Effect of the chemical reaction and radiation

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Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1056–1066 www.elsevier.com/locate/cnsns

Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction F.S. Ibrahim

a,*

, A.M. Elaiw b, A.A. Bakr

b

a b

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

Received 24 May 2006; received in revised form 12 July 2006; accepted 12 September 2006 Available online 7 November 2006

Abstract Analytical solutions for heat and mass transfer by laminar flow of a Newtonian, viscous, electrically conducting and heat generation/absorbing fluid on a continuously vertical permeable surface in the presence of a radiation, a first-order homogeneous chemical reaction and the mass flux are reported. The plate is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. Graphical results for velocity, temperature and concentration profiles of both phases based on the analytical solutions are presented and discussed. Ó 2006 Elsevier B.V. All rights reserved. PACS: 44.05.+e; 44.20.+b; 44.25.+f; 44.30.+v; 44.40.+a; 47.15.x; 47.15.Cb; 47.15.Fe; 47.70.n Keywords: Chemical reaction; Free convection; Heat and mass transfer; Heat source; Mass diffusion; Radiation absorption

1. Introduction Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. In processes such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneously. Possible applications of this type of flow can be found in many industries. For example, in the power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid. *

Corresponding author. Tel.: +20 88 2363015; fax: +20 88 2342708. E-mail addresses: fi[email protected] (F.S. Ibrahim), [email protected] (A.A. Bakr).

1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.09.007

F.S. Ibrahim et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1056–1066

1057

We are particularly interested in cases in which diffusion and chemical reaction occur at roughly the same speed. When diffusion is much faster than chemical reaction, then only chemical factors influence the chemical reaction rate; when diffusion is not much faster than reaction, the diffusion and kinetics interact to produce very different effects. The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reaction. Due to the fast growth of electronic technology, effective cooling of electronic equipment has become warranted and cooling of electronic equipment ranges from individual transistors to main frame computers and from energy suppliers to telephone switch boards and thermal diffusion effect has been utilized for isotopes separation in the mixture between gases with very light molecular weight (hydrogen and helium) and medium molecular weight. The effect of radiation on MHD flow and heat transfer problem have become more important industrially. At high operating temperature, radiation effect can be quite significant. Many processes in engineering areas occur at high temperature 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. Bestman [1] examined the natural convection boundary layer with suction and mass transfer in a porous medium. His results confirmed the hypothesis that suction stabilises the boundary layer and affords the most efficient method in boundary layer control yet known. Abdus Sattar and Hamid Kalim [2] investigated the unsteady free convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate. Makinde [3] examined the transient free convection interaction with thermal radiation of an absorbing-emitting fluid along moving vertical permeable plate. Recently, Ibrihem et al. [4] have studied nonclassical thermal effects in Stokes’ second problem for micropolar fluids by used perturbation method. Muthucumaraswamy and Ganesan [5] studied effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. Deka et al. [6] studied the effect of the firstorder homogeneous chemical reaction on the process of an unsteady flow past an infinite vertical plate with a constant heat and mass transfer. Chamkha [7] studied the MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Soundalgekar and Patti [8] studied the problem of the flow past an impulsively started isothermal infinite vertical plate with mass transfer effects. The effect of foreign mass on the free-convection flow past a semi-infinite vertical plate were studied [9]. Chamkha [10] assumed that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. Raptis [11] investigate the steady flow of a viscous fluid through a very porous medium bounded by a porous plate subjected to a constant suction velocity by the presence of thermal radiation. Raptis and Perdikis [12] studied the unsteady free convection flow of water near 4 C in the laminar boundary layer over a vertical moving porous plate. The spite of all these studies, the unsteady MHD free convection heat and mass transfer for a heat generating fluid with radiation absorption has received little attention. Hence, the main objective of the present investigation is to study the effects of radiation absorption, mass diffusion, chemical reaction and heat source parameter of heat generating fluid past a vertical porous plate subjected to variable suction. It is assumed that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. It is also assumed that temperature over which are superimposed an exponentially varying with time. 2. Mathematical analysis Consider unsteady two-dimensional flow of a laminar, viscous, electrically conducting and heat-absorbing fluid past a semi-infinite vertical permeable moving plate embedded in a uniform porous medium and subjected to a uniform transverse magnetic field in the presence of thermal and concentration buoyancy effects. It is assumed that there is no applied voltage which implies the absence of an electrical field. The fluid properties are assumed to be constant except that the influence of density variation with temperature has been considered only in the body-force term. The concentration of diffusing species is very small in comparison to other chemical species, the concentration of species far from the wall, C1, is infinitesimally small [5] and hence the

1058


Soret and Dufour effects are neglected. The chemical reactions are taking place in the flow and all thermophysical properties are assumed to be constant of the linear momentum equation which is approximated according to the Boussinesq approximation. Due to the semi-infinite plane surface assumption, the flow variables are functions of y* and the time t* only. Under these assumptions, the equations that describe the physical situation are given by ov ¼ 0; oy

ð1Þ

ou o 2 u rB2o m ou þ u þ gbT ðT T 1 Þ þ gbc ðC C 1 Þ; þv ¼ m 2 K ot oy oy q

ð2Þ

oT k o2 T Q 0 oT þ v ¼ ðT T 1 Þ þ Ql ðC C 1 Þ; ot oy qcp oy 2 qcp

ð3Þ

oC o2 C oC þ v ¼ D K l ðC C 1 Þ; ot oy oy 2

ð4Þ

where x*, y*, and t* are the dimensional distances along and perpendicular to the plate and dimensional time, respectively. u* and v* are the components of dimensional velocities along x* and y* directions, respectively, T* is the dimensional temperature, C* is the dimensional concentration, Cw and Tw are the concentration and temperature at the wall, respectively. C1 and T1 are the free stream dimensional concentration and temperature, respectively. q is the fluid density, m is the kinematic viscosity, cp is the specific heat at constant pressure, r is the fluid electrical conductivity, Bo is the magnetic induction, K* is the permeability of the porous medium, Q0 is the dimensional heat absorption coefficient, Ql is the coefficient of proportionality for the absorption of radiation, D is the mass diffusivity, g is the gravitational acceleration, and bT and bc are the thermal and concentration expansion coefficients, respectively and Kl is the chemical reaction parameter. The magnetic and viscous dissipations are neglected in this study. The third and fourth terms on the RHS of the momentum equation (2) denote the thermal and concentration buoyancy effects, respectively. Also, the second and third terms on the RHS of the energy equation (3) represents the heat and radiation absorption effects, respectively. It is assumed that the permeable plate moves with a variable velocity in the direction of fluid flow. In addition, it is assumed that the temperature and the concentration at the wall as well as the suction velocity are exponentially varying with time. Under these assumptions, the appropriate boundary conditions for the velocity, temperature and concentration fields are u ¼ up ; u ¼ 0;

T ¼ T w þ eðT w T 1 Þen t ; C ! C1;

T ! T 1;

C ¼ C w þ eðC w C 1 Þen t ;

y ¼ 0;

y ! 1;

ð5Þ

where up is the wall dimensional velocity, n* is constant. It is clear from Eq. (1) that the suction velocity at the plate surface is a function of time only. Assuming that it takes the following exponential form:

v ¼ V o ð1 þ eAen t Þ;

ð6Þ

where A is a real positive constant, e and eA are small less than unity, and Vo is a scale of suction velocity which has non-zero positive constant. On the introducing the dimensionless quantities: u¼

u ; Vo

v¼

v ; Vo

y¼

V oy ; m

t¼

V 2o t ; m

up ¼

up ; Vo

n¼

n m ; V 2o

h¼

T T1 ; Tw T1

C¼

C C1 Cw C1

ð7Þ

In view of the above non-dimensional variables, the basic field Eqs. (2)–(4) can be expressed in non dimensional form as


1059

ou ou o2 u 1 ð1 þ eAent Þ ¼ 2 ðM þ Þu þ Grh þ GmC; ot oy oy K

ð8Þ

oh oh 1 o2 h ð1 þ eAent Þ ¼ /h þ Ql C; ot oy Pr oy 2

ð9Þ

oC oC 1 o2 C ð1 þ eAent Þ ¼ cC: ot oy Sc oy 2

ð10Þ

The boundary conditions are u ¼ up ; u ! 0;

h ¼ 1 þ eent ; C ¼ 1 þ eent ; on y ¼ 0; h ! 0; C ! 0; as y ! 1;

ð11Þ

where Gr is the Grashof number, Gm is the solutal Grashof number, Pr is the Prandtl number, M is the magnetic field parameter, K is the permability parameter, c is the Chemical reaction parameter, Sc is the Schmidt number, / is the heat source parameter and Ql is the absorption of radiation parameter mgbðT w T 1 Þ mgbc ðC w C 1 Þ ; Gm ¼ ; V 3o V 3o m Qm mQ ðC w C 1 Þ Sc ¼ ; / ¼ 0 2 ; Ql ¼ l : D qcp V o ðT w T 1 ÞV 2o

Gr ¼

Pr ¼

lcp ; k

M¼

rB2o m ; qV 2o

K¼

K V 2o ; m2

c¼

K lm ; V 2o

The mathematical statement of the problem is now complete and embodies the solution of Eqs. (8)–(10) subject to boundary conditions (11). 3. Method of solution Eqs. (8)–(10) represent a set of partial differential equations that can not be solved in enclosed form. However, it can reduced to a set of ordinary differential equations in dimensionless form that can be solved analytically. this can be done by representing the velocity, temperature and the concentration as uðy; tÞ ¼ uo ðyÞ þ eent u1 ðyÞ þ Oðe2 Þ; hðy; tÞ ¼ ho ðyÞ þ eent h1 ðyÞ þ Oðe2 Þ;

ð12Þ

2

nt

Cðy; tÞ ¼ C o ðyÞ þ ee C 1 ðyÞ þ Oðe Þ: Substituting Eqs. 12 into Eqs. (8)–(10), equating the harmonic and non-harmonic terms, and neglecting the higher order of O(e2) , and simplifying we obtains the following pairs of equations uo, ho, Co and u1, h1, C1. 1 ð13Þ u00o þ u0o M þ uo ¼ Grho GmC o ; K h00o þ Prh0o Pr/ho ¼ Ql PrC o ;

ð14Þ

C 00o þ ScC 0o SccC o ¼ 0

ð15Þ

subject to the boundary conditions uo ¼ up ; uo ! 0;

h0 ¼ 1; h0 ! 0;

C o ¼ 1; Co ! 0

on y ¼ 0; as y ! 1

for O (1) equations, and 1 u001 þ u01 M þ þ n u1 ¼ Au0o Grh1 GmC 1 ; K

ð16Þ

ð17Þ

h001 þ Prh01 Prð/ þ nÞh1 ¼ PrAh0o Ql PrC 1 ;

ð18Þ

C 001

ð19Þ

þ

ScC 01

SCðc þ nÞC 1 ¼

AScC 0o

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with the boundary conditions u1 ¼ 0;

h1 ¼ 1;

C1 ¼ 1

on y ¼ 0;

h1 ! 0;

C1 ! 0

as y ! 1:

ð20Þ

Without going into detail, the solutions of Eqs. (13,14,15) and (17,18,19) subject to Eqs. (16), (20) can be shown to be uðy; tÞ ¼ A2 em1 y þ A3 em2 y þ A4 em3 y þ eent fA10 em1 y þ A11 em2 y þ A12 em3 y þ A13 em4 y þ A14 em5 y þ A15 em6 y g; hðy; tÞ ¼ e

m2 y

þ A1 ðe

m1 y

e

m2 y

nt

Þ þ ee fA6 e

m1 y

þ A7 e

m2 y

þ A8 e

m4 y

þ A9 e

m5 y

g;

Cðy; tÞ ¼ em1 y þ eent fem4 y þ A5 ðem1 y em4 y Þg:

ð21Þ ð22Þ ð23Þ

The physical quantities of interest are the wall shear stress sw and the local surface heat transfer rate qw. These are defined by ou sw ¼ l ¼ qV 2o u0 ð0Þ oy y ¼0

therefore, the local friction factor Cf is given by sw Cf ¼ ¼ u0 ð0Þ qV 2o ¼ m1 A2 m2 A3 m3 A4 þ eent ðm1 A10 m2 A11 m3 A12 m4 A13 m5 A14 m6 A15 Þ: The local surface heat flux is given by oT qw ¼ k ; oy y ¼0 where k is the effective thermal conductivity, together with the definition of the local Nusselt number qw x Nux ¼ Tw T1 k

ð24Þ

ð25Þ

ð26Þ

one can write Nu 0 ¼ hð0Þ ¼ m2 þ A1 ðm1 m2 Þ þ eent ðm1 A6 þ m2 A7 þ m4 A8 þ m5 A9 Þ; Rex

ð27Þ

where Rex ¼ V mo x is the local Reynolds number. 4. Results and discussion Numerical evaluation of the analytical results reported in the previous section was performed and a representative set of results is reported graphically in Figs. 1–12. These results are obtained to illustrate the influence of the chemical reaction parameter c, the absorption radiation parameter Ql, the Schmidt number Sc, the heat absorption coefficient /, the magnetic field parameter M and permeability parameter K on the velocity, temperature and the concentration profiles, while the values of the physical parameters are fixed at real constants, A = 0.5, e = 0.2, the frequency of oscillations n = 0.1, scale of free stream velocity up = 0.5, Prandtl number Pr = 0.71 and t = 1.0. Figs. 1–3 displays results for the velocity, temperature and concentration distributions, respectively. It is seen, that the velocity, temperature and concentration increases with decreasing the chemical reaction parameter c, Also, we observe that the magnitude of the stream wise velocity increases and the inflection point for the velocity distribution moves further away from the surface. For different values of the absorption radiation parameter Ql, the velocity and temperature profiles are plotted in Figs. 4 and 5. It is obvious that an increase in the absorption radiation parameter Ql results an increasing in the velocity and temperature profiles within the boundary layer, as well as an increasing in


1061

2.5

2.0 γ = 20,15,10,5

u

1.5

1.0

0.5

0.0 0

1

2

y

3

4

5

Fig. 1. Velocity profile against spanwise coordinate y for different values of (the chemical reaction parameter c) with Gr = 2, Gm = 1, Sc = 0.6, K = 0.5, M = 0.2, / = 1 and Ql = 2.

1.2 γ = 20,15,10,5

θ

0.9

0.6

0.3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

y Fig. 2. Temperature profile against spanwise coordinate y for different values of (the chemical reaction parameter c) with Gr = 2, Gm = 1, Sc = 0.6, K = 0.5, M = 0.2, / = 1 and Ql = 2.

1.5

1.0

C

γ = 20, 15, 10, 5

0.5

0.0 0.0

0.5

1.0

y

1.5

2.0

2.5

Fig. 3. Concentration profile against spanwise coordinate y for different values of (the chemical reaction parameter c) with Gr = 2, Gm = 1, Sc = 0.6, K = 0.5, M = 0.2, / = 1 and Ql = 2.

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F.S. Ibrahim et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1056–1066 2.0 Q l = 1,2,3,4

u

1.5

1.0

0.5

0.0

0

2

4

6

8

10

y Fig. 4. Velocity profile against spanwise coordinate y for different values of (absorption radiation parameter QL) with Gr = 10, Gm = 2, Sc = 0.4, K = 0.5, M = 2, / = 2 and c = 0.5.

1.5

Q l=1,2,3,4

θ

1.0

0.5

0.0 0

2

4

6

8

y Fig. 5. Temperature profile against spanwise coordinate y for different values of (absorption radiation parameter QL) with Gr = 10, Gm = 2, Sc = 0.4, K = 0.5, M = 2, / = 2 and c = 0.5.

3.5 3.0 2.5 2.0

u

Sc=0.8,0.6,0.4,0.2

1.5 1.0 0.5 0.0 0

3

6

9

12

15

y Fig. 6. Velocity profile against spanwise coordinate y for different values of (the Schmidt number Sc) with Gr = 4, Gm = 2, Ql = 2, K = 2, M = 2, / = 2 and c = 0.2.


1063

1.5 1.2 0.9

θ

Sc=0.8,0.6,0.4,0.2

0.6 0.3 0.0 0

5

y

10

15

Fig. 7. Temperature profile against spanwise coordinate y for different values of (the Schmidt number Sc) with Gr = 4, Gm = 2, Ql = 2, K = 2, M = 2, / = 2 and c = 0.2.

1.5

C

1.2 0.9 Sc=0.8,0.6,0.4,0.2

0.6 0.3 0.0 0

5

y

10

15

Fig. 8. Concentration profile against spanwise coordinate y for different values of (the Schmidt number Sc) with Gr = 4, Gm = 2, Ql = 2, K = 2, M = 2, / = 2 and c = 0.2.

1.2 1.0 0.8

u

φ=3,2,1,0

0.6 0.4 0.2 0.0 0

1

2

3

4

5

6

7

8

y Fig. 9. Velocity profile against spanwise coordinate y for different values of (heat absorption parameter /) with Gr = 4, Gm = 2, Sc = 0.8, K = 0.5, M = 2, Ql = 0.5 and c = 5.

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1.5 1.2

θ

0.9 φ= 3,2,1,0

0.6 0.3 0.0 0

2

4

y

6

Fig. 10. Temperature profile against spanwise coordinate y for different values of (heat absorption parameter /) with Gr = 4, Gm = 2, Sc = 0.8, K = 0.5, M = 2, Ql = 0.5 and c = 5.

14 12 10

M=5,2,1,0

u

8 6 4 2 0 0

3

6

9

12

y Fig. 11. Velocity profile against spanwise coordinate y for different values of (Magnetic parameter M) with c = 0.1, Gr = 4, Gm = 2, Sc = 0.2, K = 2, / = 2 and Ql = 2.

1.5 K=2.5,2,1

u

1.0

0.5

0.0 0

2

4

6

8

y Fig. 12. Velocity profile against spanwise coordinate y for different values of (permeability parameter K) with c = 0.1, Gr = 4, Gm = 2, Sc = 0.2, M = 2, / = 2 and Ql = 2.


1065

the momentum and thermal thickness. This is because the large Ql-values correspond to an increased dominance of conduction over absorption radiation thereby increasing buoyancy force (thus, vertical velocity) and thickness of the thermal and momentum boundary layers. Figs. 6–8 display the effects of the Schmidt number Sc on the velocity, temperature and concentration profiles, respectively. As the Schmidt number increases, the concentration decreases. This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid velocity. The reductions in the velocity, temperature and concentration profiles are accompanied by simultaneous reductions in the momentum and concentration boundary layers thickens. These behaviors are clearly shown in Figs. 6–8. Figs. 9 and 10 illustrate the influence of the heat absorption coefficient / on the velocity and temperature profiles, respectively. Physically speaking, the presence of heat absorption (thermal sink) effects has the tendency to reduce the fluid temperature. This causes the thermal buoyancy effects to decrease resulting in a net reduction in the fluid velocity. These behaviors are clearly obvious from Figs. 9 and 10 in which both the velocity and temperature distributions decrease as / increases. It is also observed that both the hydrodynamic (velocity) and the thermal (temperature) boundary layers decrease as the heat absorption effects increase. For different values of the magnetic field parameter M, the velocity profile are plotted in Fig. 11. It is obvious that the effect of increasing values of magnetic field parameter results in a decreasing velocity distribution across the boundary layer. Fig. 12 illustrate the variation of velocity distribution across the boundary layer for various values of the permeability parameter K. The velocity increases with a decrease in permeability parameter K. 5. Concluding remarks The plate velocity was maintained at a constant value and the flow was subjected to a transverse magnetic field. The resulting partial differential equations were transformed into a set of ordinary differential equations using two-term series and solved in closed-form. Numerical evaluations of the closed-form results were performed and some graphical results were obtained to illustrate the details of the flow and heat and mass transfer characteristics and their dependence on some of the physical parameters. It was found that the velocity profiles increased due to decreases in chemical reaction parameter, the Schmidt number, heat absorption coefficient, magnetic field and permeability parameters while it increased due to increases in absorption radiation parameter Ql. However, an increase temperature profile is a function of an increase in absorption radiation parameter Ql. Whereas an increase in chemical reaction parameter c, the Schmidt number Sc and heat absorption coefficient / led to a decrease in the temperature profile on cooling. Also, it was found that the concentration profile increased due to decreases in the chemical reaction parameter c and the Schmidt number Sc. Appendix qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 Sc þ Sc þ 4cSc ; m2 ¼ Pr þ Pr2 þ 4Pr/ ; m1 ¼ 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1þ 1þ4 M þ Sc þ Sc2 þ 4ðc þ nÞSc ; m3 ¼ ; m4 ¼ 2 K 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 2 Pr þ Pr þ 4Prðn þ /Þ ; m6 ¼ 1þ 1þ4 M þ þn ; m5 ¼ 2 2 K Gm GrA1 GrðA1 1Þ ; A3 ¼ 2 ; 1 m2 m2 ðM þ K1 Þ m1 ðM þ K Þ m1 ASc PrðAA1 A5 Ql Þ ; A6 ¼ 2 ; A4 ¼ up ðA2 þ A3 Þ; A5 ¼ 2 m1 Scm1 Scðc þ nÞ m1 Prm1 Prð/ þ nÞ PrAm2 ð1 A1 Þ PrQl ðA5 1Þ ; A8 ¼ 2 ; A9 ¼ 1 ðA6 þ A7 þ A8 Þ; A7 ¼ 2 m2 Prm2 Prð/ þ nÞ m4 Prm4 Prð/ þ nÞ

A1 ¼

m21

Ql ; Prm1 Pr/

A2 ¼

m21

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AA2 ml GrA6 GmA5 AA3 m2 GrA7 AA4 m3 ; ; A11 ¼ 2 ; A12 ¼ 3 m21 m1 ðN þ nÞ m2 m2 ðN þ nÞ m2 m3 ðN þ nÞ A5 Gm Gm GrA8 GrA9 A13 ¼ 2 ; A14 ¼ 2 ; A9 ¼ ðA10 þ A11 þ A12 þ A13 þ A14 Þ: m4 m4 ðN þ nÞ m5 m5 ðM þ K1 þ nÞ

A10 ¼

ð28Þ References [1] Bestman AR. Natural convection boundary layer with suction and mass transfer in a porous medium. Int J Energy Res 1990;14:389–96. [2] Abdus Sattar MD, Hamid Kalim MD. Unsteady free-convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate. J Math Phys Sci 1996;30:25–37. [3] Makinde OD. Free convection flow with thermal radiation and mass transfer past amoving vertical porous plate. Int Comm Heat Mass Transfer 2005;32:1411–9. [4] Ibrihem FS, Hassanien IA, Bakr AA. Nonclassical thermal effects in stokes’ second problem for micropolar fluids. ASME J Appl Mech 2005;72:468–74. [5] Muthucumaraswamy R, Ganesan P. Effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. J Appl Mech Tech Phys 2001;42:665–71. [6] Deka R, Das UN, Soundalgekar VM. Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction. Forschung im Ingenieurwesen 1994;60:284–7. [7] Chamkha AJ. MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence of heat generation/ absorption and a chemical reaction. Int Comm Heat Mass Transfer 2003;30:413–22. [8] Soundalgekar VM, Patti MR. Stokes problem for a vertical plate with constant heat flux. Astrophys Space Sci 1980;70:179–82. [9] Gebhart B, Pera L. The nature of vertical natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion. J Heat Mass Transfer 1971;14:2025–50. [10] Chamkha AJ. Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption. Int J Eng Sci 2004;42:217–30. [11] Raptis A. Radiation and free convection flow through a porous medium. Int Comm Heat Mass Transfer 1998;25:289–95. [12] Raptis A, Perdikis C. Free convection flow of water near 4 °C past a moving plate. Forschung im Ingenieurwesen 2002;67:206–8.