Chemical Reaction and Viscous Dissipation Effects on MHD free

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 12 (2017), pp. 8297-8322 © Research India Publications http://www.ripublication.com

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective flow Past a Semi-Infinite Moving Vertical Porous Plate with Radiation Absorption K.V.B.Rajakumar a K.S. Balamuruganb Ch.V. Ramana Murthy, M. Umasenkara Reddyd a

b

Research Scholar in Rayalaseema University, Kurnool, Andhra Pradesh, India. Department of Mathematics, RVR&JC College of Engineering, Guntur, (AP), India. c Department of Mathematics, Sri Vasavi Institute of Engineering & Technology, Nanadamuru (AP), India. d Department of Mathematics, Kallam Haranadhareddy Institute of technology, Guntur, India. (Corresponding Author) K.V.B.Rajakumar Abstract In this paper the Chemical reaction and Viscous Dissipation effects on MHD free convective flow past a semi-infinite moving vertical porous plate with radiation absorption was investigated. The flow was unsteady and restricted to the laminar domain. The dimensionless governing equations for this investigation are solved analytically by using multiple regular perturbation law. The effects of various physical parameters on velocity, temperature and concentration fields are presented graphically. With the aid of these, the expression for the skin friction, Nusselt number and Sherwood number profiles was done with the help of tables. It was found that an increases in chemical reaction, Schmidt number, Prandtl number, thermal radiation, leads to decrease in both velocity and temperature. However an increase in both chemical reactions, Schmidt number, leads to decreases in concentration. And also an increase in porous medium, solutal Grashof number and Radiation parameter leads to increases in both velocity and temperature. Keywords: Radiation absorption, viscous dissipation, MHD, chemical reaction, multiple regular Perturbation law.

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K.V.B.Rajakumar, et al

Particularly in the Magneto Hydrodynamic with free convection flow has been received considerable attention to many researchers in last few years. In addition, Magneto Hydrodynamic is the branch of continuum mechanics which deals with the flow of electrically conducting fluid in electrical and magnetic fields. The flow of an electrically conducting fluid in the presence of a magnetic field is important in various area of technology. There are many applications of MHD free convection flows in fiber and granular insulation, geothermal systems etc. Throughout the last few decades a great deal of research work has been done about the Magneto-Hydrodynamic effects. Magneto hydrodynamic flow of heat and mass transfer processes action in abounding of the automated applications: such as cooling of geothermal systems, aerodynamic processes, actinic catalytic reactors and processes, electromagnetic pumps, and Magneto Hydrodynamic power generators etc. Abounding studies accept been agitated out to investigate the magneto hydrodynamic past free convective fluid flow. Magneto hydrodynamic free convection flow of an electrically conducting fluid in different porous geometries is of considerable interest to the technical field due to its frequent occurrence in industrial, technological and geothermal applications. As an example, the geothermal region gases are electrically conducting and undergo the influence of magnetic field. Also, it has applications in nuclear engineering in connection with reactors cooling. The study of Magneto Hydrodynamic flow through porous media has been the subject of considerable improvement in the last few decades. In nuclear engineering, the magneto hydrodynamic flow in a porous medium is required for the design of a layer of liquid metal around a thermonuclear fusion–fission hybrid reactor. In metallurgy, a permanent magnetic field can be applied during the solidification process to modify the intensity of the interdendritic flow of the metallic liquid in a porous medium. Helmy [1] studied MHD unsteady free convection Flow past a vertical porous plate. Gurivireddy.P. et al [2] analyzed the thermo diffusion effect on an unsteady simultaneous convective heat and mass transfer flow of an incompressible, electrically conducting, heat generating/ absorbing fluid along a semi-infinite moving porous plate embedded in a porous medium with the presence of pressure gradient, thermal radiation and chemical reaction. In this article the governing equations are solved by using a regular perturbation method. I observed that an increase in both Pr and R it leads to decreases in thermal boundary layer thickness. However, concentration decreases as Kr, Sc increase but it increases with an increase in both Soret and heat absorption parameters. Hady et al. [3] studied the problem of free convection flow along a vertical wavy surface embedded in electrically conducting fluid saturated porous media in the presence of internal heat generation or absorption effect. Jeftasunzu 𝑒𝑡 𝑎𝑙 [4] investigated the radiative magneto hydrodynamic transient heat and mass transfer flow by free convection past a vertical plate with viscous dissipation in the presence of chemical reaction, when the temperature of the plate oscillates frequently about a constant mean temperature and the plate being embedded in a porous medium. Thakar

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8299 and Ram [5] also studied the Magneto Hydrodynamic free convection heat transfer of water at 4000C through a porous medium. . Srinivasa Raju et al [6] studied numerical investigation of transient magneto hydrodynamic free convection flow past an infinite vertical plate embedded in a porous medium with viscous dissipation. The governing equations are solved by using finite element method. In this paper I observed that the velocity increases with the increase in thermal Grashof number and solutal Grashof number, The velocity decreases with an increase in the magnetic parameter, The velocity increases with an increase in the permeability of the porous medium parameter, An increase in the Eckert number increases the velocity and temperature, Increasing the heat absorption parameter reduces both velocity and temperature. And also the velocity as well as concentration decreases with an increase in the Schmidt number influence on the concentration boundary layer and wall mass transfer rate. There are several investigations on the effects of chemical reaction on fluid flow in different physical situations. Analysis of a chemical reaction in heat and mass transfer problems are important for chemical industries due to its huge applications in several branches of science and engineering, such as in drying, evaporation, reservoir engineering, manufacturing of ceramics, food processing. The order of chemical reaction depends on many factors, for instance foreign mass, active fluid and stretching of sheet and so on. R. Srinivasa Raju [7] studied the effects of chemical reaction and combined buoyancy effects on an unsteady MHD mixed convective flow along an infinite vertical porous plate in the presence of hall current. Jithender Reddy et al [8] Effects of chemical reaction and radiation on an unsteady transient MHD free convective and heat transfer of a viscous, incompressible, electrically conducting and fluid past a suddenly started infinite vertical porous plate taking into account the viscous dissipation were presented. The dimensionless governing partial differential equations are solved by using finite element method. Rajeswari et al. [9] have investigated chemical reaction, heat and mass transfer on nonlinear MHD boundary layer flow through a vertical porous surface in presence of suction. Cintaginjala et al. [10] investigated the effects of chemical reaction and thermo-diffusion on hydro magnetic free convective fluid flow past an infinite vertical plate in the presence of heat sink. In the way of analysis, it is supposed that the magnetic field of uniform strength is applied and induced magnetic field is neglected and also we observe that how various parameters affect the flow past an infinite vertical plate. Ibrahim et al. [11] analyzed the effects of the chemical reaction and radiation absorption on the unsteady Magneto Hydrodynamic free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction. Mythili et al [12] studied Heat generating/absorbing and chemically reacting Casson fluid flow over a vertical cone and flat plate saturated with non-Darcy porous medium. Jadish [13] studied the thermal-diffusion impact on MHD three dimensional natural convective Couette flows by employing perturbation technique with the presence of chemical reaction and the absence of Diffusion terms. Chamkha [14] studied the Magneto hydrodynamic flow of

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a numerical of uniformly stretched vertical permeable surface in the presence of heat generation or absorption on chemical reaction. R. Srinivasa Raju et al [15] studied both numerical and analytical solutions for the effect of chemical reaction on unsteady, incompressible, viscous fluid flow past an exponentially accelerated vertical plate with heat absorption and variable temperature in a magnetic field. The governing equations are solved by using finite element method. In this paper, I observed that when the Grashof number for heat and mass transfer increased the fluid velocity increase. The velocity, temperature distribution within the boundary layer decreases with the increase in values of the Prandtl number. The presence of heat absorption effects caused the reductions in the fluid temperature which resulted in decreasing in the fluid velocity. Chemical reaction considerably influence on the concentration boundary layer and wall mass transfer rate. R. Srinivasa Raju et al [16] Unsteady MHD natural convective, heat and mass transfer, electrically conducting Casson fluid flow over on an vertical surface taken in to the account with angle of inclination, chemical reaction, viscous dissipation and constant heat flux. The governing equations are solved by using finite element method. In this paper I observed that the velocity decreases with an increase in magnetic field parameter, Schmidt number, chemical reaction parameter, Casson fluid parameter, angle of inclination parameter and Prandtl number while it increases with an increase in Eckert number and Soret number. Temperature increases with an increase in Eckert number while it decreases with increasing values of Prandtl number. Concentration increases with increase in Soret number while it decreases with an increase in Schmidt number and chemical reaction parameter. A detailed numerical study was carried out for unsteady Hydro Magnetic natural convection heat and mass transfer with chemical reaction over a vertical plate in rotating system with periodic suction by Parida et al. [17]. Jasmine Benazir et al [ 18] studied the effects of double dispersion, non-uniform heat source/sink and higher order chemical reaction on unsteady, free convective, MHD Casson fluid flow over a vertical cone and flat plate saturated with porous medium. The extensively validated and unconditionally stable numerical solutions are obtained for the governing equations of two dimensional boundary layer flows by using the finite difference scheme of Crank-Nicolson type. Viscous dissipation plays an important role in changing the temperature distribution, just like an energy source, which affects the heat transfer rates considerably. In fact, the shear stresses can induce a considerable amount of heat generation. In the flow of fluids, mechanical energy is degraded into heat and this process is called viscous dissipation. The viscous dissipation effects are important in geophysical flows and also in certain industrial operations are usually characterized by the Eckert number. In the literature, extensive research work is available to examine the effect of natural convection on flow past a plate. . Gebhart et al. [19] have considered the effects of viscous dissipation for external natural convection flow over a surface. Maharajan et al. [20] have reported the influence of viscous dissipation effects in natural convective flows, showing that the heat transfer rates are reduced by an increase in the dissipation

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8301 parameter. Vajravelu et al. [21] studied a Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation. Srinivasa Raju et al. [22] have studied on applied magnetic field on transient free convective flow of an incompressible viscous dissipative fluid in a vertical channel. Emmanuel et al. [23] have studied on the effectiveness of viscous dissipation and joule heating on steady MHD and slip flow of a Bingham fluid over a porous rotating disk in the presence of hall and ion-slip currents. Hemant poonia et al. [24] investigated the heat and mass transfer effects on an unsteady two-dimensional laminar mixed convective boundary layer flow of, electrically conducting fluid, along a vertical flat plate with suction, embedded in porous medium, in the existence of transverse magnetic field, by taking into account the effects of the viscous dissipation. The governing equations of the motion are solved numerically by using multiple regular perturbation technique. In this paper, it is found that, the temperature as well as velocity increases due to increase in viscous dissipation. Also, when the Schmidt number is increased then it leads to decreases in concentration level. Suneetha et al [25] studied the effects of radiation and viscous dissipation on Magneto Hydrodynamic free convection flow past a vertical plate with varied boundary conditions. Radiation absorption and viscous dissipation effects on Magneto Hydrodynamic free convective flow past a semi-infinite moving vertical porous plate with chemical reaction and heat generation finds useful application in many of engineering problems such as Magneto Hydrodynamic power generator, plasma studies, nuclear reactors, geothermal extractors and boundary layer control in the field of aeronautics, Magneto Hydrodynamic pumps, Magneto, polymer technology and aerodynamic heating and accelerators, in cooling of electronic devices like mobiles, computers etc. and solar panels and purification of crude oil in petroleum industries, design of flow meters, heat exchangers, space vehicle, and propulsion, electromagnetic pumps etc. Meteorologists can use this study to understand dynamics of climatic changes and thunder storms. Israel-Cookey et al. [26] investigated the influence of viscous dissipation and radiation on unsteady Magneto Hydrodynamic free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. The governing equations motion was solved numerically by using multiple regular perturbation technique. From this paper, it is observed that increase in Eckert number it leads to increase in the velocity where as increases in magnetic field and radiation it leads to decrease in the velocity. Also, an increase in the Eckert number leads to an increase in the temperature, whereas increase in radiation and magnetic field parameters lead to a decrease in the temperature. Raptis.A et al. [27] considered the effect of radiation on 2D steady Magneto Hydrodynamic optically thin grey gas flow along infinite vertical plates taking into account the induced magnetic field. In this paper, the governing equations for the problem were solved numerically by using regular perturbation technique. It is observed that, as the radiation parameter increase it leads to decreases in the velocity and temperature. As Gr number increases then it leads to increase in


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velocity. Further, the velocity decreases with the increase of the Schmidt number in this examination viscous dissipation effect was not consider. Seth et al. [28] observed Natural convection Heat and Mass Transfer flow with Hall current, rotation, radiation and heat absorption past an accelerated moving vertical plate with ramped temperature. Rajput et al [29]. The exact solution of free convection in unsteady MHD Couette flow of a viscous incompressible and electrically conducting fluid between the two vertical parallel plates with the presence of thermal radiation and in the absence of thermal diffusion and diffusion thermo were analyzed. Motivated by the above studies, the main objective of this paper is to study Chemical reaction and Viscous Dissipation effects on unsteady MHD free convective flow past a semi-infinite moving vertical porous plate with radiation absorption. The equations of momentum, energy and concentration species diffusion which govern the flow field are solved numerically by using multiple regular perturbation law. This can be established by representing the velocity, temperature and concentration of the fluid in the region of the plate. The innovation of this study is to be consideration of a double diffusion fluid along with radiation absorption and viscous dissipation, the heat source of a radiating and chemical reaction fluid through a porous plate in a conducting field.

2. MATHEMATICAL FORMULATION: Consider the two dimensional unsteady laminar flow of a viscous incompressible and heat generation/absorption electrically conducting fluid past a semi infinite vertical moving porous plate in a uniform of a pressure grading has been considered with double-diffusive free convection, chemical reaction, thermal radiation effects and thermal diffusion. The following assumptions are listed below 1. Let the x*-axis be taken along the porous plate in the upward direction and y*axis in the direction perpendicular to the flow. 2. Let the x*and y* are the dimensional distance along the perpendicular to the plate and t* is the dimensional time. The physical model and coordinate system of the flow problem is shown in Figure A: u* and v* are the components of dimensional velocities along x*and y* directions. C* and g*are the dimensional concentration, acceleration due to gravity, 𝑄0 (𝑇 ∗ − 𝑇∞∗ ) is the amount of heat generated per unit volume. 𝑄0 Is the constant where either 𝑄0 < 0 or 𝑄0 > 0, g is acceleration due to gravity. 𝑔𝛽(𝑇 ∗ − 𝑇∞∗ ) Is the body force ∗ due to non-uniform temperature, 𝑔𝛽 ∗ (𝐶 ∗ − 𝐶∞ ) is the body force due to non uniform concentration. It is Assume that the porous plate moves with a constant velocity in the direction of fluid flow. 3. Assumed transverse magnetic field of the uniform strength B0 is to be applied normal to the plate.

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8303 4. The radiate heat flux in the x*- direction is considered negligible in comparison with that in y* - direction. 5. Viscous dissipation, Radiation absorption and Darcy resistance terms are taken into account the constant permeability porous medium. Since the plate is semiinfinite in length the entire flow variable are functions of y and t only. 6. The fluid has constant thermal conductivity and kinematic viscosity, the Boussinesqu’s approximation has been taken for the flow. 7. The homogeneous chemical reaction is of first order with rate constant K r between the diffusing species and the fluid is considered.

Figure A: The physical model in the problem From the above assumptions, the unsteady flow is governed by the following partial differential equations.


8304 Equation of continuity: *

u* v  0 x* y*

(1)

Equation of Momentum: * u* * u  2u* * u* 1 p* * * * *   u *   B 2 u* u  v     g  (T  T )  g  C*C * 2 * * *  x  0 t y K* x y*





(2)

Equation of Energy: *

* * T * * T K 2T * Q0 1 qr*  u u * * *  u  v   ( T  T )    R C*C*  K C p y* C p y* y* t* x* y* C p y*2 C p

T *



*



(3)

Concentration species diffusion equation: * C*  2C* * C * C* * *  u v  D 2  K r (C  C ) y* * t* y* y

(4)

The boundary conditions for velocity, Temperature and Concentration field are given by









** * * * * * * * * n t * * * * n t at y  0 u  0 T T  T T e ,C  C   C  C e  w w  w w   **  * * n t * * * * As y  u U (1 e ), T T , C C  0  

(5)

∗ Where 𝑇𝑤∗ is the wall dimensional temperature, 𝐶𝑤∗ is the concentration 𝐶∞ is the free ∗ stream dimensional concentration.𝑈0 , 𝑛 are the constant.

Since the motion is two dimensional and length of the plate is large enough so all the u * physical variables are independent of x − axis. Therefore, *  0 (6) x from the equation of the continuity, it is clear that the suction velocity is to be assumed as, v*  0

(7)

From the equation of momentum equation gives 

*  *  2 * 1 p* dU   U   B0 U   x* k  dt*

The radiative heat flux term by using the Rosseland approximation is given by

(8)

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8305 4 4 * T * * qr   *s 3ke y*

(Gurivireddy.P.et al [2])

(9)

We assume that the temperature difference within the flow is sufficiently small such 4 that 𝑇 ∗ can be expressed as a linear function of the temperature; this is expanding in Taylor series about 𝑇∞∗ and neglecting higher order terms to give: 3

T *  4T *T *  3T * 

(10)



By using equations (6), (7), (8), (9) and (10) then the governing equations (2), (3) and (4) can be write as follows: * u* dU 2u* * )  g  *  C*C*    U * u*    B 2 U * u*     g  (T *  T         2 * *    0   t dt K*  y*

(11)

* *3 2 * * *   2T *  Q 16 s T  T  u u * * * 0  2  T  T   R * *2 C * *  y*   C p 3 C p ke y p y y  

(12)

* T k  * C t p





C* 2C* *) D  K r (C*  C 2 * t y*

C*  C* 

(13)

Now Non-dimensional quantities are defined as V 2t* V y* u* U* ,U   , y  0 , t  0 , U0 U0   2 * K 2 * V n * * * T * , C *  C *  C C* C* K  2 , n  0 , T  T   Tw  w    V0

u









      

(14)

After substituting the boundary conditions and non-dimensional variables in the governing equations (11), (12) and (13) and taking into account equation (14) then we get u dU  2u   2  Gr  GmC  N U  u  t dt y

(15)

2 u  1  4 R      Pr  1  2    Ec t 3  y  y

(16)

2 C 1  C   Sc   Kr C t y 2

u R C y a

(17)


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Boundary conditions (5) are given by the following non-dimensional form

At y 0 u 0,  1 ent , C 1 ent   As y  u U  ,  0, C 0 

(18)

Where   * g  Cw*  C*   g Tw*  T*   B02 U 02 4 T*   M , Gm  , Gr  ,R  , Sc  , Ec  2 2 2 V0 V0 U 0 V0 U 0 ke k D C p Tw*  T*   (19) 3

Ra 

R*  Cw*  C*  V02 Tw*  T* 

, Pr 

C p k

Q0 k 1   , S  M  , Kr  1 2 ,   ,    Pr 1 1 4 R  2 3   V0 CP K V0

   

The mathematical statement of the problem is now complete and embodies the solution of Eqs. (15), (16) and (17) subject to the conditions (18).

3. METHOD OF SOLUTION: The problem posted in Esq. (15), (16) and (17) subject to the boundary conditions presented in Esq. (18) is non-linear partial differential equations and these cannot be solved in closed-form. So these equations can be reduced into the system of partial differential equations to set of ordinary differential equations, which can be solved by using multiple regular perturbation law. This is more economical and flexible from analytical point of view. Finding multiple regular perturbation method is an art rather than a science. In research it is useful to be responsive to suggestions from the physics. There is certainly no routine method appropriate to all problem, or even classes of problems. Instead one needs a determination to exploit the smallness of the parameter. Obtaining good numerical values for the solution is not the only quest of a perturbation method. One can hope that the analysis will reveal some physical insights through the simplified physics of the limiting problem. The behaviors of velocity, temperature, concentration, Skin-friction coefficient, Nusselt number and Sherwood numbers has been discussed in detail for variation of thermo physical parameters. The solution is assumed to be as

u  u0  y    ent u1  y       0  y    ent1  y   c  c0  y    e nt c1  y  

(20)

Substituting Esq. (20) in Equation (15), (16) and (17) and equating the harmonic and

 

non-harmonic terms, and neglecting the higher order terms o  2

we obtain the

following pairs of equations for (u0 ,θ0 ,C0) and (u1 ,θ1 ,C1) u0"  S u0  S  Gr0  GmC0

(21)

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8307 u1"   S  n u1    S  n   Gr1  GmC1

(22)

1"   n   1  2Ec u0' u1'  Ra C1

(23)

0"  0   Ec

u0 u0  RaC0 y y

(24)

C0"  Sc Kr C0  0

(25)

C1"  Sc  Kr  n  C1  0

(26)

The corresponding boundary conditions can be written as u0  0, 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   

(27)

Here primes denote differentiation w.r.t. y the equations (21) – (24) are still coupled and non-linear, whose exact solutions are not possible. So we expand u , u   0 1 0 1 First we solve equation (25) and equation (26) by using equation (27). Then we get C0  e

C1  e











ScKr y

(28)



Sc Kr  n  y

(29)

Now using multi parameter perturbation technique and assuming Ec  1. u  u  Ecu  0( )2 ,  0 00 01   0  00  Ec01  0( )2   u  u  Ecu  0( )2  1 10 11  1  10  Ec11  0( )2 

(30)

By using equations (30) in equations (21), (22), (23), (24) and equating the coefficients of like powers of Ec neglecting those of(𝐸𝑐)2 𝑎𝑛𝑑 0(𝜖)2 , we get the following set of differential equation "  Su  S  G   G C u00 r 00 m 0 00

(31)


8308 u"01  Su01  Gr01

(32)

"   S  n u    S  n  G   G C u10 r 10 m 1 10

(33)

"   S  n  u  G  u11 r 11 11

(34)

"   n      R C 10 a 1 10

(35)

' u' 11"   n   11  2u00 10

(36)

"     R C 00 a 0 00

(37)

''    u' u' 01 01 00 00

(38)

The corresponding boundary conditions are at y  0; u00  0, u01  0, u10  0, u11  0, 00  1, 01  0,10  1, 11  0   As y   u00  1, u01  0, u10  1, u11  0,00  0,01  0,10  0, 11  0 

(39)

Solve (31) – (38) subject to boundary Condition (39), we get u00  1  N3 e d y  N4 e fy  A1 eQy

(40)

u01  N11e dy  N12e2 fy  N13e2Qy  N14e d  f  y  N15e f Q y  N16ed Q y  N17 e  dy  A3e Qy

(41)

u10  1  N18e m y  N4ely  A4 e q y

(42)

u

11



N 29e d  m y  N30e d l  y  N31e  d  q  y  N32e  d  f  y  N33e  f l  y  N34e l  q  y  N35e Q  m y  N36e

 l  Q  y

 N37 e

 q  Q  y

 N38e

 my

 A6e

(43)

 qy

00  (1  N1 ) e d y  N1 e f y

(44)

10  (1  N2 ) e my  N2 ely

(45)

01  N5e dy  N6e2 fy  N7e2Qy  N8e d  f  y  N9el Q y  N10e d Q y  A2e dy

(46)



11



N 20e d  m y  N 21e d l  y  N 22e d  q  y  N 23e  d  f  y  N 24e  f l  y  N 25e  f  q  y  N 26e Q  m y  N 27 e

 l  Q  y

 N 28e

 q  Q  y

(47)

 A5e  my

By virtue of equations (11),(12),(13) we obtain for the velocity, temperature and

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8309 concentration as follows  f  Q  y   d  f  y  dy  fy  Qy  dy 2 fy 2Qy   EcN11e  EcN12 e  EcN13e  EcN14 e  EcN15e u  1  N 3e  N 4e  A1e  ( d Q ) y  dy  Qy  EcN16 e  EcN17 e  EcA3e    1  N e my  N ely  A e qy  EcN e d  m  y  EcN e  d  m  y  N e d  q  y  EcN e  d  f  y  18 4 4   29 30 31 32   f l  y  f q  y  Q  m y   l Q  y   q Q  y  dy  nt   e  EcN e  EcN e  EcN e  EcN e  EcN e  EcN e   33 34 35 36 37 38  S  n  y    EcA6 e    

(48)

 (1  N1 )e  dy  N1e  fy  EcN 5e  dy  EcN 6e 2 fy  EcN 7 e 2Qy  EcN 8e  d  f  y  EcN 9e  f Q  y   d  Q  y   dy  EcN10 e  EcA2 e     (49)   d  m y   d l  y d q y d  f  y  my  ly  N 2 e  EcN e  EcN e  EcN e  EcN e nt  (1  N 2 )e  20 21 22 23  e   EcN e   f l  y  EcN e   f  q  y  EcN e  Q  m y  EcN e  l Q  y  EcN e   q Q  y  EcA e  my    24 25 26 27 28 5

 

C e

 fy

  ent e

ly

(50)

Given the velocity field in the boundary lawyer, we can now calculate the skin friction at the wall as  mN3  fN 4  QA1  dEcN11  fEcN12  2QEcN13   d  f  EcN14   f  Q  EcN15   d  Q  EcN16  u   y y 0   dEcN17  QEcA3 

w 

 mN18lN 4  qA4   d  m  EcN   d  l  EcN   d  q  EcN   d  f  EcN   f  l  EcN  29 30 31 32 33    ent   qy     f  q  EcN   Q  m  EcN   l  Q  EcN   q  Q  EcN  mEcN  EcA e 6   34 35 36 37 38

(51)

We calculate the heat transfer coefficient in terms of Nusselt number as follows. Nu 

 d1  N1 )  fN1  dEcN5  2 fEcN 6  2QEcN 7   d  f  EcN8   f  Q  EcN9   d  Q  EcN10      y y 0  dEcA2 

 m(1  N 2 )  lN 2   d  m  EcN   d  l  EcN   d  q  EcN   d  f  EcN   f  l  EcN  20 21 22 23 24   ent     f  q  EcN   Q  m  EcN   l  Q  EcN   q  Q  EcN  mEcA  25 26 27 28 5  

(52)

Similarly the mass transfer coefficient in terms of Sherwood number, as follows S

h



c   f   ent l y y 0

(53)

4. RESULTS AND DISCUSSION: The formulation of the chemical reaction, Radiation absorption, thermal diffusion and thermal radiation, heat source and viscous dissipation effects on MHD convective flow and mass transfer of an incompressible, viscous fluid along a semi

8310


infinite vertical porous moving plate in a porous medium has been performed in the preceding section. This enables us to carry out the numerical calculation for the distribution of the velocity, temperature and concentration across the boundary layer for various values of the parameters. In the present study we have to select t=1.0, n=0.5, ϵ=0.2, A=0.5, Up=0.5 while Kr, Pr, Gr, Gm, ξ, k, M, Ra and R are varied over a range, which listed in the figure legends. For different value of permeability of the porous medium K, the velocity profile is plotted in Figure 1: Here we find that as the values of permeability of the porous medium K increases then it leads to the velocity increase. The variety of velocity and concentration profiles with y for various esteems in chemical reaction parameter Kr is appeared in the Figure 2: and Figure 14: these figures mirror that with increment in Kr there is diminishing in fluid velocity and concentration. For different value of Sc on the velocity is shown in the Figure 3: and Figure 15: Here we find that as the values of Schmidt number Sc increase then it leads to velocity and concentration decreases. The values of the Schmidt number are chosen to represent the presence of various species Oxygen (0.60), Carbon dioxide (Sc=0.94), the values of other parameters are chosen arbitrarily. The numerical estimations of the rest of the parameters are picked self-assertively. For different values of Ra on the velocity and temperature show in the Figure 4: and Figure 9: here the values of Ra increase then it leads to velocity and temperature decreases. Figures 5: and Figure 13: depicts the effect of Eckert number Ec on the velocity temperature profiles .It is revealed that velocity scores increases and diminishing in temperature with the increasing of the Eckert number Ec. Eckert number physically is a measure of frictional heat in the system. The variation in velocity profile with y for different values in Modified Grashof number is shown in Figure 6: The modified Grashof number Gm defines the ratio of the species buoyancy force to the viscous hydrodynamic force. As expected, the fluid velocity increases as Grashof number Gm increases. The influence of Magnetic parameter on velocity profiles is as shown in the Figure 7: various values of M increases then it leads to velocity increases. The variation in velocity profile with y for various values in Grashof numbers are shown in Figure 8: This figure reflects that with increase in Gr there is increase in fluid velocity due to improvement of the buoyancy force. For different values of the Prandtl number Pr, the Temperature profile is plotted in figure 10: Here we find that as the values of Prandtl number Pr increase then the Temperature profile decreases. To be realistic, the numerical values of Prandtl number Pr are chosen as Pr=0.71, and Pr=7.00, which correspond to air and water at 20 degree Celsius’s respectively. In the Figure 11: here we found that the values of R increase then it leads to temperature increases. In the Figure 12:the temperature decreases with increase of heat generator parameter ξ. Numerical values of the Skin-friction coefficient  w Nusselt number 𝑁𝑢 Sheer wood number 𝑆ℎ are shown below t=1,n=0.5,ϵ=0.03 ,A=0.5,ξ=0.3 From the

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8311 Table: 1 depict the effect of the Prandtl number Pr Schmidt number Sc, heat generation parameter (ξ ), Eckert number (Ec), radiation absorption(Ra) ,chemical reaction (Kr) and reaction rate R on the Skin friction coefficient 𝜏𝑤 and Nusselt number respectively. It is observed from this table that as Pr and Ra increase then it leads to skin friction coefficient 𝜏𝑤 and Nusselt number Nu decreases. Whereas R increases it leads to skin friction and Nusselt number increases. Also Skin friction increases with increase of M, ξ, GM, Gm, and Ec. But Skin friction decreases with increase of Sc. From the Table: 3 shows that the effect of the Schmidt number Sc and chemical reaction Kr for different values on Sherwood Sh. It is observed from the table that as Sherwood Sh decreases with increase of Sc and Kr.

5. VALIDATION OF THE RESULTS:

8312


Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8313


8314

Figure 15: Effect of Sc for different values on Concentration

Table 3: Effect of Sc and Kr on Sherwood numbers Sc

Kr

𝑆ℎ

0.42

2

-0.9319

0.60

2

-1.1138

0.94

2

-1.3941

0.60

2

-1.1138

0.60

4

-1.5738

0.60

6

-1.9270

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8315 Table 2: Effect of Pr,Sc,ξ,Ec,Ra,Kr,M,KGr,Gm and R on Skin friction  w and Nussle number N w Pr

SC

Ec

Kr

R

Ra

Gr

Gm

K

M

ξ

0.71 1 7

0.60

0.001

2

1

1

2

4

0.2

0.2

0.3

0.42 0.81 0.94 0.003 0.004 0.005 4 6 8 2 3 4 0.4 0.6 0.9 1 3 4 2 3 5 0.4 0.6 0.8 0.1 0.4 0.5 0.5 0.7 0.9

w 4.6258 4.5777 4.0439 4.6389 4.5985 4.5806 4.7829 4.8614 4.9400 4.5462 4.4777 4.4233 4.6772 4.7076 4.7292 4.6929 4.6705 4.6370 4.6185 4.6331 4.6403 4.0250 4.3254 4.9262 4.1975 4.0558 3.9908 4.6092 4.6589 4.6753 4.6267 4.6342 4.6443

Nw -18.4197 -23.3714 -113.7771 -18.4197 -18.4206 -18.4216 -13.7434 -11.3788 -9.9071 -16.4340 -17.0953 -18.0871 -

8316


CONCLUSION 1) Velocity and temperature profiles decrease with the increase of Thermal radiation parameter 2) Concentration profiles decrease near the plate with the increase of chemical reaction parameter Kr and Schmidt number Sc. 3) An increase in K, Gm,  and R leads to increase in both velocity and temperature. 4) Velocity profile decrease with the increase of Eckert number (Ec) and Magnetic field parameter M. 5) Sherwood Sh profile decrease with the increase of chemical reaction parameter Kr and Schmidt number Sc. Skin friction profile decreases with the increase of Eckert number (Ec).However, Nusselt number remains unchanged.

ACKNOWLEDGMENTS The authors express their gratitude to the referees for their constructive review of the paper and for valuable comments.

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[2]. Gurivireddy, Raju M.C, Mamatha and Varma S.V.K. (2016) Thermal Diffusion Effect on MHD Heat and Mass Transfer Flow past a Semi Infinite Moving Vertical Porous Plate with Heat Generation and Chemical Reaction Applied Mathematics, 7, 638-649. [3]. Hady. F. M, Mohamed. R. A, Mahdy. A, MHD Free convection flow along a vertical wavy surface with heat generation or absorption effect, Int. Comm. Heat Mass Transfer 2006; 33:1253-1263. [4]. Jeftasunzu and Prabhakar Reddy.B, Thermal Radiation and Viscous Dissipation Effects on MHD Heat and Mass Diffusion Flow Past a Surface Embedded in a Porous Medium With Chemical Reaction International Journal of Mathematics And its Applications 2016; 4: 91-103: 2347-1557. [5]. Takhar, H.S. and Ram, P.C. Magneto Hydrodynamics Free Convection Flow of Water at 400C through a Porous Medium. International Communications in Heat and Mass Transfer, 1994; 21: 371-376.

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8317 [6]. Siva Reddy, R. Srinivasa Raju.,Transient MHD free convective flow past an infinite vertical plate embedded in a porous medium with viscous dissipation, Meccanica, 2016; 51 Issue 5, pp. 1057 – 1068. [7]. R. Srinivasa Raju, Transfer effects on an unsteady MHD mixed convective flow past a vertical plate with chemical reaction. Engineering transactions eng. trans. 2017; 65(2):pp221–249, 2017 [8]. G. Jithender Reddy, J. A. Rao, R. Srinivasa Raju., Chemical reaction and radiation effects on MHD free convection from an impulsively started infinite vertical plate with viscous dissipation. Int. J. Adv. Appl. Math. andMech.2015; 2(3): pp 164 - 176. [9]. Rajeswari R, Jothiram B., Nelson V.K., Chemical reaction, heat and mass transfer on Nonlinear MHD boundary layer flow through a vertical porous surface in the presence of suction, Applied Mathematical Sciences, 2009;3:pp 49–52. [10]. Ibrahim. F. S, Elaiw. A. M., Bakr. A. A, Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a Semiinfinite vertical permeable moving plate with heat source and suction, Communications in Non-linear Science and Numerical Simulation 2008; 13: pp1056-1066. [11]. Mythili, Siva raj, Rashid, Heat generating/absorbing and chemically reacting Casson fluid flow over a vertical cone and flat plate saturated with non-Darcy porous medium, International Journal of Numerical Methods for Heat and Fluid Flow 2017;27 :156 - 173. [12]. Prakash Jagdish, Balamurugan, Vijayakumar. Thermo-diffusion and chemical reaction effects on MHD three dimensional free convective Couette flow. Walailak J.Sci. Tech ,2015; 12(9)805–30. [13]. Chamkha. A.J, 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 and Mass transfer 2003; 30:413-422. [14]. Srinivasa Raju R., Jithender Reddy G., Anand Rao Analytical and Numerical study of unsteady MHD free convection flow over an exponentially moving vertical plate with heat absorption, International Journal of Thermal Sciences, 107: 303–315, 2016. [15]. R. Srinivasa Raju, Reddy, B. Mahesh; Reddy, Jithender, Influence of angle of inclination on unsteady MHD Casson fluid flow past a vertical surface filled by porous medium in presence of constant heat flux, chemical reaction and viscous

8318


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Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8319 flow with Hall current. Rotation, radiation and heat absorption past an accelerated moving vertical plate with ramped temperature. Appl. Fluid Mech. 2015; 8(1): 7–20. [28]. Rajput, Sahu. Natural convection in unsteady hydro magnetic Couette flow through a vertical channel in the presence of thermal radiation. Int. J. Appl. Math. Mech 2012; 8(3)35–56.

APPENDICES

d i

 

, f  ScKr , Q  s , l  Sc( Kr  n), m 

n  ,q  s  n 

N1 

  GrN1  Gm   Ra  Ra Gr (1  N1 ) N4  N2  2 N3  2 2 f   , l  (n   ) , d2  S d S , ,

N5 

2 fdN3 N 4 d 2 N32  f 2 N 42  sA12 N8  N  N  6 7 2 2 ( d  f ) 2   , 4d   , 4f   , 4S   ,

N9 

2 fdA1 N 4Q (Q  f ) 2  

N13 

GrN8 GrN9 GrN10 GrN 7 N14  N16  N15  2 2 (d  f )  S , (Q  d )2  S , (Q  f )  S , 3S ,

N17 

2dmN3 N18 GrA2 Gr (1  N 2 ) GrN  Gm N18  2 N19  2 2 N 20  2 (d )  S , m  ( S  n) , l  ( S  n) , (d  m)2  (n   ) ,

N 21 

2dlN3 N19 2dqN3 A4 2 fmN 4 N18 N 22  N 23  2 2 ( d  l )  ( n   ) , ( d  q )  ( n   ) , ( f  m)2  (n   ) ,

N 24 

2 flN 4 N19 2dqN 4 A4 2QmN18 A1 N 25  N 26  2 2 ( f  l )  ( n   ) , ( f  q )  ( n   ) , (Q  m)2  (n   ) ,

N 27 

GrN 20 2QlN19 A1 2QqA4 A1 N 29  N 28  2 2 (d  m)2  ( S  n) , (Q  l )  (n   ) , (Q  q)  (n   )

N30 

GrN 21 GrN 22 N31  2 ( d  l )  ( S  n) , ( d  q ) 2  ( S  n) ,

,

N10 

2dA1 N3Q GrN 6 GrN N11  2 5 N12  2 d S , (Q  d )   4f 2 S ,


8320

N32 

GrN 23 GrN 25 GrN 24 N33  N34  2 2 ( f  m)  ( S  n) ( f  l )  ( S  n) , ( f  q ) 2  ( S  n) ,

N35 

GrN 26 GrN 27 GrN 28 N36  N37  2 2 (Q  q)  ( S  n) , (Q  l )  ( S  n) , (Q  q)2  ( S  n) ,

N38 

GrA5 m  ( S  n) , A1  (1  N3  N4 ) A2  (1  N5  N6  N7  N8  N9  N10 ) , 2

A3  ( N11  N12  N13  N14  N15  N16  N17 ) A4  (1  N18  N19 ) , , A5  ( N20  N21  N22  N23  N24  N25  N26  N27  N28 ) ,

A6  ( N29  N30  N31  N32  N33  N34  N35  N36  N37  N38 ) ,

Chemical Reaction and Viscous Dissipation Effects on MHD free Convective… 8321 Nomenclature A

Real positive constant

T

Dimensional free stream temperature (K)

B0

Magnetic induction (A.m-1)

qr*

Radiation heat flux density (W.m2 )

C

Non dimensional Concentration

Ra

Thermal radiation parameter

Cp

Specific heat for constant pressure(J. kg −1.K)

G m

Solutal Grashof number

D

Chemical molecular diffusivity (m2.S1 )

Gr

local temperature Grashof number

n

Mining

C

Concentration in the fluid far away from the plate( kg.m3 )

k

Thermal conductivity(W.m-1 .K-1)

Sc

Schmidt number

g*

acceleration due to gravity (m.S-2)

w

Skin-friction coefficient

Tw

Fluid temperature at walls (K)

R*

Radiation parameter

u

components of velocity vector in x direction

V0

Scale of suction velocity (m.S-1)

K

Permeability of porous medium

U0

Scale of free stream velocity

Kr

Chemical reaction parameter (m.S-1)

U *

Free stream velocity (K)

M

Magnetic parameter

T

Temperature of the fluid (K)

β

Spin gradient viscosity (K-1)

ke*

mean absorption coefficient

*

Coefficient of volumetric expansion (m3.kg-1)

Greek Symbols


8322

 s*

Stefan-Boltzmann constant(Wm-2K-4)

ɳ

Heat generation parameter

u *p

Direction of fluid flow



Fluid dynamic viscosity

N

Dimensionless material parameter



Density of the fluid( kg.m-3)

Ec

Eckert number



Fluid kinematic viscosity( m2.S-1)

Nu

Nusselt number



Non dimensional temperature K

Pr

Prandtl number



Scalar constant(