Cmjv05i01p0023 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.5 (1), 23-34 (2014)

Chemical Reaction and Radiation Effect on the Unsteady MHD Free Convection Flow Past a Semi Infinite Vertical Permeable Moving Plate with Heat Source and Suction R. Lakshmi1, G.V. Ramana Reddy2 and K. Jayarami Reddy3 1

Department of Mathematics, DVR and Dr. H. S. MIC College of Technology, Kanchikacherla, INDIA. 2 Department of Mathematics, K. L. University, Guntur, INDIA. (Received on: January 9, 2014) 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 plate 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 perturbation technique. Graphical results for velocity, temperature and concentration profiles of both phases based on the analytical solutions are presented and discussed qualitatively. Keywords: Heat Transfer, MHD, permeability parameter, heat absorption parameter, chemical reaction parameter and skinfriction.

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

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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. The effects 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 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. Several researchers have analyzed the free convection and mass transfer flow of a viscous fluid through porous medium. In light of these facts, Heat flow and mass transfer over a vertical porous plate with variable suction and heat absorption/generation have been studied by many workers. Gebhar1 has shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux in the plate. Soundalgekar2 analyzed the effect of viscous dissipative heat on the two dimensional unsteady, free convective flow past a vertical porous plate when the temperature oscillates in time and there is constant suction at the plate. Elbashbeshy3 studied heat and mass transfer along a vertical plate under the combined buoyancy effects of thermal and species diffusion, in the presence of magnetic field. Soundagekar et al.4 analyzed the problem of free

convection effects on stokes problem for a vertical plate under the action of transversely applied magnetic field with mass transfer. Kim5 investigated unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction by assuming that the free stream velocity follows the exponentially increasing small perturb action law. Chamkha6 extended the problem of Kim5 to heat absorption and mass transfer effects. Chen7 studied heat and mass transfer in MHD flow by natural convection from a permeable, inclined surface with variable wall temperature and concentration. Israel Cookey et al.8 investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in porous medium with time dependent suction. Perdikis and Rapti9 studied the unsteady MHD flow in the presence of radiation. Rahman and Sattar10 analyzed the MHD convective flow of a micro polar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption. Raji Reddy and Srihari11 studied numerical solution of unsteady flow of a radiating and chemically reacting fluid with time-dependent suction. Ramana Reddy et al.12 studied an unsteady MHD free convection flow and mass transfer near a moving vertical plate in the presence of thermal radiation. Singh et al.13 studied the heat transfer over stretching surface in porous media with transverse magnetic field. Singh et al.14 and15 also investigated MHD oblique stagnation-point flow towards a stretching sheet with heat transfer for steady and unsteady cases. Elbashbeshy et al.16 investigated the effects of thermal radiation and magnetic field on

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unsteady boundary layer mixed convection flow and heat transfer problem from a vertical porous stretching surface. Ahmed Sahin studied influence of chemical reaction on transient MHD free Convective flow over a vertical plate. Ramana Reddy et al.17 have investigated an unsteady MHD free convective flow past a semi-infinite vertical porous plate. In 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.

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 C¥ is infinitesimally small and hence the Soret and Dufour effects are neglected. The chemical reactions are taking place in the flow and all thermo physical 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 Continuity Equation:

2. FORMATION OF THE PROBLEM

Energy Equation:

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

v  0 y 

(1)

Momentum Equation: u  u       v  g  T  T   g   C   C t  y 

y  2



 K

u 

 B02 u 

(2) T  T  1  v  t  y C p

 k  T2  Q0 T T    Ql  CC   y  (3)

Mass diffusion Equation: C  C   2C   v  K r C C  D t y y2





(4)

The boundary conditions for the velocity, temperature and concentration fields are

u   u p ,

    T   T     T w  T   e n t , C '  C     C w  C   e n t

u   0,

T   T  ,

C   C  ,

 u

at as

y  0 y  

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R. Lakshmi, et al., J. Comp. & Math. Sci. Vol.5 (1), 23-34 (2014)

where x , y  and t  are the dimensional distances along and perpendicular to the plate and dimensional time respectively. u and v are the velocity components in the x , y  directions respectively, T  is the dimensional temperature, the C  is dimensional concentration, Cw and Tw are the concentration and temperature at the wall, respectively. C and T are the free stream dimensional concentration and temperature, respectively.  is the fluid density,  is the kinematic viscosity, C p is the specific heat at constant pressure,  is the fluid electrical conductivity, B0 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  and  are the thermal and concentration expansion coefficients, respectively and Kr 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, u p and n are the velocity and the constants. From Equation (1), it is clear that the suction velocity at the plate surface is either a constant and or a function of time. Hence the suction velocity normal to the plate is assumed in the form

v  v0 (1  Aent )

where A is a real positive constant,  and  A is small values less than unity, and v0 is scale of suction velocity which is non zero positive constant. The negative sign indicates that the suction is towards the plate. In order to write the governing equations and the boundary conditions in dimensional following non-dimensional quantities are introduced.

u u v t v02  n T   T C   C y ,u  ,v  ,t  , u p  p , n  2 ,  ,C   v0 v0  v0 v0 Tw  T Cw  C v0 y 

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

(6)

(7)

u u 2u  1  1  Aent   Gr  GmC  2   M   u t y y  K

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  1  2  1   Aent      Ql C (9) t y Pr y 2 C C 1  2C (10)  1   Aent    C t y Sc y 2

hood of the fluid in the neighborhood of the plate as u ( y, t )  u0 ( y )   e nt u1 ( y )  O  2   ............

 ( y, t )   0 ( y )   e nt1 ( y )  O  2   ............ C ( y , t )  C0 ( y )   e nt C1 ( y )  O  2   ............

The corresponding boundary conditions for t  0 are transformed to: u  u p ,   1   ent , C  1   e nt

u  0,   0,

as y  

C 0

y0

(11)

where M , K , Gr, Gm, Pr,  , Sc, , and Ql are the magnetic field parameter, permeability parameter, thermal Grashof number, Solutal Grashof number, Prandtl number, Chemical reaction number, Schmidt number, heat absorption parameter and absorption of radiation parameter respectively. Gr 

 Q  C   C  g  (Tw  T ) g   (Cw  C ) , Gm  , Ql  l w v03 v03 Tw  T  v02

 C p  B02 K v 2 K    Q0 M , K  2 0 ,   r2 , Pr  , Sc  ,  , 2  v0  v0 k D  C p v02

3. SOLUTION OF THE PROBLEM Equations (8) – (10) are coupled, non – linear partial differential Equations and these cannot be solved in closed form. However, it can be 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 concentration of the fluid in the neighbor-

(12)

Substituting (12) in Equations (8) – (11) and equating the harmonic and non – harmonic terms, and neglecting the higher order terms

  , we obtain

of O 

1  u0  u0   M   u0  Gr0  GmC0 K   0  Pr 0  Pr 0   Pr Ql C0

(13)

C0  ScC0  Sc C0  0

(15)

1   u1  u1   M   n  u0  Gr1  GmC1  Au0 K  

(16)

(14)

1 Pr 1  Pr   n 1   A Pr0  Pr Ql C1 (17) C1 ScC1  Sc    n  C1   AScC0

(18)

where prime denotes ordinary differentiation with respect to y. The corresponding boundary conditions can be written as u0  U p , u1  0,0  1,1  1, C0  1, C1  1

at y  0

u0  1, u1  1,0  0,1  0, C0  0, C1  0 as y  

(19) Solving Equations (13) – (18) under the boundary conditions (19), we obtain the velocity, temperature and concentration distributions in the boundary layer as

u  y, t   A7 exp(-m3 y )  A3 exp(-m2 y)  A6 exp(- m1 y )   A27 exp(-m1 y )  A19 exp(-m2 y )  A21 exp(-m3 y )   - A28 exp(-m4 y) - A22 exp(-m5 y )  A29 exp(-m6 y) 

 exp(nt ) 

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  y, t   A2 exp(-m2 y )  A1 exp(-m1 y )   exp( nt )  A14 exp(-m5 y )  A15 exp(-m1 y )  A12 exp(-m4 y)  C  y, t   exp(-m1 y )   exp(nt )  A8 exp(-m1 y )  A9 exp(-m4 y) 

The skin-friction, Nusselt number and Sherwood number are important physical parameters for this type of boundary layer flow. SKIN FRICTION Knowing the velocity field, the skin – friction at the plate can be obtained, which in non –dimensional form is given by  u   u  u  C f  w2      0   e nt 1   v0  y  y  0  y y  y  0   m A  m2 A19  A21m3   C f    m3 A7  m2 A3  A6 m1   ent  1 27    A28 m4  m5 A22  A29 m6   

NUSSELT NUMBER Knowing the temperature field, the rate of heat transfer coefficient can be

obtained, which in non –dimensional form is given, in terms of the Nusselt number, is given by  T    y         y0   Nu   x  Nu Re x 1        0   ent 1   y  y y  y 0 Tw  T    y0 

Nu Re x 1  m2 A2  A1m1   ent  m5 A14  m1 A15  m4 A12 

where Re x 

v0 x



is the local Reynolds

number. SHERWOOD NUMBER Knowing the concentration field, the rate of mass transfer coefficient can be obtained, which in non –dimensional form, in terms of the Sherwood number, is given by

 C      C   C  y  y0 C  Sh   x  Sh Re x 1       0   e nt 1   y  y  0  Cw  C   y  y  0  y

Sh Re x 1   m1   ent  m4 A9  m1 A8   Here the constants are not given due to shake of brevity. 2. 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. 1–13. These results are obtained to illustrate the influence of the chemical reaction

parameter , 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, = 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. For different values of the magnetic field parameter M, the velocity profiles are

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plotted in Fig-1. It is obvious that the effect of increasing values of magnetic field parameter results in a decreasing velocity distribution across the boundary layer. For various values of Grashof number and modified Grashof number, the velocity profiles are plotted in Figs. 2 and 3. The Grashof number Gr signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer. It is observed that there is a rise in the velocity due to the enhancement of thermal buoyancy force. The modified Grashof number Gm defines the ratio of the species buoyancy force to the viscous hydrodynamic force. As expected, the fluid velocity increases and the peak value is more distinctive due to increase in the species buoyancy force. The velocity distribution attains a distinctive maximum value in the vicinity of the surface. It is noticed that the velocity increases with increasing values of the Solutal Grashof number. Fig-4 illustrates the variation of velocity distribution across the boundary layer for various values of the permeability parameter K. The velocity increases with increases in permeability parameter K. For different values of the absorption radiation parameter Ql, the velocity and temperature profiles are plotted in Figs. 5 and 6. 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 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. 7 and 8 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. 7 and 8 in which both the velocity and temperature distributions are decreases. 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 chemical reaction parameter , for the velocity and concentration profiles are plotted in Figs. 9 and 10 respectively. It is seen, that the velocity and concentration decreases with an increasing the chemical reaction parameter , 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. Figs. 11 and 12 display the effects of the Schmidt number Sc on the temperature and concentration profiles, respectively. As the Schmidt number increases, the temperature and concentration decreases. This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid temperature. For different values of the Prandtl number on the temperature profiles are plotted in Fig.13. It is obvious that the effect of increasing values of Prandtl number results in a decreasing temperature distribution across the boundary layer.

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0.9

 =2;Gr=4;Gm=2;n=0.1;t=1.0;A=0.5; Pr=0.71;Ql=2;Sc=0.6;up=0.5; =0.2; =0.5;K=0.5;

0.8

0.7

M=1, 2, 3, 4

0.6

0.5

Velocity

0.7

0.4

0.3

0.2

0.1

0.5

1.5

2.5 y

3.5

4.5

Gr=1, 2, 3, 4

0.4

0.3

 =2;M=1;Gm=2;n=0.1;t=1.0;A=0.5; Pr=0.71;Ql=2;Sc=0.6;up=0.5; =0.2; =0.5;K=0.5;

0.8

Fig-1. Velocity profiles for different values of magnetic parameter

0.5

1.5

2.5 y

3.5

4.5

Fig-2. Velocity profiles for different values of Grashof number 1

1.4

 =2;M=1;Gr=5;n=0.1;t=1.0; A=0.5;Pr=0.71;Ql=2;Sc=0.6; up=0.5;=0.2; =0.5;K=0.5;

1.2

 =2;M=1;Gr=5;n=0.1;t=1.0; A=0.5;Gm=2;Ql=2;Sc=0.6; up=0.5; =0.2; =0.5;Pr=0.71

0.9 0.8

Gm=1, 2, 3, 4

0.7 K=0.1, 0.2, 0.3, 0.4

Velocity

0.6 0.8

0.6

0.5 0.4 0.3

0.4

0.2 0.2

0.1 0

0.5

1.5

2.5 y

3.5

4.5

Fig-3. Velocity profiles for different values of Solutal Grashof number

0.5

1.5

2.5 y

3.5

4.5

Fig-4. Velocity profiles for different values of permeability parameter

1.4

2.5

 =2;M=1;Gr=5;n=0.1;t=1.0;A=0.5;Gm=2; K=0.5;Sc=0.6;up=0.5; =0.2; =0.5;Pr=0.71;

1.2

 =2;M=1;Gr=5;n=0.1;t=1.0;A=0.5;Gm=2; K=0.5;Sc=0.6;up=0.5; =0.2;Pr=0.71; =2;

2 1

Ql=1, 2, 3, 4

Temperature

Velocity

0.8

0.6

1.5

1 Ql=1, 2, 3, 4

0.4 0.5 0.2

0.5

1.5

2.5 y

3.5

4.5

Fig-5. Velocity profiles for different values of absorption radiation parameter

0.5

1.5

2.5 y

3.5

4.5

Fig-6. Temperature profiles for different values of absorption radiation parameter

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1.2

M=2;Gr=4;Gm=2;n=0.1;t=1.0;A=0.5;Pr=0.71; Ql=2;Sc=0.60;up=0.5; =0.2; =0.5;K=0.5;

0.6

0.5

0.4

0.3

0.6

0.2

0.1

0.5

1.5

2.5 y

3.5

4.5

-0.2

Fig-7. Velocity profiles for different values of heat source parameter

1.2

1.5

2.5 y

3.5

4.5

Sc=0.22;M=1;Gr=5;n=0.1;t=1.0;A=0.5;Gm=2; K=0.5;Pr=0.71;up=0.5; =0.2;Ql=2;  =2;

=5, 10, 15, 20

0.6

1 Concentration

0.5 Velocity

0.5

1.4

 =2;M=1;Gr=5;n=0.1;t=1.0;A=0.5;Gm=2; K=0.5;Sc=0.6;up=0.5; =0.2;Ql=2;Pr=0.71;

0.7

0.4 0.3

0.8

0.6

=1, 2, 3, 4

0.4

0.2

0.1

0.5

1.5

2.5 y

3.5

4.5

Fig-9. Velocity profiles for different values of chemical reaction parameter

5 y

Fig-10. Concentration profiles for different values of chemical reaction parameter

1.4

1.2

=0.2;M=1;Gr=5;n=0.1;t=1.0;A=0.5; Gm=2;K=0.5;Pr=0.71;up=0.5; =0.2;Ql=2;  =2;

1 Concentration

Temperature

Fig-8. Temperature profiles for different values of heat source parameter

0.8

 =5, 10, 15, 20

0.4

0.2

=2;M=1;Gr=5;n=0.1;t=1.0;A=0.5; Gm=2;K=0.5;Sc=0.6;up=0.5;=0.2;Ql=2;Pr=0.71;

0.8

Temperature

Velocity

 =5,10,15,20

0.8

0.6

0.4

0.8

0.6

Sc=0.22, 0.30, 0.60, 0.78

0.4

Sc=0.22, 0.30, 0.60, 0.78 0.2

0.2

Fig-11. Temperature profiles for different values of Schmidt number

5 y

Fig-12. Concentration profiles for different values of Schmidt number

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R. Lakshmi, et al., J. Comp. & Math. Sci. Vol.5 (1), 23-34 (2014)

1.5

1 Pr=0.71, 1.00, 3.00, 7.00

Temperature

0.5

-0.5

-1

 =0.2;M=1;Gr=5;n=0.1;t=1.0;A=0.5;Gm=2; K=0.5;Sc=0.22;up=0.5;=0.2;Ql=2;  =2;

-1.5

-2 0

Fig-13. Temperature profiles for different values of Prandtl number

5. CONCLUSIONS 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 perturbation technique method. 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 decreases due to an increase in chemical reaction parameter, magnetic field and heat absorption parameters while it increased due to increases in absorption radiation parameter Ql, Grashof number, modified Grashof number and permeability parameter. However, an increase temperature profile is a function of an increase in absorption radiation parameter Ql. whereas an increase in chemical reaction parameter , the Schmidt number Sc and heat absorption

coefficient Ql 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. REFERENCES 1. Gebhar.B., Effects of viscous dissipative in natural convection, J. Fluid Mech., 14, 225-232, (1962). 2. Soundalgekar. V.M., Viscous dissipative effects on unsteady free convective flow past an vertical porous plate with constant suction, Int. J. Heat Mass Transfer, 15, 1253-1261, (1972). 3. Elbashbeshy. E. M. A., Heat and mass transfer along a vertical plate with variable surface tension and concentration in the presence of magnetic Field, International Journal of Engineering Science, 35, 515-522, (1997). 4. Soundagekar.V. M., Gupta. S. K and Birajdar. S. S., Effects of mass transfer and free convection effects on MHD

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