]

1 f ðgÞ ¼ f  efg ; f   M Pr  1; Pr þ 1; Prefg =f2 hðgÞ ¼   ; M Pr þ 1; Pr  1; Pr=f2

ð23Þ ð24Þ

where M(a, b, z) denotes the confluent hypergeometric function (see [38]) as follows:

Mða; b; zÞ ¼ 1 þ

1 X an zn ; b n! n¼1 n

ð25Þ

where

an ¼ aða þ 1Þða þ 2Þ . . . ða þ n  1Þ;

bn ¼ bðb þ 1Þðb þ 2Þ . . . ðb þ n  1Þ:

ð26Þ

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Using (18) we have from Eq. (23) as f ð0Þ ¼ f0 ¼ f  1f with f > 0 (0 < f < 1) and f > 1 correspond to injection and suction, respectively. Using Eqs. (23) and (24), the skin friction coefficient f00 (0) and local Nusselt h0 (0) are given by

f 00 ð0Þ ¼ f;

ð27Þ

  M Pr  1; Pr þ 1; Pr=f2 h0 ð0Þ ¼ fPr  : M Pr  1; Pr þ 1; Pr=f2

ð28Þ

3. The method of solution The coupled ordinary differential equations (16) and (17) is of third-order in f and second-order in h which have been reduced to a system of five simultaneous equations of first-order for five unknowns. In order to solve this system of equaTable 1 Comparison of results of skin friction coefficient for A* = 0.0, B* = 0.0, f = 1.0 with previously published data. f00 (0)

a

Sharidan et al. [33]

Present result

0.8 1.2 2.0

1.261042 1.377722 1.587362

1.261043 1.377724 1.587366

Table 2 Comparison of results of the wall temperature gradient with Grubka and Bobba [6], Ishak et al. [32] and exact solution [32] for A* = 0.0, B* = 0.0, Nr = 0.0. h0 (0)

a

f

Pr

0

0.5

0.72 1.0 10.0 0.01 0.72 1.0 3.0 10.0 100.0 0.72 1.0 10.0 1.0

1.0

2.0

1.0

0.5 1.0 2.0

Grubka and Bobba [6]

Ishak et al. [32] 0.4570 0.5000 0.6452 0.0197 0.8086 1.0000 1.9237 3.7207

0.0197 0.8086 1.0000 1.9237 3.7207 12.2940

– 1.4944 2.0000 16.0842 0.8095 1.3205 2.2224

Exact solution [32]

Present results

0.457026833 0.500000000 0.645161289 0.019706354 0.808631350 1.000000000 1.923682594 3.720673901 12.29408326 1.494368413 2.000000000 16.08421885

0.457026833 0.500000000 0.645161290 0.019706795 0.808631352 1.000000000 1.923682561 3.720673903 12.29408344 1.494368414 2.000000000 16.08421882 0.809511470 1.320522071 2.222355356

Table 3 Grid convergence test for h0 (0) when A* = 0.0, B* = 0.0, Nr = 0.0, a = 0.0, f = 0.5. Exact solution [32] (1)

No. of grids

Dg

Present results (2)

% Absolute error = ð1Þð2Þ ð1Þ  100

0.72

0.457026833

101 201 401 801 1601

0.2 0.1 0.05 0.025 0.0125

0.457034418 0.457034418 0.457034418 0.457034418 0.457034418

0.0016596 0.0016596 0.0016596 0.0016596 0.0016596

1.0

0.500000000

101 201 401 801 1601

0.2 0.1 0.05 0.025 0.0125

0.500000083 0.500000083 0.500000083 0.500000083 0.500000083

0.0000166 0.0000166 0.0000166 0.0000166 0.0000166

10.0

0.645161289

101 201 401 801 1601

0.2 0.1 0.05 0.025 0.0125

0.645161285 0.645161285 0.645161285 0.645161285 0.645161285

0.0000006 0.0000006 0.0000006 0.0000006 0.0000006

Pr

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D. Pal / Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

tions numerically we require five initial conditions but two initial conditions on f and one initial condition on h are known. However, the values of f0 and h are known at g ? 1. Thus, these two end conditions are utilized to produce two unknown initial conditions at g = 0 by using shooting technique. The most crucial factor of this scheme is to choose the appropriate finite value of g1. Thus to estimate the value of g1, we start with some initial guess value and solve the boundary value problem consisting of Eqs. (16) and (17) to obtain f00 (0) and h0 (0). The solution process is repeated with another large value of g1 until two successive values of f00 (0) and h0 (0) differ only after desired significant digit. The last value of g1 is taken as the finite value of the limit g ? 1 for a particular set of physical parameters for determining velocity f(g) and temperature h(g) in the boundary layer. After knowing all the five initial conditions we solve this system of simultaneous equations using fifth-order Runge–Kutta–Fehlberg integration scheme. The value of g1 was selected to vary from 3 to 25 depending upon the physical parameters such as Prandtl number, non-uniform heat source/sink parameter, thermal radiation parameter and unsteadiness parameter so that no numerical oscillations would occur. Thus, the coupled nonlinear boundary value problem of third-order in f and second-order in h has been reduced to a system of five simultaneous equations of first-order for five unknowns as follows:

1 0.9

Pr=1.0, ζ=0.5, Nr=1.0 A*= −0.05, B*= −0.05

0.8 0.7

f’(η)

0.6 0.5

α=3, 2, 1, 0

0.4 0.3 0.2 0.1 0

0

2

4

6

η

8

10

12

14

Fig. 2. Velocity profiles for different values of g for different values of a.

1 0.9

Pr=1.0, α=0.5, Nr=1.0 * * A = −0.05, B = −0.05

0.8 0.7

f’(η)

0.6 0.5 0.4

ζ=0.5, 1.0, 1.5, 2

0.3 0.2 0.1 0

0

2

4

6

8 η

10

12

14

Fig. 3. Velocity profiles for different values of g for different values of f.

16

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D. Pal / Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

f10 ¼ f2 ; f20 ¼ f3 ;

  1 f30 ¼ f1 f3 þ f22 þ a f2 þ gf3 ; 2

ð29Þ

f40 ¼ f5 ; f50 ¼ 

 

 Pr 1 ðf2 A þ f4 B Þ f1 f5  f2 f4  a f4 þ gf5 þ ; ð1 þ NrÞ 2 Pr

where f1 = f, f2 = f0 , f3 = f00 , f4 = h, f5 = h0 and a prime denotes differentiation with respect to g. The boundary conditions now become

f2 ¼ 0;

f4 ¼ 0

f 3 ¼ b;

f5 ¼ c

f 4 ¼ 1;

at g ¼ 0;

ð30Þ

as g ! 1:

ð31Þ

1 0.9

Pr=1.0, ζ=0.5, Nr=1.0 A*= −0.05, B*= −0.05

0.8 0.7 0.6 θ(η)

f 2 ¼ 0;

0.5

α=0, 1, 2, 3

0.4 0.3 0.2 0.1 0

0

2

4

6

8

η

10

12

14

16

18

Fig. 4. Temperature profiles vs. g for various values of a.

1 0.9

Pr=1.0, α=0.5, Nr=1.0, * * A =−0.05, B =−0.05

0.8

−−−−Ishak et al. results when Pr=1.0, ζ=0.5, Nr=0.0, A*=0.0, B*=0.0

0.7 0.6 θ(η)

f1 ¼ 0;

0.5 0.4

ζ=0.5, 1.0, 1.5, 2

0.3 0.2 0.1 0

0

2

4

6

8 η

10

12

Fig. 5. Temperature profiles vs. g for various values of f.

14

16

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D. Pal / Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

Since f3(0) and f5(0) are not prescribed so we have to start with the initial approximations as f3(0) = b0 and f5(0) = c0. Let b and c be the correct values of f3(0) and f5(0), respectively. The resultant system of five ordinary differential equations is integrated using fifth-order Runge–Kutta–Fehlberg method and denote the values of f3 and f5 at g = g1 by f3(b0, c0, g1) and f5(b0, c0, g1), respectively. Since f3 and f5 at g = g1 are clearly function of b and c, they are expanded in Taylor series around b  b0 and c  c0, respectively by retaining only the linear terms. The use of difference quotients is made for the derivatives appeared in these Taylor series expansions. Thus, after solving the system of Taylor series expansions for db = b  b0 and dc = c  c0, we obtain the new estimates b1 = b0 + db0 and c1 = c0 + dc0. Next the entire process is repeated starting with f1(0), f2(0), b1, f4(0) and c1 as initial conditions. Iteration of the whole outlined process is repeated with the latest estimates of b and c until prescribed boundary conditions are satisfied. Finally, bn = bn1 + dbn1 and cn = cn1 + dcn1 for n = 1, 2, 3, . . . are obtained which seemed to be the most desired approximate initial values of f3(0) and f5(0). In this way all the five initial conditions are determined. Now it is possible to solve the resultant system of five simultaneous equations by fifth-order Runge–Kutta–Fehlberg integration scheme so that velocity and temperature fields for a particular set of physical parameters can easily be obtained. The results are provided in several tables and graphs.

1 0.9

ζ=0.5, Nr=0.0, A*=0.0, B*=0.0

0.8

I − II − III − IV −

0.7

θ(η)

0.6 0.5

α=0, Pr=0.72 α=0, Pr=1.0 α=0, Pr=10.0 α=1.0, Pr=0.72

I

0.4

II III

0.3

IV

0.2 0.1 0

0

2

4

6

8

η

10

12

14

16

18

Fig. 6. Temperature profiles vs. g for various values of a and Pr.

1 0.9

ζ=2.0, α=1.0, Nr=1.0 A*= −0.5, B*= −0.5

0.8 0.7

θ(η)

0.6 0.5 0.4

Pr=0.72, 1, 2, 3, 5, 10

0.3 0.2 0.1 0

0

1

2

3 η

4

5

Fig. 7. Temperature profiles vs. g for various values of Pr.

6

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4. Discussion of the results Boundary layer problem for momentum and heat transfer in flow of Newtonian fluid over a stretching sheet in a porous medium were solved numerically using Runge–Kutta–Fehlberg method with shooting technique. An iteration process is employed and continued until the desired results are obtained within the following convergence criterion

fiþ1  fi 6 106 ; f iþ1

ð32Þ

where f stands for u, h and i refers to iteration level. In order to check the accuracy of the numerical solution procedure used, a comparison of skin friction coefficient is made with that of Sharidan et al. [33] for a = A* = B* = 0.0 and f = 1.0 and results are tabulated in Table 1. The comparison of present results for wall temperature gradient h0 (0) for a = 0.0, 1.0 and for various values of Pr and f with those of Grubka and Bobba [6], and Ishak et al. [32], in the absence of non-uniform heat source/ sink (i.e. A* = B* = 0.0) is shown in Table 2. From these Tables 1 and 2, it is noted that the comparison in the above cases is found to be in excellent agreement and thus verifies the accuracy of the numerical method used in the present work. The

1 0.9

ζ=2.0, α=1.0, Nr=1.0 Pr=1.0, B*= −0.5

0.8 0.7

θ(η)

0.6 0.5

A*= −4.0, −2.0, 0.0, 2.0, 4.0

0.4 0.3 0.2 0.1 0

0

1

2

3 η

4

5

6

Fig. 8. Variation of temperature profiles with g for various values of A*.

1 0.9

ζ=2.0, α=1.0, Nr=1.0 Pr=1.0, A*=−0.5

0.8 0.7

θ(η)

0.6 0.5 0.4 0.3

*

B = −4.0, −2.0, 0.0, 0.5, 0.6

0.2 0.1 0

0

5

10 η

15

Fig. 9. Temperature distribution vs. g for various values of B*.

20

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D. Pal / Commun Nonlinear Sci Numer Simulat 16 (2011) 1890–1904

slight deviation in the values may be due to the use of Runge–Kutta–Fehlberg method which has fifth-order accuracy, thus the present results are more accurate. Further, present results coincides very well with the exact solutions [32] as seen from Table 2. Hence this confirms that the numerical method adopted in the present study gives very accurate results. Further, the impact of some important physical parameters on skin friction coefficient f00 (0) and wall temperature gradient h0 (0) may be analyzed from Tables 1 and 2, respectively. It is to noted that the effect of increasing the unsteadiness parameter a is to decrease the skin friction coefficient f00 (0) whereas the wall temperature gradient increases with unsteadiness parameter. It is also observed that the effect of suction/injection parameter f is to increase the wall temperature gradient for both suction (f > 1) and injection (f < 1) cases. Grid convergence test for local Nusselt number is presented in Table 3. It is clearly seen that there is absolute convergence in the value of h0 (0) as there is increase in the value of number of grids from 101 to 1601 i.e. the grid size decreases from 0.2 to 0.0125. Since the change in h0 (0) is only 0.0016596 for Pr = 0.72 which decreases as the value of Pr is increased, so the value of grid size is fixed to 0.1 (i.e. No. of grids = 201) for the entire work of the present paper.

1 0.9

ζ=2.0, α=1.0, Pr=1.0 * * A = −0.5, B = −0.5

0.8 0.7

θ(η)

0.6 0.5 0.4

Nr=0.5, 1.0, 2.0, 3.0, 5.0, 10.0

0.3 0.2 0.1 0

0

5

10 η

15

20

Fig. 10. Temperature distribution vs. g for various values of Nr.

Fig. 11. Variation of the local Nusselt number h0 (0) with Pr for various values of a.

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Fig. 2 is a plot of velocity distribution with g for various values of unsteadiness parameter a. It is clearly observed from this figure that the velocity profile decreases with increase in the unsteadiness parameter a. This is due to the fact that the momentum boundary layer thickness decreases as a increases. Since the fluid flow is merely due to stretching of the sheet so velocity decreases with increase in g till it satisfies the boundary condition at g = 1. The velocity profiles presented in Fig. 3 illustrates the variation of velocity profiles with g for various values of suction/injection parameter f. It is seen from this figure that the velocity decreases with increase in the value of the suction parameter (f > 1) whereas reverse trend is seen for the injection parameter (f < 1). When f = 1 it indicates the case of no suction or injection at the surface. The decrease in the value of velocity due to increase in the value of f > 1 indicates that Vw < 0, which represents the mass transfer at the surface due to suction whereas reverse trend in velocity profiles is seen when f < 1 (i.e. Vw > 0) which indicates the mass transfer at the surface is due to injection. Fig. 4 presents the behaviour of the temperature profiles for various values of unsteadiness parameter a. It can be seen that temperature h(g) decreases with increase in the value of unsteadiness parameter a. This is due to the fact that the fluid flow is caused by stretching of the sheet and the stretching sheet temperature is greater than

Fig. 12. Variation of the local Nusselt number h0 (0) with A* for various values of a.

Fig. 13. Variation of the local Nusselt number h0 (0) with B* for various values of a.

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the free stream temperature (i.e. Tw > T1), hence the temperature decreases with increasing g and the boundary layer thickness decreases with increase in a. Fig. 5 depicts the variation in temperature profiles in the presence of non-unform heat source/sink (present case) and in absence of non-uniform heat source/sink [32] for various values of suction/injection parameter in the thermal boundary layer. The imposition of the wall suction (f > 1) have the tendency to reduce the thermal boundary layer thickness which results in reduction in temperature profiles. However, the opposite effect is produced by imposition of wall fluid injection or blowing (f < 1). Further, it is observed that the value of the temperature is much higher in the case of non-uniform heat source/sink than in absence of it. Figs. 6 and 7 show the effect of Prandtl number on the temperature profiles in the boundary layer. The temperature is increased as Pr decreases which is seen from these figures whereas temperature decreases with increase in a (see Fig. 6). This is due to the fact that thermal boundary layer decreases with increase in both a and Pr. Temperature profiles for the selected values of space-dependent and temperature-dependent parameters for heat source/sink are depicted in Fig. 8. It is observed from this figure that the temperature in the thermal boundary layer increases with increase in A* for a given value of B*. The heat absorption sink (A* < 0, B* < 0) leads to decrease in the thermal boundary layer whereas the boundary layer thickness increases with increase in A* > 0. Fig. 9 is aimed to shed light on the effect of B* on the temperature distribution in the boundary layer for a fixed value of A*. It is noted that the temperature decreases with heat absorption sink (A* < 0, B* < 0) parameter which is due to the fact that the thermal boundary layer thickness decreases with increase in the heat absorption sink (B* < 0) parameter. Further, it is observed that the boundary layer thickness increases as B* > 0 increases which results in higher value of temperature in the thermal boundary layer. Fig. 10 is plotted to demonstrate the influence of thermal radiation parameter Nr for fixed values of non-uniform heat source/sink and other physical parameters. It is noted that the temperature in the boundary layer increases with increase in the value of the thermal radiation parameter. This is due to the fact that, the divergence of the radiative heat flux @qr/@y increases as the Rosseland radiative absorptivity K* decreases (see expression for Nr) which in turn increases the rate of radiative heat transfer to the fluid which causes the fluid temperature to increase. In view of this fact, the effect of radiation becomes more significant as Nr ? 1 and the radiation effect can be neglected when Nr = 0. Fig. 11 is the graph for the local Nusselt number, in terms of h0 (0), with Pr for A* = 2.0, B* = 0.5 and for various values of the unsteadiness parameter a. It is seen from this figure that the local Nusselt number increases with increase in the Prandtl number, due to the fact that Pr decreases the thermal boundary layer thickness as seen from Fig. 7, which in turn is the cause for higher wall temperature. Further, it is observed that as unsteadiness parameter a increases, there is also increase in the local Nusselt number for any particular value of Pr. Fig. 12 illustrates the variation of the local Nusselt number with A* for fixed values of B* = 0.5, Pr = 1.0 and various values of unsteadiness parameter a. It is evident from this figure that for given values of Pr and B*, the local Nusselt number is decreased with increase in A*. Also, Fig. 12 reveals that for a given values of A*, the local Nusselt number increases with increasing the value of unsteadiness parameter a. The variation of the local Nusselt number with B* for a fixed value of A* = 0.5 and for various values of a is displayed in Fig. 13 which reveals that the increase in the value of B* results in lowering the value of local Nusselt number, whereas reverse trend is seen on local Nusselt number by increasing the values of the unsteadiness parameter for any fixed value of B*. The local Nusselt number in terms of h0 (0) as a function of unsteadiness parameter a for two values of Pr = 0.7 and 7.0, and for various values of thermal radiation parameter Nr are depicted in Fig. 14. It is clearly seen from this figure that the local Nusselt number decreases with

Fig. 14. Variation of the local Nusselt number h0 (0) with a for various values of Pr and Nr.

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Nr for all values of Nr and a. Also, it is noted that higher local Nusselt number results in for Pr = 7.0, which is due to the fact that boundary layer thickness decreases as the Prandtl number increases. Further, this figure reveals that the local Nusselt number increases with increase in the value of unsteadiness parameter a. 5. Conclusion Numerical solutions have been obtained to study the flow and heat transfer in a laminar flow of an incompressible Newtonian fluid past an unsteady stretching sheet. Thermal radiation and non-uniform heat source/sink terms have been included in the energy equation. The effects of various physical parameters such as a, Nr, A*, B* and Pr on the heat transfer characteristics were examined. Numerical computations show that the present values of skin friction coefficient and local Nusselt number are in close agreement with those obtained by previous investigators in the absence of suction/injection, non-uniform heat generation/absorption and unsteadiness parameter. In the light of the present investigation, following conclusions may be drawn: (i) The velocity and temperature decrease with increase in the value of unsteadiness parameter which is consistent with the fact that the momentum and thermal boundary layer thickness decreases with increase in the unsteadiness parameter. (ii) The velocity decreases with increase in f in the momentum boundary layer. (iii) The temperature decreases with increase in the suction/injection parameter f. (iv) The wall temperature gradient increases with increasing unsteadiness parameter and Prandtl number. (v) The temperature decreases with increasing Prandtl number, while it increases with increasing in the value of A*, B* and Nr in the thermal boundary layer. (vi) The local Nusselt number increases with increase in the value of unsteadiness parameter a for fixed values of Pr, A* and B* while it decreases with Nr for fixed values of unsteadiness parameter a.

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