Effects of Radiation on MHD Boundary Layer Flow of Combined Heat and Mass Transfer … · 2020-01-22 · non- Newtonian flows with heat transfer analysis, MHD and non- linear slip

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Volume-7, Issue-1, January-February 2017

International Journal of Engineering and Management Research

Page Number: 137-150

Effects of Radiation on MHD Boundary Layer Flow of Combined Heat

and Mass Transfer over a Moving Inclined Plate in a Porous Medium

with Suction and Viscous Dissipation in Presence of Hall Current and

Chemical Reaction

V.Subhakanthi1, N. Bhaskar Reddy

2

1,2Department of Mathematics, SVU, Tirupati, INDIA

ABSTRACT This paper analyzes the chemical reaction and

radiation effects on heat and mass flow over a moving

inclined plate in porous medium with suction and viscous

dissipation in presence of Hall current. A suitable similarity

transformation is used to transform the nonlinear system of

partial differential equations into a system of ordinary

differential equations.to solve the resultant system a well

tested numerical technique Runge-Kutta fourth order is

used along with shooting technique. The behavior of

primary and secondary velocities, temperature and

concentration for variations in thermo physical parameters

are presented in graphs. transfer in magnetohydrodynamic

boundary layer The values of skin friction coefficient,

Nusselt number and Sherwood number are also computed

and are reported in tables.

Keywords-- heat and mass transfer-MHD- radiation-

viscous dissipation-chemical reaction

I. INTRODUCTION

The free convection processes involving the

combined mechanism of heat and mass transfer are

encountered in many natural processes, in many

industrial applications and in many chemical processing

systems. The study of free convective mass transfer flow

has become the object of extensive research as the

effects of heat transfer along with mass transfer effects

are dominant features in many engineering applications

such as rocket nozzles, cooling of nuclear reactors, high

sinks in turbine blades, high speed aircrafts, chemical

devices and process equipments.

The study of MHD flows have stimulated more

attention due its important applications in different

subject areas such as astrophysical, geophysical and

engineering problems. Free convection in electrically

conducting fluids through an external magnetic field has

been a subject of considerable research interest of a large

number of scholars for a long time due to its

miscellaneous applications in the fields of nuclear

reactors, geothermal engineering, liquid metals and

plasma flows etc. Fluid flow control under magnetic

forces is also applicable in MHD generators and a host

of magnetic devices used in industries. Jha[1] explained

the problem of MHD free convection and mass transfer

flow past an impulsively moving vertical plate through

porous medium when the vertical plate moves with

uniform acceleration and applied magnetic field is fixed

with the moving plate. Pioneer work on convective flow

in porous media are presented in the form of books and

monographs by Ingham and Pop[2], Ingham et al. [3],

Vafai[4] and Nield and Bejan [5].

A two dimensional steady MHD mixed

convection and mass transfer flow over a semi- infinite

porous inclined plate in the presence of thermal radiation

with variable suction and thermophoresis was studied by

Alam et al. [6]. Orthan Aydm and Ahmet Kaya[7]

considered MHD mixed convective heat transfer flow

about an inclined plate. Gnaneswara Reddy and Bhaskar

Reddy[8] presented mass transfer and heat generation

effects on MHD free convection flow over an inclined

vertical surface in a porous medium. Recently, Hitesh

Kumar[9] done his work on the heat transfer MHD

boundary layer flow through a porous medium.

The role of thermal radiation is of major

importance in some industrial applications such as glass

production and furnace design and in space technology

applications, cosmical flight aerodynamics, propulsion

systems, plasma physics and craft re-entry,

aerothermodynamics which operate at high

temperatures. Solving the governing equations become

quite complicated when radiation is taken into account

and hence many difficulties arise while solving such

equations. Viskanta and Grosh [10] 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.

They used Rosseland approximation for the radiative

flux vector to simplify the energy equation. Suneetha et



al. [11] studied the effects of thermal radiation on

unsteady hydro magnetic free convection flow over an

impulsively started vertical plate with variable surface

temperature and concentration. Gnaneswara Reddy and

Bhaskar Reddy[12] reported the radiation and mass

transfer effects on unsteady MHD free convection flow

past a vertical porous plate with viscous dissipation by

using finite element method. Recently, Narahari and

Ishak[13] carried out an analysis to study the effects of

thermal radiation on unsteady free convection flow of an

optically thick fluid past a moving vertical plate with

Newtonian heating. Their interesting cases are impulsive

movement of the plate, uniformly accelerated movement

of the plate and exponentially accelerated movement of

the plate.

The viscous dissipation heat in the natural

convective flow is important, when the flow field is of

extreme size or at low temperature or in high

gravitational field.. such effects are also important in

geophysical flows and also in certain industrial

operations and are usually characterized by the Eckert

number. When the viscosity of the fluid is high, the

dissipation term becomes important. For many cases,

such as polymer processing which is operated at a very

high temperature, viscous dissipation cannot be

neglected. An extensive work on the viscous dissipative

heat effects on the study free convection and on

combined free and forced convection flows has been

done by Ostrach [14-18]. Numerical analysis of steady

non- Newtonian flows with heat transfer analysis, MHD

and nonlinear slip effects was examined by Ellahi and

Hameed [19]. Ellahi et al. [20] explained the influence of

slip on steady flows in viscous fluid with heat and mass

transfer. Recently, Vajravelu[21] investigated unsteady

convective boundary layer flow of a viscous fluid at a

vertical surface with different fluid properties.

It may be noted that when the density of an

electrically conducting fluid is low and /or applied

magnetic field is strong (Sutton and Sherman[22]), the

effects of Hall current become significant. It plays an

important role in determining flow features of the fluid

flow problems because induces secondary flow in the

fluid. Therefore it is of considerable interest to study the

effects of Hall current on MHD fluid flow problems.

Sato[23], Sherman and Sutton[22] have analyzed the

Hall effects on the steady hydromagnetic flow between

two parallel plates. These effects in the unsteady cases

were reported by Pop[24]. Seth et al. [25] present the

effects of Hall current on unsteady hydromagnetic

natural convection transient flow of a viscous,

incompressible, electrically conducting and heat

absorbing fluid past an impulsively moving vertical plate

fixed in a fluid saturated porous medium, under

boussinesq approximation, taking into the effects of

thermal diffusion when temperature of the plate has a

temporarily ramped profile. Flow through a porous

medium bounded by a vertical surface in presence of

Hallcurrent was considered by Sudhakar[26].

Gopichand[27] analyzed the unsteady stretching surface

in porous medium and explained the viscous dissipation

and radiation effects on MHD flow over it.

Another important aspect, which influences

heat transfer processes is the suction /injection. It is well

known that the effects of injection on the boundary layer

flow area of interest in reducing the drag force. Suction

and heat transfer characteristics were addressed by

Youn[28]. The effects of suction or injection on the free

convection boundary layers induced by a heated vertical

plate fixed in a saturated porous medium with an

exponential decaying heat generation were presented by

Ali[29]. Suction or blowing of a fluid through the

bounding surface can significantly change the flow field.

In general, suction tends to increase the skin friction,

whereas injection acts in the opposite manner. In many

engineering activities such as in the design of thrust

bearing and radial diffusers, and thermal oil recovery the

process of suction/ blowing plays a significant role

because of its importance.

Bhattacharya [30] explained the effects of

radiation and heat source/sink on unsteady MHD

boundary layer flow and heat transfer over a shrinking

sheet with suction /injection. The flow is permeated by

an externally applied magnetic field normal to the plane

of the flow in his work. The self similar equations

corresponding to the velocity, temperature and

concentration fields are obtained, and then solved

numerically by finite difference method using quasi

linearization technique. Lin et al. [31] investigates study

laminar boundary layer flow of power law fluids past a

flat surface with suction or injection and magnetic

effects. Recently, Cao et al. [32] analyzed the MHD

Maxwell fluid over a stretching plate with suction or

injection in the presence of nano particles.

Heat and mass transfer problems in the

presence of chemical reaction are of importance in many

processes, and have therefore received a considerable

amount of attention in recent times. Possible applications

can be found in processes such as drying, distribution of

temperature and moisture over agricultural fields and

groves of fruit trees, damage of crops due to freezing,

evaporation at the surface of a water body and energy

transfer in a wet cooling tower, and flow in a desert

cooler. In many chemical engineering processes,

chemical reactions take place between a foreign mass

and the working fluid which moves due to the stretching

of a surface. The order of the chemical reaction depends

on several factors. One of the simplest chemical reaction

is the first-order reaction in which the rate of reaction is

directly proportional to the species concentration .

Deka et al. [33] reported the effect of first order

homogeneous chemical reaction on the process of an

unsteady flow over an infinite vertical plate with a

constant heat and mass transfer. Muthucumaraswamy

and Ganesan [34] studied the flow characteristics in an

unsteady upward motion of an isothermal plate by taking

chemical reaction and injection into account. Reddy et

al. [35] analyzed the effects of radiation and chemical

reaction on an unsteady hydromagnetic natural



convection flow over a moving vertical plate in a porous

medium.

Owing to the above mentioned studies, the

author made an attempt to investigate the combined

effects of chemical reaction, thermal radiation and Hall

current on the hydro magnetic free convective flow of

heat and mass transfer over a moving inclined plate in a

porous medium with suction and viscous dissipation.

The governing boundary layer equations (4.2.11) to

(4.2.14) subject to the boundary conditions (4.2.15) are

solved numerically by using Runge-Kutta fourth order

method along with shooting technique.

II. MATHEMATICAL ANALYSIS

A two dimensional steady laminar MHD

viscous incompressible electrically conducting and

chemically reacting fluid along a moving inclined plate

with an acute angle γ embedded in a porous medium, in

the presence of suction is considered. x- direction is

taken along the leading edge of the inclined plate and y

is normal to it and extends parallel to x -axis. Let wT

(> T ) be the uniform plate temperature, where T∞ is

the temperature of the fluid far away from the plate. Let

u, v and w be the velocity components along the x and y

axis and secondary velocity component along the z axis

respectively in the boundary layer region. Let wC be the

concentration of the fluid at the surface of the plate and

C be the free stream concentration. The flow is

subjected to the effect of thermal radiation and a

transverse magnetic field of strength B0, which is

assumed to be applied in the positive y direction, normal

to the surface. The induced magnetic field is also

assumed to be small compared to the applied magnetic

field so it is neglected. All the fluid properties are

assumed to be constant except for the density variations

in the buoyancy force term of the linear momentum

equation. The Hall effects and viscous dissipation are

taken into account .Joule heating term is neglected. The

sketch of the physical configuration and coordinate

system are shown in Fig 1.

Figure1 Physical configuration and coordinate system

Under the above assumptions the boundary layer

equations describing the flow field under consideration

are

The boundary conditions for the velocity,

temperature and concentration fields are

bx+C=C=C

,ax+T=T=T,0=w,V=v,ax=u

∞w

∞w at 0y

CCTTwu ,,0,0 as y

(6)

where u, v and w be the velocity components along the

x- axis and y- axis and secondary velocity component

along the z - axis respectively in the boundary layer

region. T and C are the temperature and concentration of

the fluid respectively. g- the gravitational acceleration,

T and c - the coefficients of thermal and

concentration expansions,γ - the acute angle or

inclination parameter, 0B - the magnetic field induction,

m- the hall parameter, - the kinematic viscosity, k -

permeability of the porous medium, - thermal

diffusivity, pc - the specific heat at constant pressure,

rq - the radiative heat flux, D- the mass diffusivity,

1k - chemical reaction rate, wT and wC - the temperature

and concentration of the fluid at the surface of the plate,

T - the temperature of the fluid far away from the plate

and C - the free stream concentration .

The second and third terms on the right hand

side of equation (4) are the viscous dissipative heat and

radiative heat flux respectively. The second term on right

hand side of the equation (5) is the species chemical

reaction.

0=y∂

v∂+

x∂

u∂ (1)

2

2

y∂

u∂υ=

y∂

u∂v+

x∂

u∂u +

( ) ( )( )

( ) uk′

υmw+u

m+1ρ

BσγcosCCβg+γcosTTβg 2

2

0

∞c∞T

(2)

( )( ) w

k′

υwmu

m+1ρ

Bσ+

y∂

w∂υ=

y∂

w∂v+

x∂

w∂u

2

2

0

2

2

(3)

+y∂

T∂α=

y∂

T∂v+

x∂

T∂u 2

2

y∂

q∂

cρ

1_

y∂

w∂+

y∂

u∂

cρ

υ r

p

22

p

(4)

CCk

y

CD

y

Cv

x

Cu 12

2

(5)



Continuity equation (1) is identically satisfied by the

stream function yx, , defined as

xv

yu

, (7)

By using Rosseland approximation, the radiative heat

flux rq is given by

y

T

kqr

4

*

*

3

4 (8)

Where* 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 (4.2.6)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 (9)

To transform equations (2) to (4) into a set of

ordinary differential equations, the following similarity

transformations and dimensionless variables are

introduced

Substituting the equations (7) to (10) into the

equations (2) to (5) we obtain

γcosφG+γcosθG+f′′f2

1+f′′′ cr

0=f′kgm+1

mMf′

m+1

M022

(11)

0112

1002200

kggm

Mf

m

Mmfgg

(12)

0Pr2

11

2

0

2

gfEcfR

(13)

02

1 ScKrScf (14)

The corresponding boundary conditions are

1,1,1,0, 0 fgFf w at 0

00 gff as (15) (15)

where prime ( ' ) denotes differentiation with respect to η.

η - the similarity parameter, f is the

dimensionless stream function,, - the

dimensionless temperature, - the dimensionless

concentration, ψ – the stream function, M- the magnetic

field parameter 0g - the secondary velocity parameter,

rG - the local thermal Grahsof number, cG - the

local solutal Grahsof number, K- the permeability

parameter, γ - the inclination parameter, m – Hall

current parameter, R- radiation parameter, Pr - the

Prandtl number, Ec - the Eckert number, Sc - the

Schmidt number, Kr - the chemical reaction parameter,

wF - the suction parameter.

III. SOLUTION OF THE PROBLEM

The governing boundary layer equations (11) to

(14) subject to the boundary conditions (15) are solved

numerically by using Runge-Kutta fourth order method

along with shooting technique. First of all higher order

non-linear differential equations (11) to (14) 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.[36]). The resultant initial value problem is solved

by employing Runge-kutta fourth order technique.

Numerical results are reported in figures 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 ,

0 and 0 are also sorted out and numerical

values are presented in a tabular form.

IV. RESULTS AND DISCUSSION

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 viz., M- the

magnetic field parameter 0g - the secondary velocity

parameter, rG - the local thermal Grahsof number,



cG - the local solutal Grahsof number, K- the

permeability parameter, γ - the inclination parameter, m

– Hall current parameter, R- radiation parameter,, Pr -

the Prandtl number, Ec - the Eckert number, Sc - the

Schmidt number, Kr - the chemical reaction parameter,

wF - the suction parameter. The results are presented in

Figures from 4.2 – 1.38. Numerical results for the skin –

friction, Nusselt number and Sherwood number are

reported in Tables 1 and 2. A parametric study is carried

out to demonstrate the effects of governing parameters

on velocity, temperature and concentration profiles.

Fig. 2 and Fig. 3 show the effects of thermal

Grashof number Gr and solutal Grash of number Gc on

the velocity respectively. As shown the velocity

increases as Gr and Gc increases. Physically Gr > 0

means heating of the fluid or cooling of the boundary

surface, Gr < 0 means cooling of the fluid or heating of

the boundary surface and Gr = 0 corresponds to the

absence of free convection current. The effect of

inclination parameter on the velocity of the fluid is

shown in Fig. 4. It is noticed that increasing the

inclination parameter results a decrease in the velocity.

Fig. 5 displays the effect of magnetic field paramater on

the velocity of the fluid. The presence of a magnetic

field in an electrically conducting fluid induces a force

called Lorentz force, which opposes the flow. This

resistive force tends to slow down the flow, so the effect

of increase in M is to decrease the velocity. Fig. 6

illustrates the effect of Hall parameter m on the velocity.

It is observed that the velocity of fluid increases on

increasing the Hall parameter. Fig.7 shows the effect of

the thermal conductivity on the velocity of the fluid. It is

seen that velocity decreases on increasing the thermal

conductivity. Fig. 8 represents the effect of radiation

parameter on the velocity, and it is noticed that the effect

of radiation parameter on the velocity of the fluid is

slight.

The effect of the Prandtl number on the velocity

of the fluid is illustrated in Fig. 9. On increasing the

Prandtl number, the velocity of the fluid flow increases.

Fig. 10 depicts the effect of Eckert number on the

velocity of the boundary layer. A slight change in the

velocity is seen. The effect of Schmidt number on the

velocity of the fluid is shown in Fig. 11. A slight

decrease in the velocity of the fluid on increasing the

Schimdt number is noticed. The effect of chemical

reaction parameter on the velocity of the fluid flow is

illustrated in Fig. 12. It is found that on increasing the

chemical reaction parameter the velocity of the fluid is

decreasing. Fig. 13 shows the effect of suction parameter

on the velocity. It is observed that the velocity increases

on increasing the suction parameter. The effect of

magnetic parameter M on the secondary velocity of the

fluid is shown in Fig. 14. Increase in the secondary

velocity of the fluid is observed on increasing the

magnetic parameter. Fig. 15 shows the effect of radiation

parameter on secondary velocity of the fluid. On

increasing the radiation parameter secondary velocity of

the fluid is found to be decreased. The effect of

chemical reaction parameter on the secondary velocity of

the fluid is shown in Fig. 16. Decrease in the secondary

velocity is noticed from the figure on increasing the

chemical reaction parameter. Fig. 17 and Fig. 18

illustrates the effects of thermal and mass Grashofer

numbers Gr and Gc respectively on the temperature of

the fluid. Decrease in the temperature is noticed. The

effect of the Magnetic field parameter is shown in Fig.

19, increase in the temperature of the fluid is observed.

The effects of thermal conductivity on temperature of

the fluid is depicted in the Fig. 20. It is shown from the

figure that temperature increases on increasing the

thermal conductivity. The effects of radiation on

temperature of the fluid is illustrated in Fig. 21.

Decrease in the temperature of the fluid on increasing

the radiation is observed. Effects of Prandtl number on

temperature of the fluid is shown in Fig. 22. Increase in

the temperature is noticed. Hall parameter decreases the

temperature of the fluid as shown in Fig. 23. Effects of

Eckert number on thermal boundary layer is illustrated

in Fig. 24. Temperature of the fluid in the boundary

layer increases on increasing the Eckert number.

Chemical reaction effect on the thermal boundary layer

is depicted in Fig. 25. Thermal boundary layer thickness

increases on increasing the chemical reaction. Thermal

boundary layer thickness is increased on increasing the

inclination parameter γ as shown in Fig. 26. The effect

of suction parameter on the temperature is depicted in

Fig. 27. It is seen that temperature increases on

increasing the suction parameter.

Fig. 28 and Fig. 29 illustrate the effects of

thermal and mass Grashof numbers Gr and Gc on the

species concentration field. Concentration of the fluid in

the boundary layer decreases on increasing Gr and Gc.

Concentration of the fluid in the boundary layer

increases on increasing the magnetic parameter M as

shown in Fig. 30. The effect of hall parameter m on

concentration field was displayed in Fig. 31. A decrease

in the concentration is noticed on increasing the hall

parameter.

Fig. 32 depicts the effect of thermal

conductivity on the concentration field . Concentration

of the fluid increases on increasing the thermal

conductivity. The effect of Prandtl number on

concentration field is illustrated in Fig. 33. Effect of

Eckert number on concentration field was displayed in

Fig. 34. Decrease in the concentration on increasing the

Eckert number is noticed. Fig. 35 represents the effect of

Schmidt number on concentration field. It is interesting

to note that the chemical species concentration also

decreases within the boundary layer with an increase in

Schmidt number due to the combined effects of

buoyancy forces and species molecular diffusivity. Fig.

36 depicts the influence of chemical reaction rate on

concentration field. An increase in the value of chemical

reaction parameter decreases the concentration of

species in the boundary layer. This is due to the fact that

chemical reaction in this system results in consumption

of the chemical and hence results in decrease of

concentration. Fig.37.demonstrates the effect of



inclination parameter on the concentration .There is a

slight change in the concentration of the fluid is

observed on increasing the inclination parameter γ. Fig.

38 shows the effect of suction parameter on the

concentration. It is found that concentration increases on

increasing the suction.

Table. I shows the comparison of results of

present work with that of Ali et al.[37], and it is found

that there is a good agreement. Numerical computations

of skin friction coefficient, Nusselt number and

Sherwood number for different values of Gr, Gc, M, m,

K, Ec, Sc, γ, Fw, R and Kr =0 are reported in Table II

and Table 3.

V. CONCLUSIONS

A two dimensional steady laminar MHD

viscous incompressible electrically conducting and

chemically reacting fluid along a moving inclined plate

with an acute angle γ embedded in a porous medium, in

the presence of suction has been studied. In addition to

this, thermal radition and external magnetic field

strength are also considered. The governing boundary

layer equations are solved numerically using well tested,

highly efficient Runge-Kutta fourth order method along

with shooting technique. From the present study we

arrive at the following significant observations.

On comparing the present results with previous work, it

is found that there is a good agreement.

Increasing buoyancy ratio parameters Gr and

Gc increases the velocity, but decreases the

temperature and concentration.

Increasing the magnetic field parameter

increases the temperature and the concentration,

but reduces the velocity.

Increasing the hall parameter enhances the

velocity, but reduces the temperature and

concentration.

Increasing the thermal conductivity parameter

rises the temperature and concentration, but

decreases the velocity.

Increasing the radiation parameter results a

slight change in the velocity, but reduces the

temperature.

Increasing the prandtl number increases the

velocity, temperature as well as the

concentration.

Increasing the Eckert number results a slight

change in the velocity, increases the

temperature, but decreases concentration.

Increasing the Schmidt number decreases the

velocity and the concentration

Increasing the suction parameter results an

increase in the velocity, temperature and the

concentration.

Increasing the chemical reaction parameter

increases the temperature, but reduces the

velocity and concentration.

Increasing inclination parameter decreases the

velocity and increases the temperature and there

is a slight change in the concentration.

Secondary velocity increases on increasing the

magnetic parameter, but reduces o increasing

the radiation and chemical reaction parameters.

Skin fraction coefficient and Nusselt number

decreases where as sherewood number

increases on increasing chemical reaction

parameter.

Skin friction coefficient and Sherwood number

decreases, whereas Nusselt number increases

with an increase in the radiation parameter.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

Gr = 0.2,0.4,0.6,0.8

Gc=0.1, M = 1,m = 0.1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1,

Figure 2: Velocity profiles for different Gr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gc = 0.2,0.4,0.6,0.8

Gr= 0.1, M = 1,m = 0.1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 3: Velocity profiles for different Gc.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1

Figure 4: Velocity profiles for different γ.



0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

M = 1, 2, 3, 4

Gr= 0.1, Gc=0.1, m = 0.1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 5 :Velocity profiles for different M.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

m = 0.2,0.4,0.6,0.8

Gr= 0.1, Gc=0.1, M = 1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 6 : Velocity profiles for different m.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

k= 1,2,3,4

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 7: Velocity profiles for different k.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1,Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

R = 1,2,3,4

Figure 8: Velocity profiles for different R.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Pr = 0.7,1.7,2.7,7.7

Gr= 0.1, Gc=0.1, M = 1,

m = 0.1,k = 1, R= 1, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 9: Velocity profiles for different Pr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Ec= 0.01,0.03,0.05,0.07

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7,Sc = 0.22,

Kr = 1, = 30°

Figure 10: Velocity profiles for different Ec.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Sc= 0.2,0.6,1.0,1.4

G r= 0.1, Gc =0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Kr = 1, = 30°

Figure 11: Velocity profiles for different Sc.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Kr = 1,2,3,4

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, = 30°

Figure 12: Velocity profiles for different Kr.



0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Fw = - 0.5, 0.5, 1

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 13: Velocity profiles for different Fw.

0 1 2 3 40.000

0.002

0.004

0.006

0.008

0.010

0.012

g0

M= 0.5, 1,1.5, 2

Figure 14: Secondaryvelocity profiles

for different M.

0 1 2 3 40.000

0.002

0.004

0.006

0.008

0.010

0.012

R= 0.2, 4,8,10g

0

Gr= 0.1, M = 1,m = 0.1,

k = 1,Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 15: Secondaryvelocity profiles

for different R.

0 1 2 3 40.000

0.002

0.004

0.006

0.008

0.010

0.012

g0

Kr= 1,2,3,10

Gr= 0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, = 30°

Figure 16: Secondary velocity profiles for different Kr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr = 0.2,0.4,0.6,0.8

Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 17: Temperature profiles for different Gr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gc = 0.2,0.4,0.6,0.8

Gr= 0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 18: Temperature profiles for different Gc

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

M = 1, 2, 3, 4

Gr= 0.1, Gc=0.1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure19: Temperature profiles for different M.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

k= 1, 2, 3, 4

Figure 20: Temperature profiles for different k.



0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

R = 1, 5, 10, 15

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 21: Temperature profiles for different R.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Pr = 0.7,1.7,2.7,7.7

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 22: Temperature profiles for different Pr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

m= 0.1, 10, 20, 30

Gr= 0.1, Gc=0.1, M = 1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 23: Temperature profiles for different m.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ec= 2,4,6,8

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Sc = 0.22,

Kr = 1, = 30°

Figure 24: Temperature profiles for different Ec.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Kr= 0.0001,0.01,1,100

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, = 30°

Figure 25: Temperature profiles for different Kr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1,

Figure 26: Temperature profiles for different γ.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Fw=-1,- 0.5, 0.5, 1

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 27: Temperature profiles for different Fw.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr= 1,3,5,7

Gc=0.1, M = 1,m = 0.1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 28: Concentration profiles for different Gr.



0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gc = 1,3,5,7

Gr= 0.1, M = 1,m = 0.1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 29: Concentration profiles for different Gc.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

M= 0.1, 1, 5, 10

Gr= 0.1, Gc=0.1, m = 0.1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 30: Concentration profiles for different M.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

m= 10,90

Gr= 0.1, Gc=0.1, M = 1,k = 1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 31: Concentration profiles for different m.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

k= 0.1, 1, 5, 10

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 32: Concentration profiles for different k.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Pr = 0.7 ,1,1.5,2

Gr=0.1,Gc=0.1,M=1, = 0.1,

Pr=0.7,R=0.5,Ec=0.01,Sc=0.22,

N = 0.1, = 45 ,Kr = 0.5.

Figure 33: Concentration profiles for different Pr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Ec=1,50

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Sc = 0.22,

Kr = 1, = 30°

Figure 34: Concentration profiles for different Ec.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Sc= 0.22,0.6 1.0,1.4

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Kr = 1, = 30°

Figure 35: Concentration profiles for different Sc.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Kr= 1,5,10,15

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, = 30°

Figure 36: Concentration profiles for different Kr.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1,

Figure 37: Concentration profiles for different γ.

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Fw= -1, -0.5,0.5,1

Gr= 0.1, Gc=0.1, M = 1,m = 0.1,

k = 1, R= 1, Pr = 0.7, Ec = 0.01,

Sc = 0.22, Kr = 1, = 30°

Figure 38: Concentration profiles for different Fw.



TABLE I

Computations showing comparision of present results for 0f and - ( )0θ′ at the plate with Gr,Gc, M, m, K, Ec, Sc, γ

and Fw for, R=0, Kr =0.with that of Ali[37].

TABLE II

Variation of 0f , 0 , 0 for different Gr, Gc, M, m, k, R with

P r = 0.7,Ec = 0.01, Sc = 0.22, Kr = 1, γ = 30°and Fw = 1.

Gr Gc M m K Pr Ec Sc

γ

de

g

Fw

Present work Ali [37]

0f

- ( )0θ′ 0f

- ( )0θ′

1

2

3

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

0.1

0.1

0.1

0.1

0.1

1

2

3

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

1

1

1

1

1

1

1

2

3

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.3

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

1

1

1

1

1

2

3

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

0.7

0.7

0.7

0.7

0.7

0.7

1.7

7.7

0.7

0.7

0.7

0.7

0.7

0.7

0.7

0.7

0.7

0.7

0.7

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.05

0.1

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

0.22

0.22

0.22

0.22

0.22

0.22

0.22

0.6

2.6

0.22

0.22

0.22

0.22

0.22

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

45

60

30

30

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0.2

0.4

-0.898306

-0.398999

0.074336

-0.789894

-0.392932

0.001118

-1.14962

-1.46438

-1.73085

-1.14962

-1.14077

-1.1269

-1.14962

-1.46652

-1.73456

-1.14962

-1.1488

-1.14548

-1.14962

-1.14951

-1.14937

-1.14962

-1.15355

-1.16052

-1.14962

-1.16522

-1.18555

-1.32787

-1.28076

0.292389

0.291575

0.289413

0.224745

0.237695

0.249848

0.212403

0.205491

0.200752

0.212403

0.212579

0.212861

0.212403

0.205471

0.200725

0.212403

0.169949

0.0401351

0.212403

0.202479

0.19008

0.212403

0.212102

0.211644

0.212403

0.211814

0.211045

0.275895

0.258952

-0.743451

-0.313333

0.103076

-0.745896

-0.307984

0.12476

-1.14453

-1.46032

-1.72746

-1.14453

-1.13565

-1.12174

-1.14453

-1.46247

-1.73118

-1.14453

-1.14311

-1.13925

-1.14453

-1.14431

-1.14404

-1.14453

-1.14389

-1.14131

-1.14453

-1.16103

-1.18257

-1.32363

-1.27621

0.208039

0.235192

0.258818

0.207936

0.236809

0.26331

0.180087

0.167525

0.158989

0.180087

0.180409

0.180922

0.180087

0.16749

0.158941

0.180087

0.113843

0.003882

3

0.180087

0.16123

0.137683

0.180087

0.180181

0.180592

0.180087

0.178894

0.177333

0.303219

0.268367

Gr Gc M m k R 0f 0 0

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

0.2

0.4

0.6

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

2

3

1

1

1

1

1

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

1

1

1

1

1

1

1

1

1

1

1

1

1

2

3

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1.10482

-1.01549

-0.926519

-1.10954

-1.02946

-0.949488

-1.14962

-1.46438

-1.73085

-1.14077

-1.10914

-1.06687

-1.14962

-1.46652

-1.73456

0.214188

0.217706

0.221155

0.213806

0.216589

0.21934

0.212403

0.205491

0.200752

0.212579

0.213232

0.214162

0.212403

0.205471

0.200725

0.463457

0.464835

0.466196

0.463316

0.46442

0.465518

0.462762

0.459924

0.457976

0.462837

0.463114

0.463507

0.462762

0.459914

0.457962



TABLE III

Variation of 0f , 0 and 0 for different Pr, Ec, Sc, kr, γ, fw with Gr = 0.1,

Gc =0.1,M =1,m = 0.1,k =1 and R=1 .

0.1

0.1

0.1

0.1

0.1

0.1

1

1

1

0.1

0.1

0.1

1

1

1

1

2

3

-1.14962

-1.15006

-1.14995

0.212403

0.236925

0.230594

0.462762

0.462749

0.462752

Pr Ec Sc Kr γ fw 0f 0 0

0.7

1.7

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

0.7

0.7

0.7

0.7

0.01

0.01

0.01

0.01

0.03

0.05

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.22

0.22

0.22

0.22

0.22

0.22

0.22

0.6

1.0

0.22

0.22

0.22

0.22

0.22

0.22

0.22

0.22

0.22

0.22

1

1

1

1

1

1

1

1

1

1

2

3

1

1

1

1

1

1

1

300

300

300

300

300

300

300

300

300

300

300

300

300

450

600

600

600

600

600

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

-1

-0.5

0.5

1

-1.14962

-1.1488

-1.14548

-1.14962

-1.14957

-1.14951

-1.14962

-1.15355

-1.15597

-1.14962

-1.15257

-1.15471

-1.14962

-1.16522

-1.18555

-1.64629

-1.5062

-1.25785

-1.14962

0.212403

0.169949

0.0401351

0.212403

0.20744

0.202479

0.212403

0.212102

0.21193

0.212403

0.212179

0.212026

0.212403

0.211814

0.211045

0.392528

0.340839

0.250749

0.212403

0.462762

0.462787

0.462900

0.462762

0.462763

0.462764

0.462762

0.688359

0.84358

0.462762

0.637593

0.780226

0.462762

0.462532

0.462231

0.572475

0.543122

0.488276

0.462762



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Effects of Radiation on MHD Boundary Layer Flow of Combined Heat and Mass Transfer … · 2020-01-22 · non- Newtonian flows with heat transfer analysis, MHD and non- linear slip