Chemical Reaction Effects on Heat and Mass Transfer in MHD … · 2020-01-22 · convective heat and mass transfer flow of a viscous incompressible fluid past a semi-infinite inclined

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Volume-6, Issue-3, May-June 2016

International Journal of Engineering and Management Research

Page Number: 447-459

Chemical Reaction Effects on Heat and Mass Transfer in MHD Boundary Layer Flow past an Inclined Plate with Viscous Dissipation

and Radiation in Porous Medium

V. Subhakanthi1, N.Bhaskar Reddy2 1,2

Department of Mathematics, Sri Venkateswara University, Tirupati, A.P, INDIA

ABSTRACT This paper analyzes the chemical reaction effects on heat and mass transfer in magnetohydrodynamic boundary layer flow past an inclined plate with viscous dissipation and radiation in porous medium. A suitable similarity transformation is used to transform the non linear system of partial differential equations into a system of ordinary differential equations. To solve the resultant system an efficient numerical technique Runge-Kutta fourth order is used along with shooting technique. The behavior of the velocity, temperature, concentration for variations in the thermo physical parameters are presented in graphs. The values of skin friction coefficient, Nusselt number and Sherewood number are also computed and are reported in tables. Keywords--- heat and mass transfer-MHD- radiation – chemical reaction-similarity parameter

I. INTRODUCTION The MHD boundary layer theory has a powerful place in the development of the magnetohydrodynamics. It has many applications in engineering problems such as geophysics, astrophysics, boundary layer control in the field of aerodynamics. So that the study of mixed convection flow and heat transfer for electrically conducting fluids over a surface has attracted much interest of researchers. Pioneer work is done by Ostrach [1] on convection flow and obtained a similarity solution of transient free convection flow over a semi infinite vertical plate using the integral method. Yih [2] presented the free convection effect on MHD combined heat and mass transfer of a moving permeable vertical surface. Non similar analytic solution for MHD flow and heat transfer in a third-order fluid past a stretching sheet was examined by Sajid et al. [3]. More recently the problem of magnetohydrodynamic flow over infinite surfaces has become more important due its applications in areas like nuclear fusion, chemical engineering, medicine, and high-speed, noiseless printing. MHD flow in the vicinity of infinite plate has

been investigated intensively by many investigators [4-9]. Investigation (theoretical and experimental) of natural convection flow under the influence of gravitational force over a solid body with different geometries embedded in a fluid-saturated porous medium is of considerable importance due to frequent occurrence of such fluid flow in nature as well as in science and technology viz. geothermal reservoirs, drying of porous solids, thermal insulators, heat exchanger devices, enhanced recovery of petroleum resources, underground energy transport etc. Nakayama and Koyama [10] carried out an analysis to study the free and forced convection flow in Darcian and non- Darcian porous medium. Similarity solution for free convection flow from a vertical plate fixed in a fluid-saturated porous medium was studied by Cheng and Minkowycz [11]. Makinde [12] examined MHD mixed convection flow and mass transfer over a vertical porous plate with constant heat flux embedded in a porous medium. Recently, Zeeshan and Ellahi [13] obtained series solutions for nonlinear partial differential equations with slip boundary conditions for non-Newtonian magnetohydrodynamic fluid in porous space. Viscous dissipation changes the temperature distributions by playing a role like energy source, which leads to influence the heat transfer rates. The effect of viscous dissipation depends on whether the plate is being cooled or heated. Alam et al. [14, 15] studied the combined effects of viscous dissipation and Joule heating on steady magneto hydrodynamic free convective heat and mass transfer flow of a viscous incompressible fluid past a semi-infinite inclined radiate isothermal permeable moving surface in the presence of thermophoresis. Singh [16] reported the effect of viscous dissipation on heat and mass transfer in MHD boundary layer flow over an inclined plate in porous medium. The radiative effects have dominent applications in physics and engineering. In space technology and high temperature processes the radiative heat transfer effects on different flows are very



important and very little is known about the effects of radiation on the boundary layer of a radiative-MHD fluid past a body. Viskanta and Grosh [17] were one of the initial investigators to study the effects of thermal radiation on temperature distribution and heat transfer in an absorbing and emitting media flowing over a wedge. To simplify the energy equation they used Rosseland approximation for the radiative flux vector. The radiative effects on heat transfer in nonporous and porous medium utilizing the Rosseland or other radiative flux model was investigated by number of researchers such as Raptis [18], Hall et al. [19], Bakier [20], Raptis and Perdikis [21] and Rao [22] etc. Recently Mishra et al. [23] done their work on the effect of radiation on free convection heat and mass transfer flow through porous medium in a vertical channel with heat absorption/ generation. Heat and mass transfer problems with chemical reactions are of importance in many processes, and therefore have received a considerable attention in recent years. In many chemical engineering processes, a chemical reaction between a foreign mass and the fluid does occur. These processes take place in numerous industrial applications, such as the polymer production, the manufacturing of ceramics or glassware, food processing. . Sharma and Nabajyoti Dutta[24] analyzed the effect of chemical reaction and thermal radiation effects on MHD boundary layer flow past a moving vertical porous plate. magnetohydrodynamic boundary layer flow of heat and mass transfer over a moving vertical plate with suction and chemical reaction is investigated by Ibrahim, and Makinde[25].

However, the interaction of chemical reaction and thermal radiation with MHD boundary layer flow of heat and mass transfer in porous medium in the presence of viscous dissipation received little attention. Hence, the object of the present paper is to study the combined effect of thermal radiation and a first-order chemical reaction on heat and mass transfer effects on MHD boundary layer flow past an inclined plate with viscous dissipation in porous medium in the presence of a uniform transverse magnetic field. The dimensionless governing equations of the flow, heat and mass transfer are solved numerically using Runge-Kutta fourth order method along with shooting technique. Numerical results are reported in figures for various values of the physical parameters of interest.

II. MATHEMATICAL ANALYSIS

A steady two dimensional hydromagnetic

convective flow of a viscous, incompressible electrically conducting fluid past an inclined plate with an acute angle to the vertical is considered in porous medium. x-axis is taken along the leading edge of the inclined plate and y -axis normal to it, i.e., the plate starts at x = 0 and extends parallel to the x axis and is of semi infinite length. A magnetic field of uniform strength is introduced normal to the direction of the flow. The uniform plate temperature and concentration are maintained at wT and wC . The plate temperature is

higher than the temperature of the fluid far away from the plate and concentration at the surface of the plate is greater than the free stream concentration. A steady flow parallel to the plate with free stream velocity is assumed. The convective flow starts under the simultaneous action of the buoyancy forces used by the variations in density due to temperature and species concentration differences. The magnetic Reynold number is assumed to be small, so that the induced magnetic field is neglected. The Hall effects term is neglected. The effects of viscous dissipation, radiation and chemical reaction have been taken into the account. Then, under the Boussinesq and the usual boundary layer approximations, the governing equations for the Darcy type flow, following Schlichting [16] and Nield and Bejan[17], are given by

Fig. 1 Physical configuration and coordinate system

0=∂∂

+∂∂

yv

xu (1)

( ) ( ) ( )∞∞∞ −′−−−+−+∂∂

=∂∂

+∂∂ Uu

ku

BCCgTTg

yu

yuv

xuu CT

υρ

σγβγβυ

20

2

2

coscos

(2)

∂∂

−+

∂∂

+∂∂

=∂∂

+∂∂

yq

cu

kcyu

cyT

yTv

xTu r

ppp ρυ

ρυα 12

2

2

2 (3)

( )∞−−∂∂

=∂∂

+∂∂ CCk

yCD

yCv

xCu 12

2

(4)

The boundary conditions of velocity, temperature and concentration are

ww CCTTvu ==== ,,0,0 at 0=y ∞∞∞ === CCTTUu ,,

as ∞→y (5)

where u, v,T and C are the fluid x-component of velocity, y-component of velocity, temperature and concentration respectively,υ - the kinematics viscosity, ρ - the density, σ - the Electric conductivity of the fluid, B0- the magnetic induction, Tβ and Cβ - the coefficients of Thermal and concentration expansions respectively,



α - the thermal diffusivity, pc - the specific heat at constant pressure, rq - the radiative heat flux, Dm - the mass diffusivity g- the gravitational acceleration, wT -the temperature of the hot fluid at the surface of the plate ,

wC - the species concentration at the plate surface, ∞T - the temperature of the fluid far away from the plate,

∞C - the free stream concentration, k′ - the permeability of the porous medium, k- the thermal conductivity , K1- chemical reaction rate and γ- the acute angle.

Continuity equation (1) is identically satisfied by the stream function ( )yx,ψ , defined as

xv

yu

∂∂

−=∂∂

=ψψ , (6)

By using Rosseland approximation, the radiative heat flux rq is given by

yT

kqr ∂

∂−=

4

*

*

34σ

(7)

where *σ is the Stefan – Boltzman constant and *k is the mean absorption coefficient. It should be noted that by using Rosseland approximation , the present analysis is limited to optically thick fluids. If the temperature differences within the flow are sufficiently small then equation (7) can be linearized by expanding 4T in a Taylor series about the free stream temperature

∞T which after neglecting the higher order terms takes the form

434 34 ∞∞ −≅ TTTT (8)

To transform equations (2) - (4) in to a set of ordinary differential equations, the following similarity transformations and dimensionless variables are

introduced.x

Uyυ

η ∞= , ( )ηυψ fxU∞= ,

( )ηfUu ′= ∞ , [ ]ffxUv −′= ∞ ηυ

21

( )

−−

=∞

∞

TTTT

w

ηθ , ( )

−−

=∞

∞

CCCC

w

ηφ ,

( )2∞

∞−=U

xTTgG wTr

β,

( )

2∞

∞−=U

xCCgG wcc

β

∞

=U

xBM

ρσ 20 ,

ρυ

=Pr ,

*

3*

316

kkTR ∞= σ ,

mDSc υ= , ( )∞

∞

−=

TTcUE

wpc

2

,∞

=U

xkKr 1 ,

( )∞∞

−=

TTkcUxNwp

υ,

∞′=

Ukxυδ , (9)

Substituting the equations (6) to (9 ) into the equations (2) to (5) we obtain

( ) 0coscos21

=++′−++′′+′′′ δδγφγθ MfGcGrfff

(10)

( ) ( ) 0Pr)(PrPr211 22 =′+′′+′+′′+ fNfEcfR θθ (11)

021

=−′+′′ φφφ ScKrScf (12)

The transformed boundary conditions are

1,1,0,0 ===′= φθff at 0=η

0,0,1 ===′ φθf as ∞→η (13)

where prime ( ´ ) denotes differentiation with respect to η. η is the similarity parameter, ( )ηf - the dimensionless stream function, ( )ηθ - the dimensionless temperature, ( )ηφ - the dimensionless concentration, ψ – the stream function, rG - the local thermal Grahsof number, cG - the local solutal Grahsof number, M- the magnetic field parameter, δ - the permeability parameter, γ - the inclination parameter , R- radiation parameter, , Pr - the Prandtl number , Ec - the Eckert number, N - viscous dissipation parameter , Sc - the Schmidt number , Kr - the chemical reaction parameter .

III. SOLUTION OF THE PROBLEM

The governing boundary layer equations (10) to (12) subject to the boundary conditions (13) are solved numerically by using Runge-Kutta fourth order method along with shooting technique. First of all higher order non-linear differential equations (10) to (12) are converted into simultaneous linear differential equations of first order and they are further transformed into initial value problem by applying the shooting technique(Jain



et al. [27]). The resultant initial value problem is solved by employing Runge-kutta fourth order technique. Numerical results are reported in Figures 2 - 34 for various values of the physical parameters of interest. From the process of numerical computation , the skin-friction coefficient, the Nusselt number and Sherwood number which are respectively proportional to ( )0f ′′ ,

( )0θ ′− and ( )0φ′− are also sorted out and numerical values are presented in a tabular form .

IV. RESULTS AND DISCUSSION

The governing equations (10)-(12) subject to the boundary conditions are solved as described in section 3. As a result of the numerical calculations, the dimensionless velocity , temperature and concentration are obtained and their behaviour have been discussed for variations in governing parameters. The results are presented in Figures from 2 – 34. Numerical results for the skin – friction, Nusselt number and Sherwood number are reported in Table 1 and Table 2. Fig. 2 and Fig. 3 show the effects of thermal Grashof number Gr and solutal Grashof number Gc on the velocity respectively. As shown the velocity increases as Gr and Gc increases. Physically the thermal Grashof number G r > 0 means heating of the fluid or cooling of the boundary surface and for cooling of the fluid or heating of the boundary surface Gr < 0 and the absence of free convection current corresponds to Gr = 0. The effect of magnetic parameter on velocity of the fluid is illustrated in Fig. 4. A decrease in the velocity is noticed on increasing the magnetic field parameter M, due to the fact that magnetic field exhorts a retarding force on free convective flow which retards the flow. i.e., the velocity boundary layer becomes thinner and thinner on increasing M. Fig. 5 represents the effect of porocity parameter δ on the velocity. An increase in the velocity of the fluid is observed. The variation of the velocity on increasing the inclination parameter γ is depicted in Fig. 6. It is noticed that the velocity of the fluid is decreasing. Fig. 7 demonstrates the effect of dissipation parameter due to porous medium on the velocity with an increasing N, we find that there is a decrease in the velocity. Fig. 8 shows the effect of radiation parameter on the velocity. On increasing the radiation parameter, there is an increase in the velocity. Velocity boundary layer was not effected by the Prandtl number as reported in the Fig. 9. The effects of Eckert number on velocity of the fluid in the boundary layer is represented in the Fig. 10. No change in the velocity is found for small values (0.01, 0.001) of Eckert number Ec which are of practical interest, by taking the values greater than 1 increase in the velocity is observed. A slight change in the velocity is observed on increasing the Schmidt number as shown in Fig. 11. The effects of chemical reaction parameter on momentum boundary layer is presented in the Fig. 12. It is found that the velocity slightly decreases on increasing the chemical reaction parameter.

The effects of thermal and solutal Grashof numbers Gr and Gc are illustrated in Fig. 13 and Fig. 14. A decrease in temperature of the fluid is found on increasing Gr and Gc. The effect of magnetic field parameter M on dimensionless temperature is depicted in Fig. 15. As we increasing the values of M decrease in the temperature is noticed. For different values of the porosity parameter δ the temperature profile is plotted in Fig. 16. It is observed that the thermal boundary layer thickness decreases as porosity parameter increases. Fig. 17 illustrates the effect of inclination parameter γ on temperature . A slight change in the temperature is seen on increasing γ . Fig. 18 displays the effect of dissipation parameter N on temperature of the fluid. Greater viscous dissipative heat causes rise in the temperature. The reason behind this is a rise in the value of viscous dissipation helps to improve the thermal conductivity of the fluid. The influence of radiation parameter R on temperature is shown in Fig. 19. It is noticed that temperature increases on increasing R due to increase in the radiation parameter provides more heat to the fluid that causes an enhancement in the temperature. The effect of Prandtl number on temperature of the fluid is presented in Fig. 20. It is clear that the dimensionless temperature decreases with an increase in Prandtl number Pr. The Prandtl number is inversely proportional to the thermal diffusivity of the fluid and due to this thermal boundary layer reduces. so, temperature gradient vanishes quicker for higher values of Pr . Fig. 21 displays the the effect of Eckert number on temperature of the fluid and it is noticed that temperature increases on increasing the Eckert number Ec . Fig. 22 shows the effect of Schmidt number Sc on the the thermal boundary layer thickness, a slight change in temperature on increasing Sc is noticed. Fig. 23 depicts the influence of chemical reaction parameter Kr on temperature of the fluid . It is found that temperature of the fluid is slightly rised on increasing Kr. Fig. 24 and Fig. 25 represent the effects of thermal and mass Grashof numbers Gr and Gc on concentration respectively. It is noticed that concentration decreases on increasing Gr and Gc. The effect of magnetic parameter M on concentration of the fluid is illustrated in Fig. 26. Decrease in concentration boundary layer thickness is observed on increasing the magnetic parameter. For different values of porosity parameter δ graphs for the concentration of the fluid are plotted in Fig. 27. A slight change in the thickness of the concentration boundary layer is found from the figure on increasing porosity parameter. Fig. 28 demonstrates the effect of inclination parameter γ on the concentration field. slight increase in the concentration is observed on increasing γ . The effects of dissipation parameter on the concentration boundary layer is reported in Fig. 29. Rising the dissipation heat causes reducing the thickness of the concentration boundary layer. Fig. 30 displays the effect of radiation parameter R on the concentration of the fluid. A Decrease in thickness of the concentration boundary layer is noticed on increasing R . Fig. 31 shows that Prandtl number Pr does not effect the thickness of the concentration boundary layer. The effect



of Eckert number on the concentration of the fluid is presented in the Fig. 32. A Decrease in the concentration is found on increasing the Eckert numbered. Fig. 33 demonstrates the effect of Schmidt number Sc on concentration and it is seen that concentration decreases on increasing the Schmidt number. Effect of the chemical reaction parameter on the concentration is reported in the Fig. 34. The concentration boundary layer becomes thinner on increasing the chemical reaction parameter Kr. This is due to the fact that the chemical reaction in this system results in consumption of the chemical and hence results in decrease of concentration.

V. CONCLUSIONS

From the present study we arrive at the following significant observations. By comparing the present results with that of Sing h et al. [ 26], it is found that there is a good agreement.

Increasing thermal Grashof number increases the velocity, but reduces the temperature and concentration. Increasing mass grashof number increases the velocity, but reduces the temperature and concentration. Increasing porosity parameter increases the velocity, but reduces the temperature and concentration. Increasing radiation parameter increases the velocity and temperature , but reduces the

concentration. Increasing Eckert number increases the velocity and temperature, but reduces the concentration. Increasing magnetic parameter increases the velocity, but reduces the temperature and concentration. Increasing dissipation parameter increases temperature and concentration, but decreases the velocity .Increasing Prandtl number slightly increases the concentration, but reduces the velocity and temperature. Increasing Schmidt number results a decrease in the velocity and concentration, but its influence is slight in the temperature. Increasing inclination parameter increases the temperature and concentration, but reduces the velocity. Increasing chemical reaction parameter enhances the temperature , but reduces the velocity and concentration. Skin friction coefficient and the Sherwood number increases with the increase in the Radiation parameter while Nusselt number decreases. Skin fraction coefficient and Nusselt number decreases where as sherewood number increases on increasing chemical reaction parameter.

.



Table1. showsthe variations of skin friction rate, nusselt number and sherwood number for different parameters.

• Increasing radiation parameterR increases skinfriction coefficient and Sherwood number, but reduces the Nusselt number

• Increasing chemical reaction parameter Kr increasesSherwood number, but reduces the skin friction coefficient and Nusselt number.

• Increasing dissipation parameter N, increases Sherwood number, but reduces skin friction coefficient and nusselt number.

• Increasing inclination parameter γ decreases skin friction coefficient, Nusselt number as well as Sherwood number.

• Increasing buoyancy parameters Gr and Gc increases skin friction coefficient, Nusselt number and Sherwood number.

• Increasing magnetic parameter decreases skin friction coefficient, Nusselt number and Sherwood number.

Table 2. shows the comparison of present work with that of Singh et al.[26]. From the table it is clear that there is a good agreement of present results with that of Singh et al.[26].

Table1: Variation of ( )0f ′′ , ( )0θ ′− and ( )0φ′− for different values of Gr, Gc, M, Pr, R,

Kr, γ and N

Table 2:comparison of present results for ( )0f ′′ , ( )0θ ′− and ( )0φ′− at the plate for different values of Gr, Gc , M,

Pr, Ec, Sc, N and γ for Kr = 0 , R = 0 with that of Singh et al. [26].

Gr Gc M Pr R Kr γ N ( )0f ′′ - ( )0θ ′ - ( )0φ ′ 0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

1 1 1 1 1 1 1 1.5 2.0 1 1 1 1 1 1 1 1 1 1 1

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.9 7.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1.5 2.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1.5 2.0 2.0 0.5 0.5 0.5 0.5 0.5

45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 60o 90o 45o 45o 45

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.4 0.6

0.282752 0.38481

0.486086 0.375073 0.375073 0.470329 0.231404 0.183974 0.156977 0.231404 0.231172 0.225703 0.231611 0.231736 0.231821 0.22875

0.226445 0.224545 0.231611 0.202367 0.131566 0.231847 0.229454 0.227616

0.26876 0.272716 0.276285 0.272143 0.272143 0.275448 0.266625 0.262643

0.2601 0.266625 0.271239 0.391472 0.262536 0.260061 0.258401 0.26239

0.262376 0.262179 0.262536 0.261605 0.259349 0.25805

0.249072 0.240241

0.391036 0.393195 0.395309 0.392854 0.392854 0.394765 0.389937 0.387701 0.386416 0.389937 0.389927 0.389705 0.389946 0.389952 0.389956 0.497741 0.591098 0.674002 0.389946 0.38933 0.38783

0.389957 0.497765 0.591131

Gr Gc M Pr Ec Sc N γ P.k.sing h et.al [ 26] Present work ( )0f ′′ - ( )0θ ′ - ( )0φ ′ ( )0f ′′ - ( )0θ ′ - ( )0φ ′



REFERENCES [1] S. Ostrach, An analysis of Laminar Free Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force. Technical Note, NACAReport, Washington, 1952. [2] K. A. Yih, Free Convection effect on MHD Coupled Heat and Mass Transfer of Moving Permeable Vertical Surface. Int. Commun, Heat Mass Transfer, 26, 95-104, 1999. [3] M. Sajid, T. Hayat and S. Asghar, Non-Similar Analytic Solution for MHD Flow and Heat Transfer in a Third- Order Fluid over a Stretching Sheet. InternationalJournal of Heat and Mass Transfer, 50, 1723-1736, 2007. [4] A. S. Gupta, Laminar Free Convection flow of an Electrically Conducting Fluid FrombVertical Plate with Uniform Surface Heat Flux and Variable Wall Temperature in the Presence of a Magnetic field. Zeitschrift fur Angewandte Mathematik und Physik , 13, 4, 324–333, 1962. [5] H. S. Takhar, A. A. Raptis, and C. P. Perdikis, MHD asymmetric flow past a semi-infinite moving plate. Acta Mechanica, 65 , 1–4, 287–290, 1987. [6] I. Pop, M. Kumari, and G. Nath, Conjugate MHD flow past a flat plate. Acta Mechanica, 106, 3-4, 215–220, 1994. [7] H. S. Takhar and G. Nath, Similarity solution of unsteady boundary layer equations with a magnetic field. Meccanica , 32 , 2, 157–163, 1997. [8] I. Pop and T. Y. Na, A Note on MHD flow over a stretching permeable surface. Mechanics Research Communications , 25 , 3, 263–269, 1998.

[9] H. S. Takhar, A. J. Chamkha and G. Nath, Unsteady flow and heat transfer on a semi infinite flat plate with an aligned magnetic field. International Journal of Engineering Science , 37, 13, 1723–1736, 1999. [10] A. Nakayama and H. Koyama, A general similarity transformation for combined free and forced convection flows within a fluid saturated porous medium. ASME J. Heat Transfer, 109, 1041-1045, 1987. [11] P. Cheng and W. J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J.Geophys. Res., 82 , 2040-2044, 1977. [12] O.D. Makinde, On MHD boundary-layer flow and mass transfer past a vertical plate in A porous medium with constant heat flux. Int. J. Numer. Methods Heat Fluid Flow, 19 ,546-554, 2009. [13] A. Zeeshan and R. Ellahi, Series solutions for nonlinear partial differential equations with slip boundary conditions for non-Newtonian MHD fluid in porous space. Applied Mathematics & Information Sciences. 7, 1, 253-261, 2013. [14] M.S. Alam, M.M. Rahman , M. A. Sattar, Effects of variable suction and thermophoresis on steady MHD com bined free forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermal radiation, Int. J. Therm. Sci. 47, 6, 758-765, 2008. [15] M. S. Alam , M. M. Rahman , M. A. Sattar, On the effectiveness of viscous dissipation and Joule heating on steady magnetohydrodynamic heat and mass transfer flow over an inclined radiate isothermal permeable

0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

1 1 1 1 1 1 1 1.5 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.9 7.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.06 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.2 0.6 2.6 0.22 0.22 0.22 0.22 0.22 0.22

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.4 0.6 0.1 0.1 0.1

45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 30 45 60

0.285508 0.386253 0.485873 0.286288 0.389048 0.491221 0.234691 0.186665 0.159254 0.234691 0.234353 0.227563 0.234713 0.234757 0.234800 0.234691 0.233772 0.229627 0.235164 0.236139 0.237083 0.257736 0.234691 0.204587

0.278131 0.283853 0.28932 0.278216 0.284125 0.289787 0.275034 0.269133 0.265511 0.275034 0.281863 0.439544 0.274521 0.273494 0.272468 0.275034 0.274413 0.27445 0.265999 0.24722 0.229222 0.276465 0.275034 0.273119

0.262422 0.265106 0.26771 0.262467 0.265264 0.268009 0.261048 0.257986 0.256215 0.261048 0.261029 0.260676 0.261049 0.261052 0.261054 0.261048 0.279626 0.36543 0.261076 0.261134 0.26119 0.261678 0.261048 0.260222

0.282752 0.38481 0.486086 0.375073 0.375073 0.470329 0.231404 0.183974 0.156977 0.231404 0.231172 0.225703 0.231622 0.231644 0.231666 0.231611 0.226584 0.220376 0.231847 0.229454 0.227616 0.231611 0.202367 0.131566

0.26876 0.272716 0.276285 0.272143 0.272143 0.275448 0.266625 0.262643 0.2601 0.266625 0.271239 0.391472 0.26228 0.261767 0.261254 0.262536 0.262276 0.261971 0.25805 0.249072 0.240241 0.262536 0.261605 0.259349

0.391036 0.393195 0.395309 0.392854 0.392854 0.394765 0.389937 0.387701 0.386416 0.389937 0.389927 0.389705 0.389947 0.3899 0.389949 0.389946 0.575737 0.856657 0.389957 0.497765 0.591131 0.389946 0.38933 0.38783



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Chemical Reaction Effects on Heat and Mass Transfer in MHD … · 2020-01-22 · convective heat and mass transfer flow of a viscous incompressible fluid past a semi-infinite inclined