Radiation effects on a certain MHD free-convection flow

Chaos, Solitons and Fractals 13 (2002) 1843±1850

www.elsevier.com/locate/chaos

Radiation e€ects on a certain MHD free-convection ¯ow Ahmed Y. Ghaly Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Hiliopolis, Cairo, Egypt Accepted 15 August 2001

Abstract Free-convection heat and mass transfer due to the simultaneous action of buoyancy, radiation and transverse magnetic ®eld is investigated near an isothermal sheet. The sheet is linearly stretched in the presence of a uniform free stream of constant velocity, temperature and concentration. A parametric study is performed to illustrate the in¯uence of the radiation parameter, magnetic parameter, Prandtl number, Grashof number and Schmidt number on the pro®les of the velocity components, temperature and concentration. Numerical results show that the radiation have signi®cant in¯uences on the velocity and temperature pro®les, Nusselt number and local shear stress. The results indicate that the velocity, ¯uid's temperature and local shear stress decrease as the radiation parameter increases. The Nusslet number increases as the radiation parameter increases. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction Boundary layer ¯ow on continuous moving surface is a signi®cant type of ¯ow occurring in several engineering applications. Aerodynamic extrusion of plastic sheets, cooling of an in®nite metallic plate in a cooling bath, the boundary layer along a liquid ®lm in condensation processes and a polymer sheet or ®lament extruded continuously from a dye, or a long thread traveling between a feed roll and a wind-up roll, are examples for practical applications of continuous ¯at surfaces. As examples on stretched sheets, many metallurgical processes involve the cooling of continuous strips or ®laments by drawing them through a quiescent ¯uid and that in the process of drawing, when these strips are stretched. Sakiadis [1,2] was the ®rst to study boundary layer ¯ow over a stretched surface moving with a constant velocity. He employed a similarity transformation and obtained a numerical solution for the problem. Erickson et al. [3] extended the work of Sakiadis to account for mass transfer at the stretched surface. Chen and Char [4] investigated the e€ects of variable surface temperature and variable surface heat ¯ux on the heat transfer characteristics of a linearly stretching sheet subject to blowing or suction. Elbashbeshy [5] investigated heat transfer over a tretching surface with variable and uniform surface heat ¯ux subject to injection and suction. Vajravelu and Hadjinicalaou [6] have considered hydromagnetic convective heat transfer from a stretching surface with a uniform free stream and in the presence of internal heat generation or absorption e€ects. Chamkha [7] have considered hydromagnetic three-dimensional convective heat transfer from a stretching surface with heat generation or absorption. In all the above studies, the authors have taken the stretching sheet to be oriented in the horizontal direction. All the above investigations are restricted to MHD ¯ow and heat transfer problems. As mentioned by Vajravelu and Hadjinicalaou [6], the rate of cooling and, therefore, the desired properties of the end product can be controlled by the use of electrically conducting ¯uids and the application of magnetic ®eld. The use of magnetic ®elds has been also used in the process of puri®cation of molten metals from non-metallic inclusions. In addition, due to the signi®cance of the study of ¯ow and heat transfer caused by a stretching surface in many practical manufacturing processes such as extrusion processes, glass blowing, hot rolling, manufacturing of plastic and rubber sheets, crystal growing, continuous coating and ®bers spinning [7]. The e€ect of radiation on MHD ¯ow and heat transfer problems have become more important industrially. At high operating temperature, radiation e€ect can be quite signi®cant. Many processes in engineering areas occur at high 0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 1 9 3 - X

1844

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

temperatures and a knowledge of radiation heat transfer becomes very important for the design of the pertinent equipment. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. Takhar et al. [8] studied the radiation e€ects on MHD freeconvection ¯ow of a gas past a semi-in®nite vertical plate. Recently, the radiation e€ect on heat transfer over a stretching surface was studied by Elbashbeshy [9]. The e€ect of radiation on heat and mass transfer over a stretching sheet in the presence of a magnetic ®eld have not been studied in recent years. Hence, we propose investigating the radiation e€ect on steady MHD free-convection ¯ow near isothermal stretching sheet with mass transfer. A solution for the velocity, heat and mass transfer characteristics in the ¯ow is obtained. 2. The governing equations and transformations Here, the x-axis is taken to be along the plate in the vertically upward direction. The y-axis is taken as being normal to be plate and a strong magnetic ®eld B0 is considered along y-axis. Two equal and opposite forces are introduced along the x-axis, so that the sheet is stretched keeping the origin ®xed (see Fig. 1). The ¯uid is considered to be a gray, absorbing emitting radiation but non-scattering medium and the Rosseland approximation is used to describe the radiative heat ¯ux in the energy equation. The radiative heat ¯ux in the x-direction is considered negligible in comparison to the y-direction. Under the usual boundary layer approximation, the ¯ow, heat and mass transfer in the presence of radiation are governed by the following equations: ou ov ‡ ˆ 0; ox oy

…1†

u

ou ou o2 u ‡ v ˆ m 2 ‡ gb…T ox oy oy

u

oT oT k o2 T ‡v ˆ ox oy qcp oy 2

u

oC oC o2 C ‡v ˆD 2: ox oy oy

T1 † ‡ gb …C

C1 †

rB20 u; q

1 oqr ; qcp oy

…2† …3† …4†

where u and v are the velocity components in x and y-directions, respectively, m is the kinematic viscosity, g is the acceleration due to gravity, b is the volumetric coecient of thermal expansion, b is the volumetric coecient of expansion with concentration, r is the electrical conductivity, q is the density of the ¯uid, T and C are the temperature and concentration, respectively, k is the thermal conductivity, cp is the speci®c heat at constant pressure and qr is the radiative heat ¯ux and D is the thermal molecular di€usivity. The appropriate boundary condition of Eqs. (1)±(5) are: u ˆ cx; u ! u1 ;

v ˆ 0;

T ˆ Tw ;

T ! T1 ;

C ˆ Cw

C ! C1

at y ˆ 0;

as y ! 1:

Fig. 1. Sketch of the physical model.

…5†

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

1845

where c > 0, the temperature and the species concentration at the plate are Tw …6ˆ T1 † and Cw …6ˆ C1 †, T1 and C1 being the temperature and species concentration of the free stream. By using the Rosseland approximation, we have [10] qr ˆ

4r oT 4 ; 3k  oy

…6†

where r is the Stefan±Boltzmann constant and k  is the mean absorption coecient. By using Eq. (6), the energy Eq. (3) becomes: u

oT oT k o2 T 4r o2 T 4 ‡v ˆ ‡  : 2 ox oy qcp oy 3k qcp oy 2

…7†

Introducing the following non-dimensional parameters cx cyR ; y ˆ ; u1 u1 u vR u ˆ ; v ˆ ; u1 u1 T T1 C hˆ ; uˆ Tw T1 Cw

x ˆ

…8† C1 ; C1

one can obtain the governing equation in dimensionless form as (with dropping the bars) ou ov ‡ ˆ 0; ox oy u

…9†

ou ou o2 u ‡ v ˆ 2 ‡ Gr h ‡ Gc u ox oy oy

Mu;

 2 ! 2 oh oh 1 o2 h 4 3o h 2 oh ; ‡ ‡ 3r…1 ‡ rh† …1 ‡ rh† u ‡v ˆ ox oy Pr oy 2 3FPr oy 2 oy u

ou ou 1 o2 u ‡v ˆ ; ox oy Sc oy 2

…10† …11† …12†

with the boundary conditions u ˆ x; v ˆ 0; h ˆ 1; u ˆ 1 at y ˆ 0; u ! 1; h ! 0; u ! 0; as y ! 1;

…13†

where =qc M ˆ rB20p  is the magnetic parameter, R ˆ u1 = cm is the Reynolds number, Gr ˆ gb…Tw T1 †=cu1 is the Grashof number, Gc ˆ gb …Cw C1 †=cu1 is the modi®ed Grashof number, Pr ˆ lcp =k is the Prandtl number, Sc ˆ m=D is the Schmidt number, 3 F ˆ kk  =4r T1 is the radiation parameter, l ˆ qm is the viscosity of the ¯uid, and r ˆ …Tw T1 †=T1 is the relative di€erence between the temperature of the sheet and the temperature far away from the sheet. Introducing the stream function w de®ned in the usual way uˆ

ow ; oy

vˆ

ow ; ox

…14†

where w…x; y† ˆ f …y† ‡ xg…y†:

…15†

1846

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

In view of Eqs. (9)±(15) and by equating the coecients of x0 and x1 , we obtain the coupled nonlinear ordinary di€erential equations f 000 ˆ f 0 g0 000

g ˆg

02

gf 00 ‡ Mf 0 00

Gr h

Gc u;

…16†

0

gg ‡ Mg ;

…17† 2

…3F ‡ 4…1 ‡ rh†3 †h00 ‡ 3Pr Fgh0 ‡ 12r…1 ‡ rh†2 h0 ˆ 0; gu0 ‡

…18†

1 00 u ˆ 0: Sc

…19†

The primes above indicate di€erentiation with respect to y only. The boundary conditions (13) in view of the Eqs. (14) and (15) are reduced to f …0† ˆ f 0 …0† ˆ g…0† ˆ g0 …1† ˆ h…1† ˆ u…1† ˆ 0; g0 …0† ˆ h…0† ˆ u…0† ˆ f 0 …1† ˆ 1:

…20†

The major physical quantities of interest are the local shear stress sx , Nusselt number Nu and the Sherwood number Sh they are de®ned, respectively by: ou sx ˆ l ; oy yˆ0

Nu ˆ

k qcp …Tw

oT ; T1 † oy yˆ0

Sh ˆ

D …Cw

oC : C1 † oy yˆ0

…21†

Using Eq. (15), quantities in Eq. (21) can be expressed as: sx ˆ lcR…f 00 …0† ‡ xg00 …0††;

Nu ˆ

cRk 0 h …0†; u1 qcp

Sh ˆ

cRD 0 u …0†: u1

…22†

The e€ect of the parameters F, Gr, Gc, Pr, Sc and M, on the functions f 00 , g00 , h0 , and u0 at the plate surface is tabulated in Table 1 for r ˆ 0:05. Table 1 Variation of f 00 ,

g00 ,

h0 and

u0 at the plate surface with F, Gr, Gc, Pr, Sc and M parameters g00 …0†

h0 …0†

u0 …0†

0.733105 0.662945 0.634787

1.04881 1.04881 1.04881

0.224391 0.297383 0.335686

0.396885 0.396885 0.396885

0.1 0.1 0.1

0.448657 0.590881 0.733105

1.04881 1.04881 1.04881

0.224391 0.224391 0.224391

0.396885 0.396885 0.396885

0.6 0.6 0.6

0.1 0.1 0.1

0.635035 0.733105 0.831174

1.04881 1.04881 1.04881

0.224391 0.224391 0.224391

0.396885 0.396885 0.396885

0.73 2.00 5.00

0.6 0.6 0.6

0.1 0.1 0.1

0.733105 0.555056 0.460663

1.04881 1.04881 1.04881

0.224391 0.480313 0.882449

0.396885 0.396885 0.396885

0.2 0.2 0.2

0.73 0.73 0.73

0.6 1.0 2.0

0.1 0.1 0.1

0.733105 0.684512 0.640639

1.04881 1.04881 1.04881

0.224391 0.224391 0.224391

0.396885 0.571914 0.900764

0.2 0.2 0.2

0.73 0.73 0.73

0.6 0.6 0.6

0.1 0.5 1.0

0.733105 0.472818 0.385118

1.04881 1.22474 p 2

0.224391 0.203986 0.186146

0.396885 0.366000 0.336324

F

Gr

Gc

Pr

Sc

M

f 00 …0†

1 2 3

0.3 0.3 0.3

0.2 0.2 0.2

0.73 0.73 0.73

0.6 0.6 0.6

0.1 0.1 0.1

1 1 1

0.1 0.2 0.3

0.2 0.2 0.2

0.73 0.73 0.73

0.6 0.6 0.6

1 1 1

0.3 0.3 0.3

0.1 0.2 0.3

0.73 0.73 0.73

1 1 1

0.3 0.3 0.3

0.2 0.2 0.2

1 1 1

0.3 0.3 0.3

1 1 1

0.3 0.3 0.3

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

1847

3. Numerical results and discussion The shooting method for linear equations is based on replacing the boundary value problem by two initial value problems. The solution of the boundary value problem is a linear combination between the solutions of the two initial value problems. The numerical computations has been done by the symbolic computation software Mathematica. The command (NDSolve) is used to solve the initial value problems. The number of grid points is 1000 and the value of ymax , the edge of the boundary layer equal to 15. The exact solution of the Eqs. (17) and (20) is given by: g…y† ˆ 1a…1

e

ay

†;

where a2 ˆ M ‡ 1. The Eqs. (16), (18) and (19) reduce to linear equations with variable coecients which could be solved by the linear shooting method to obtain f, h and u. The functions f 0 , g0 , h and u are presented in Figs. 2±5 for various values of the radiation parameter, magnetic ®eld parameter, Prandtl number, Schmidt number, Grashof number and modi®ed Grashof number. The radiation have signi®cate in¯uences on velocity pro®les, temperature pro®les. It is observed here that the radiation have signi®cate in¯uences on velocity pro®les, temperature pro®les. Fig. 2 shows the variation of f 0 for several sets of values of the dimensionless parameters F, Pr, Sc, Gr, Gc, r and M. Moreover, Fig. 2 shows that f 0 decreases with increasing the radiation parameter F, Schmidt number Sc and Prandtl number Pr. It is seen, as expected, that f 0 decreases with increasing the magnetic ®eld parameter M. As M increases, the Lorentz force, which opposes the ¯ow, also increases and leads to enchanted deceleration of the ¯ow. This result qualitatively agrees with the expectations [6], since the magnetic ®eld exerts a retarding force on the free-convection ¯ow. However, f 0 increases with an increase in Grashof number Gr, modi®ed Grashof number Gc and the parameter of relative di€erence between the temperature of the sheet and the temperature far away from the sheet r. Fig. 3 describes the behavior of g0 with changes in the values of the magnetic ®eld parameter M. It is seen that g0 decreases with increasing the magnetic ®eld parameter M. The e€ects of the dimensionless parameters Pr, M, F and r on the heat transfer are shown in Fig. 4. It is observed that dimensionless temperature increases with an increase in r and M

Fig. 2. Variation of the dimensionless velocity component f 0 with F, Pr, Sc, Gr, Gc, r and M parameters.

1848

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

Fig. 3. Variation of the dimensionless velocity component g0 with M parameter.

Fig. 4. Variation of the dimensionless temperature h with Pr, M, r and F parameters.

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

1849

Fig. 5. Variation of the dimensionless concentration component u with Sc and M parameters.

parameters. It is seen that the temperature h decreases as the radiation parameter F increases. This results qualitatively agrees with the results in [9], since the e€ect of radiation is to decrease the rate of energy transport to the ¯uid, thereby decreasing the temperature of the ¯uid. It is also observed that the temperature decreases with an increase in the Prandtl number Pr. This is in agreement with the physical fact that the thermal boundary layer thickness decreases with increasing Pr. Fig. 5 describes the behavior of u with changes in the values of the magnetic ®eld parameter M and Schmidt number Sc. It is observed that u increases with an increases in M. While u decreases with an increase in Sc. Table 1 illustrates the e€ects of the parameters F, Gr, Gc, Pr, Sc and M on the local shear stress (f 00 …0† and (g00 …0†), the Nusslet number ( h0 …0†) and the Sherwood number ( u0 …0†). We observe that the e€ect of increasing F and Pr increases the Nusslet number. Also, the Nusslet number decreases with an increase in the magnetic parameter M. Furthermore, the negative values of the wall temperature gradient, for all values of the dimensionless parameters, are indicative of the physical fact that the heat ¯ow from the sheet surface to ambient ¯uid. It can be seen that an increase in M decreases Sh while an increases in Sc increases Sh . Also, it can be seen that an increase in the radiation parameter F decreases the local shear stress parameter. The same trend can be seen with the parameters M, Pr and Sc. On the other hand, an increase in Gr and Gc increases the local shear stress parameter. 4. Conclusion In this study, the radiation e€ect on steady MHD free-convection ¯ow near isothermal stretching sheet with mass transfer is analyzed. A solution for the velocity, heat and mass transfer characteristics in the ¯ow is obtained. The boundary layer equations have been solved numerically by the shooting method. The numerical results indicate that the radiation have signi®cant in¯uences on the velocity and temperature pro®les, Nusselt number and local shear stress. References [1] Sakiadis Bc. Boundary layer behavior on continuous solid surfaces: I. boundary-layey equations for two-dimensional and axisymmetric ¯ow. AIChE J 1961;7(1):26±8.

1850

A.Y. Ghaly / Chaos, Solitons and Fractals 13 (2002) 1843±1850

[2] Sakiadis Bc. Boundary layer behavior on continuous solid surfaces: I. boundary-layey on a continuous ¯at surface. AIChE J 1961;7(1):221±5. [3] Erickson LE, Fan LT, Fox VG. Heat and mass transfer on a moving continuous ¯at plate with suction or injection. Ind Eng Chem 1966;5:19±25. [4] Chen CK, Char M. Heat transfer of a continuous stretching surface with suction or blowing. J Math Anal Appl 1988;135:568±80. [5] Elbashbeshy EMA. Heat and mass transfer along a vertical plate with variable surface tension and concentration in the presence of the magnetic ®eld. Int J Eng Sci 1997;34:515±22. [6] Vajravelu K, Hadjinicalaou A. Convective heat transfer in an electrically conducting ¯uid at a stretching surface with uniform free stream. Int J Eng Sci 1997;35:1237±44. [7] Chamkha AJ. Hydromagnetic three-dimensional free convection on a vertical stretching surface with heat generation or absorption. Int J Heat Fluid ¯ow 1999;20:84±92. [8] Takhar HS, Gorla RSR, Soundalgekar VM. Non-linear one-step method for initial value problems. Int Num Meth Heat Fluid Flow 1996;6:77±83. [9] Elbashbeshy EMA. Radiation e€ect on heat transfer over a stretching surface. Can J Phys 2000;78:1107±12. [10] Raptis A. Flow of a micropolar ¯uid past a continuously moving plate by the presence of radiation. Int J Heat Mass Transfer 1998;41:2865±6.