Unsteady MHD free convection flow of a Kuvshinski fluid past a vertical porous plate in the presence...

Unsteady MHD free convection flow of a Kuvshinski fluid past a vertical

porous plate in the presence of chemical reaction and heat source/sink

S.HARINATH REDDYa, M.C.RAJUb* and E. KESHAVA REDDYc

a,b Department of Humanities and Sciences (Mathematics), Annamacharya Institute of Technology

and Sciences (Autonomous), Rajampet -516126. A.P, INDIA.

cDepartment of Mathematics, Jawaharlal Nehru Technological University Anantapur,

Ananthapuram -515002, A.P, INDIA.

aemail: [email protected],

b*email: [email protected]

cemail: [email protected]

Keywords:

Kuvshinski fluid, chemical reaction source/sink, porous plate, MHD, free convection.

Abstract. Unsteady magneto hydrodynamic (MHD) free convection flow of a viscous,

incompressible and electrically conducting, well known non-Newtonian fluid named as Kuvshinski

fluid past an infinite vertical porous plate in the presence of homogeneous chemical reaction,

radiation absorption and heat source/sink is studied analytically. The plate is assumed to move with

a constant velocity in the direction of fluid flow. A magnetic field of uniform strength is applied

perpendicular to the plate, which absorbs the fluid with a suction that varies with time. The

dimensionless governing equations are solved analytically using two terms harmonic and non-

harmonic functions. The expressions for the fields of velocity, temperature and concentration are

obtained. With the aid of these the expressions for skin friction, Nusselt number and Sherwood

number are derived. The effects of various physical parameters on the flow quantities are studied

through graphs and tables. For the validity, we have checked our results with previously published

work and found in good agreement. Velocity decreases for an increase in visco elastic parameter α2,

heat absorption coefficient φ, the chemical reaction parameter γ , the magnetic field parameter M,

the Prandtl number Pr, the Schmidt number Sc, and increases for increase in Grashof number Gm,

the radiation absorption parameter Q1

1. Introduction

In recent times chemical reaction and radiation absorption influences the fluid flow, attracted

the attention of engineers and scientists. This type of fluid flows plays importance role in food

processing, flow in desert coolers, generating electrical power, groves of fruit trees etc. Takhar et al.

[1], discussed about radiation effects on magneto hydrodynamic (MHD) free convection flow of a

radiating gas past a semi-infinite vertical plate. Muthukumaraswamy et al. [2], studied effect of

diffusion and first-order chemical reaction on impulsively started infinite vertical plate with variable

temperature. Ibrahim et al. [3] considered the effect of chemical reaction and radiation absorption

on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate

with heat source and suction. Patil et al. [4] discussed the effect of chemical reaction on free

convection flow of a polar fluid through a porous medium in the presence of internal heat

generation. Kandaswamy et al. [5], studied the chemical reaction effect on heat and mass transfer

flow along a wedge in the presence of suction or injection. MHD free convection flows occurs in

nature frequently. Fluid flows through porous medium have been attracting the attention of many

International Journal of Engineering Research in Africa Vol. 14 (2015) pp 13-27 Submitted: 27.11.2014© (2015) Trans Tech Publications, Switzerland Revised: 29.12.2014doi:10.4028/www.scientific.net/JERA.14.13 Accepted: 29.12.2014

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researchers in the recent days based on the wide applications in many areas of science and

technological fields, namely study of ground water resources in agricultural engineering, in

petroleum technology to study the moment of ordinary gas, oil, and water through the oil reservoirs.

Ibrahim et al. [6] investigated the effect of the chemical reaction and radiation absorption on

unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat

source and sink. Prasad et al. [7] studied the radiation effect on MHD unsteady free convection flow

with mass transfer past a vertical plate with variable surface temperature and concentration. Cookey

et al. [8] analyzed the effect of unsteady MHD free convection and mass transfer flow past an

infinite heated porous vertical plate with time dependent suction. Gireesh kumar et al. [9] studied

the effect of mass transfer on MHD unsteady free convective Walters memory flow with constant

suction and heat sink. Radiative heat and mass transfer have the wide application in manufacturing

industries for the design of fins, steel rolling, nuclear power plants, gas turbines, and various

propulsion devices for air crafts, satellites and space vehicles are the examples of MHD in

Engineering. Satyanarayana et al.[10], investigated, the effect of MHD free convection heat and

mass transfer past a vertical porous plate with variable temperature. Kandaswamy et al. [11], studied

the effect of chemical reaction , heat and mass transfer on MHD flow over a vertical stretching

surface with heat source and thermal stratification. Chamkha [12], investigated, the effect of

unsteady MHD convection heat and mass transfer past a semi-infinite vertical permeable moving

plate with heat absorption. Anjali Devi et al. [13] considered the effects of homogeneous chemical

reaction, heat and mass transfer on laminar boundary flow along a semi-infinite horizontal plate.

Fluid flows through porous medium is seriously attracted by engineers and scientists.

Now a days, due to their applications in the emerging trends in science and technology, namely in

the field of agricultural engineering especially while studying water resources in the ground, to

study the moment of natural gas, oil, and water through the reservoirs in the petroleum technology.

Chen et al. [14], discussed the effect of free convection of non-Newtonian fluid along a vertical

plate embedded in porous medium. Chamkha [15], studied the effect of heat and mass transfer of a

non-Newtonian fluid flow along a surface embedded in a porous medium inform wall heat and mass

fluxes and heat generation or absorption. Panda et al. [16], considered the effect of unsteady free

convection flow and mass transfer past a vertical porous plate. Mahapatra et al. [17] , analyzed the

effect of chemical reaction on free convection flow through a porous medium surrounded by a

vertical surface. Mishra et al. [18] , investigated the effect of mass and heat transfer on MHD flow

of a visco-elastic fluid through porous medium with variable suction and heat source. Reddy et al.

[19], considered unsteady free convection MHD non-Newtonian flow through a porous medium

bounded by an infinite inclined porous plate. Raju et al. [20], investigated the effect of radiation and

mass transfer effects on a free convection flow past a porous medium bounded by a vertical surface.

Seddek et al. [21], studied the effects of chemical reaction and variable viscosity on hydro magnetic

mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation.

Ravikumar et al. [22], discussed the combined effect of heat absorption and MHD on convective

Rivlin-Erichsen flow past a semi-infinite vertical porous plate with variable temperature and

suction. Ibrahim et al. [23] analyzed the effect of the chemical reaction and radiation absorption on

the unsteady MHD free convective flow past a semi-infinite vertical permeable moving plate with

heat source and suction. Chamkha et al. [24] analyzed that unsteady MHD free convection flow past

an exponentially accelerated vertical plate with mass transfer, chemical reaction and thermal

radiation. Umamaheswar et al. [25] discussed the combined radiation and Ohmic heating effects on

14 International Journal of Engineering Research in Africa Vol. 14

MHD free convective visco-elastic fluid past a porous plate with viscous dissipation. Rao et al. [26]

put their efforts, on the study of unsteady MHD free convective heat and mass transfer flow past a

semi-infinite vertical permeable moving plate with radiation, heat absorption, chemical reaction and

Soret effects. Ravikumar et al. [27] studied the magnetic field effect on transient free convection

flow through porous medium past an impulsively started vertical plate with fluctuating temperature

and mass diffusion. Rao et al. [28] considered the unsteady MHD free convective double diffusive

and dissipative visco-elastic fluid flow in porous medium with suction. Seshaiah et al. [29] analyzed

the effects of chemical reaction and radiation on unsteady MHD free convective fluid flow

embedded in a porous medium with time-dependent suction with temperature gradient heat source.

Gireesh kumar et. al [30] investigated the effect of chemical reaction and MHD free convection

flow of Kuvshinski fluid through porous medium. Ambethkar [31] numerically investigated heat

and mass transfer effects on MHD free convection flow past an infinite vertical plate with constant

suction and heat source or sink. Recently, Vidyasagar et al. [31] investigated an unsteady MHD free

convection boundary layer flow of radiation absorbing Kuvshinski fluid through porous medium.

Radiation absorption effect on MHD free convection chemically reacting visco-elastic fluid past an

oscillatory vertical porous plate in slip regime was investigated by Raju et al. [32].

In spite of all the above studies , as per the author’s knowledge is concern , the

unsteady MHD free convection heat and mass transfer for a heat generating fluid with radiation

absorption in the presence of chemical reaction has received little attention . Further, the presence of

a non-Newtonian fluid namely Kuvshinski fluid is not studied extensively. Hence, the main

objective of the present investigation is to study the effects of chemical reaction, radiation

absorption, mass diffusion and heat source parameter of heat generating Kuvshinski fluid past a

vertical porous plate subjected to fluctuating suction. It is presumed that the plate is inserted in a

uniform porous medium and moves with a constant velocity in the flow direction in the presence of

transverse magnetic field. Beside that it is also assumed that temperature over which is

superimposed is exponentially varying with time.

2. Formulation of the problem:

Consider a two-dimensional viscous, electrically conducting and time dependent heat

absorbing Kuvshinski fluid past a semi-infinite vertical permeable moving plate embedded in a

porous medium. Let x* -axis taken along the flow in the vertical direction and y

*-axis is taken

perpendicular to it. The flow is subjected to a uniform transversely applied magnetic field of

strength B0. The presence of chemical reaction and heat source or sink is also considered along with

the thermal and concentration buoyancy effect. Initially it is assumed that the plate is moving with

uniform velocity, the temperature of the plate is maintained at , and

concentration levels are maintained at . It is assumed that there is no applied

voltage which implies the absence of electric field. The fluid properties are assumed to be constant

except that the influence of density variation with temperature has been considered only in the body-

force term. The concentration of the diffusing species is assumed to be very small in comparison to

other chemical species, therefore the concentration of species far away from the wall, C is

infinitesimally small and hence thermal diffusion and diffusion thermal effects are neglected. As the

presence of chemical reaction is taking place in the flow , all thermo physical properties are

assumed to be constant of the linear momentum equation which is approximated according to the

**

)( tn

ww eTTT **

)( tn

ww eCCC

*

pu

International Journal of Engineering Research in Africa Vol. 14 15

Boussinesq approximation. Due to the semi-infinite plane surface, the flow quantities are functions

of y* and t

* only. Under these assumptions, the equations that describe the flow pattern are given

by conservation of mass, conservation of momentum, energy and species diffusion as follows.

0*

*

y

v

(1)

2* * 2 ** * * *0

* * * *2 * *1 ( ) (C ) 1T C

Bu u uv g T T g C u

t t y y K t

(2)

* * 2 ** * * *0

1* * * *21 ( ) (C )

p p

QT T Tv T T Q C

t t y C y C

(3)

* * 2 *

* *

1* * * *21 (C )

C C Cv D K C

t t y y

(4)

Where *x ,*y , and *t are the dimensional distances along and perpendicular to the dimensional

times and plate, respectively. *u is the component of dimensional velocity along *x direction and *v is the component of dimensional velocity along

*y direction, *C is the dimensional

concentration, *T is the dimensional temperature. wC and wT are the concentration and temperature

at the wall respectively. T and C are the free stream dimensional temperature and concentration,

respectively. ρ is the density of the fluid, is the kinematic velocity, pC is the specific heat at

constant pressure , is the electrical conductivity of the fluid. B0 is the magnetic induction, K* is

the permeability of the porous medium,Q0 is the coefficient of dimensional heat absorption, *

1Q is

the coefficient of proportionality for the radiation absorption , D is the mass diffusivity , g is the

gravitational acceleration , T and C are the thermal and concentration expansion coefficients,

respectively and K1 is the chemical reaction parameter. is the coefficient of Kuvshinski fluid and

κ is the thermal conductivity of the fluid. The magnetic and viscous dissipations are neglected in

this study. The 3rd

and 4th

terms on the RHS of the momentum equation (2) denote the thermal and

concentration buoyancy effects, respectively. Also the 2nd

and 3rd

terms on the RHS of the energy

equation (3) represents the heat and radiation absorption effects respectively. It is supposed that the

permeable plate moves with a variable velocity in the direction of fluid flow. In addition, it is

assumed that the concentration and temperature at the wall as well as the suction velocity are

exponentially varying with time.

Under these assumptions, the appropriate boundary conditions for the velocity,

temperature and concentration fields are

**

puu ,

,)(*** tn

ww eTTTT ,)(*** tn

ww eCCCC at ,0* y

,0* u TT *

, ,*

CC at ,* y (5)

Where n* is constant and

*

pu is dimensional velocity of the wall. From equation (1) it is clear that

the suction velocity at the plate surface is function of time only. Let us assume that it takes the

following exponential form:


),1(**

0

* tnAeVv (6)

Where A is the positive real constant, ԑ and A are small quantities less than unity, and 0V is a

scale of suction velocity which has non-zero positive constant. By introducing the dimensionless

quantities.

0

*

V

uu

, 0

*

V

vv

,

*

0 yVy

,

*2

0 tVt

, 0

*

V

uu

p

p

,2

0

*

V

nn

,

TT

TT

w

*

,

,*

CC

CCC

w

,Pr ,pC

,)(

3

0V

TTgGr wT

,)(

3

0V

CCgGm wC

,2

0

2

0

V

BM

2

2

0

*

VKK

,

,2

0

1

V

K ,

DSc

,

,2

0

0

Vc

Q

p

,

)(

)(2

0

*

11

VTT

CCQQ

w

w

2

02

V

(7)

By using the above non-dimensional variables, the equations (2) – (4) can be expressed in non-

dimensional form as follows

2 2

2 22 2

1 1(1 ) ( )ntu u u u u

Ae Gr GmC M u Mt t y y K K t

(8)

CQyy

Aett

nt

12

2

2

2

2Pr

1)1(

(9)

Cy

C

Scy

CAe

t

C

t

C nt

2

2

2

2

2

1)1(

(10)

The boundary conditions are

,puu ,1 nte ,1 nteC at 0y ,

0, 0, 0 asu C y (11)

Where Pr is the Prandtl number , Gr is the Grashof number , Gm solutal Grashof number , k is

the permeability parameter , M is the magnetic field parameter , Sc is the Schmidt number , φ is the

heat source parameter , γ is the Chemical reaction parameter, α2 is the Visco-elastic parameter .

3. Method of solution

The above equations (8)-(10) are not possible to solve in enclosed form. But which can be reduced

to set of ordinary differential equations in dimensionless form that can be solved analytically. In this

connection we can done by representing the velocity, temperature and the concentration as follows

)()()(),( 2

10 Oyueyutyu nt , (12)

)()()(),( 2

10 Oyeyty nt , (13)

).()()(),( 2

10 OyCeyCtyC nt (14)


By substituting the equations (12)- (14) into the equations (8) – (10) and equating the harmonic and

non-harmonic terms, and neglecting the higher order of )( 2O terms , and simplifying we obtain

the following pairs of equations as follows

The O (ԑ0 ) equations

000

1

0

11

0 )1

( GmCGruK

Muu (15)

,PrPrPr 010

1

0

11

0 CQ (16)

.00

1

0

11

0 CScScCC (17)

Subject to the boundary conditions

puu 0 ,,10 ,0,10 yonC

.0,0,0 000 yasCu (18)

The O(ԑ1) equations are

1

01112

2

2

2

1

1

11

1 )1

( AuGmCGruK

n

KnMMnnuu

(19)

,PrPr)Pr(Pr 11

1

01

2

2

1

1

11

1 CQAnn (20)

.)( 1

012

21

1

11

1 CAScCnnScCScC (21)

Subject to the boundary conditions

,01,1,0 111 yonCu

.0,0,0 111 yasCu (22)

The solutions of the equations (15, 16, 17) and (19, 20, 21) subject to the conditions (18) and (22),

we get the following results.

)(

)(),(

9753111

159

191817161520

879

ymymymymymymnt

ymymym

eAeAeAeAeAeAe

eAeAeAtyu

(23)

)())1((),( 315715

543622

ymymymymntymymeAeAeAeAeeAeAty

(24)

))1((),( 131

11

ymymntymeAeAeetyC

(25)

The physical quantities of interest are the wall shear stress τ w and local surface heat transfer rate qw

and mass transfer rate Sh, are given by

τ w= *

*

y

u

at 0* y =)0(12

0 uV

Hence, the local friction factor Cf is given by


Cf = 2

0V

w

= )0(1u =)()( 1991871751631512011817599 AmAmAmAmAmAmeAmAmAm nt (26)

The local surface heat flux is given by

aty

Tkqw *

*

,0* y

Where k is the effective thermal conductivity, by the definition of local Nusselt number

k

x

TT

qNu

w

wx

(27)

We can write

)())1(()0(Re

534135672125

1 AmAmAmAmeAmAmNu nt

x

(28)

Where v

xVx

0Re is the local Reynolds number.

CC

yaty

c

xShw

0*

*

0Re

yat

y

cSh

x 11131 )1(( AmAmem nt (29)

4. Results and discussion:

In order to look in to the physical insight of the problem ,the expression obtained in previous

sections are studied with help of graphs from fig1-17.The effect of various physical parameters viz,

the chemical reaction parameter γ ,radiation of absorption parameter Q1,the Schmidt number Sc ,

heat absorption coefficient φ ,the magnetic field parameter M, the permeability parameter k, Prandtl

number Pr ,and α2. Fig-1 depicts the variations in concentration profiles for different values of

Schmidt number, from this figure it is noticed that concentration decreases as Schmidt number

increases. From fig-2 we have noticed that when concentration decreases then the chemical

radiation parameter increases. fig-3 depicts the variation in concentration profiles for different

values of α2 , from this figure we have observed that concentration decreases as α2 value increases.

Effect of α2 parameter on temperature distribution is shown in fig-4. From this figure it is noticed

that velocity increases as α2 parameter increases. From fig-5 we have noticed that temperature

decreases as heat absorption coefficient value increases.Fig-6 depicts the variations in temperature

for different values of the chemical reaction parameter, from this figure we have noticed that

temperature decreases as the chemical reaction parameter γ value increases. From the fig-7 we have

studied the effect of Prandtl number on temperature, in this figure we have noticed that temperature

decreases as Prandtl number increases.


0 2 4 6 8 10 12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

C

Sc=0.22

Sc=0.60

Sc=0.78

Sc=0.96

n=1,

t=1,A=1,Z=0.002,

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

C

n=1;z=0.002,t=1,A=1;Sc=0.22

Fig-1. Effects of Sc on concentration profiles Fig-2. Effects of γ on concentration profiles

0.785 0.79 0.795 0.8 0.805 0.81 0.815 0.82

0.43

0.435

0.44

0.445

0.45

0.455

0.46

0.465

y

C

t=1; z=0.02; Sc=.22 ; n=1; A=1;

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Pr=0.71;t=1;z=0.02;Sc=.8;n=1;A=1;Q1=1;up=0.5;k=1;

Fig-3. Effects of α2 on concentration profiles Fig-4. Effects of α2 on Temperature profiles

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

t=1; z=0.02; Sc=.22; n=1A=1; Pr=0.71; Q1=1

0 1 2 3 4 5 6

0

0.2

0.4

0.6

0.8

1

1.2

y

t=1;z=0.02;Sc=.22;n=1;A=1;Q1=1;Pr=0.71

Fig-5. Effects of on Temperature profiles Fig-6 .Effects of γ on Temperature profiles


0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

pr=0.71

Pr=1

Pr=7

t=1;z=0.02;Sc=.22;n=1;A=1;Q1=2;

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Q=1

Q=2

Q=3

Q=4

t=1; z=0.02; Sc=.22;n=1;A=1;Pr=0.71

Fig-7. Effects of Pr on Temperature profiles Fig-8. Effects of Q1 on Temperature profiles

0 1 2 3 4 5 6

0

0.5

1

1.5

2

2.5

y

U

Pr=0.71; Gr=5; Gc=5; M=1; t=1;z=0.02; Sc=.22n=1; A=1; Q1=1;up=1.5;k=1;Gm=5;

Fig-9. Effects of Sc on Temperature profiles Fig-10. Effects of α2 on velocity profiles

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

U

Pr=0.71; Gr=10; Gc=5; M=1; t=1;z=0.02;Sc=.22;n=1; A=1 ; Q1=1;up=1.5;k=1;Gm=5;

0 1 2 3 4 5 6

0

0.5

1

1.5

2

2.5

y

U

Pr=0.71;Gr=10;Gc=5;t=1;z=0.02;Sc=.22;n=1;M=1;A=1;Q1=1;up=1.5;k=1;Gm=5;

Fig-11. Effects of on velocity profiles Fig-12. Effects of γ on velocity profiles

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Sc=0.2

Sc=0.4

Sc=0.6

Sc=0.8

Pr=0.7;t=1;z=0.02;Sc=.8;n=1;A=1;Q1=1;k=1;


0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

y

U

Gm=5

Gm=10

Gm=15

Gm=20

Pr=0.71;Gc=5;M=1;t=1;z=0.02;Sc=.22;n=1;A=1;Q1=1;up=1.4;k=1;Gr=5;

0 1 2 3 4 5 6

0

0.5

1

1.5

2

2.5

y

U

M=1

M=2

M=3

M=4

Pr=0.71;Gr=10;Gc=5;t=1;z=0.02;Sc=.22;n=1;A=1;Q1=1;up=1.5;k=1;Gm=5;

Fig-13. Effects of Gm on velocity profiles. Fig-14. Effects of M on velocity profiles.

0 1 2 3 4 5 6 7 8 9 10

0

0.5

1

1.5

2

2.5

y

U

Pr=0.25

Pr=0.71

Pr=1

Pr=7

Gr=5; Gc=5; M=1; t=1; z=0.02; Sc=.22;n=1; A=1; Q1=1;up=1.5;k=1;Gm=5;

0 1 2 3 4 5 6

0

0.5

1

1.5

2

2.5

y

U

Q=1

Q=2

Q=3

Q=4

Pr=0.71; Gr=10; Gc=5; M=3; t=1; z=0.02;Sc=.22;n=1;A=1;Q1=4;up=1.5;k=1;Gm=5;

Fig-15. Effects of Pr on velocity profiles Fig-16. Effects of 1Q on velocity profiles

0 1 2 3 4 5 6

0

0.5

1

1.5

y

U

Sc=0.22

Sc=0.60

Sc=0.78

Sc=0.96

Pr=0.71; Gr=10; Gc=5; M=3; t=1; z=0.02;n=1; A=1;Q1=1; up=0.5; k=1;Gm=5;

Fig-17. Effects of Sc on Velocity profiles

From the fig-8 we have noticed that the temperature increases as radiation of absorption parameter

Q1 increases.fig-9 depicts the variations in temperature profiles for different values of Schmidt

number, from this figure it is noticed that temperature decreases as Schmidt number increases. fig-

10 exhibits that the effect of α2 on velocity, in this figure we have observed that the velocity of fluid

decreases when α2 value increases. Fig-11 shows the effect of heat absorption coefficient on


velocity, the velocity of the fluid decreases in the increase of heat absorption coefficient. Fig-12

illustrates velocity profile for different values of chemical reaction parameter, from this figure we

have observed that the velocity of the fluid decreases when chemical reaction parameter value

increases. Fig-13 shows the effect of Gm on velocity, in this figure we have observed that velocity

of the fluid increases as Gm increases. Fig-14 illustrates velocity profile for different values of the

magnetic field parameter M, from this figure we have noticed that velocity of the fluid decreases as

the magnetic field parameter M increases. Fig-15 exhibits that the effect of Pr on velocity, in this

figure we have noticed that velocity of fluid decreases when Prandtl increases. Fig-16 shows the

effect of radiation of absorption parameter on velocity, the velocity of the fluid increases as the

radiation of absorption increases. From the fig-17 we have noticed that the velocity of the fluid

decreases as Schmidt number increase.

From the table we have observed that Skin friction decreases with increase in Schmidt

number Sc, Visco-elastic parameter α2, Prandtl number Pr, Chemical reaction parameter γ, heat

source parameter φ, magnetic field parameter M, and other side skin friction increases with an

increase in radiation absorption parameter Q1, permeability parameter k, Grashof number Gr,

solutal Grashof number Gm. The rate of heat transfer Sh increases with increase in radiation

absorption parameter Q1,and decreases with an increase in Schmidt number Sc, Visco-elastic

parameter α2, Prandtl number Pr, Chemical reaction parameter γ, radiation absorption parameter

Q1.The Nusselt number decreases with an increase in Schmidt number Sc, Visco-elastic parameter

α2, Prandtl number Pr, Chemical reaction parameter γ.

5. Concluding remarks:

The plate velocity of the Kuvshinski fluid which is subjected to a transverse magnetic field is

maintained constantly. The obtained partial differential equations are expressed in two dimensional

form using non-dimensional parameters. These equations are solved analytically and graphs are

obtained to study the effects of various physical parameters on velocity, temperature, concentration

profiles. Hence it is found that

(a) Velocity decreases for an increase in α2, heat absorption coefficient φ, the chemical reaction

parameter γ, the magnetic field parameter M, the Prandtl number Pr, the Schmidt number Sc,

and increases for increase in Grashof number Gm., the radiation absorption parameter Q1.

(b) The temperature profiles increased for an in increase in α2, the radiation absorption parameter

Q1, and decreased due to increase in heat absorption coefficient φ, the chemical reaction

parameter γ, the Prandtl number Pr, the Schmidt number Sc.

(c) Concentration of the fluid decreases due to an increase in the Schmidt number Sc, the chemical

reaction parameter γ, α2.

(d) skin friction decreased due to increase in Sc, α2, Pr, γ, φ, M and increased with an increase in

Q1,A,k,Gr and Gm.

(e) Rate of heat transfer Sh increases with an increase in Q1 and A, but decreased due to increase in

Sc, α2, Pr, γ, Q1.


(f) Nusselt number decreased with an increase in Sc, α2, A, γ.

Appendix:

2

42

1

ScScScm

2

)(4 2

22

3

nnScScScm

)( 2

2

1

2

1

11

nnScscmm

AScmA

2

Pr4PrPr 2

5

m

PrPr

Pr

1

2

1

12

mm

QA

2

)Pr(4PrPr 2

22

7

nnm

)Pr(Pr

)1(Pr2

25

2

5

253

nnmm

AAmA

)Pr(Pr

)Pr(2

21

2

1

11214

nnmm

QAAAmA

)Pr(Pr

)1(Pr2

23

2

3

115

nnmm

AQA

5436 1 AAAA

2

)1

(411

9

kM

m

)

1(

)1(

5

2

5

27

kMmm

AGrA

)1

(

)(

1

2

1

28

kMmm

GmGrAA

879 AAuA p

2

)1

(411 22

2

2

11

k

n

knMMnn

m

811410 AAmGmAGrAA

)1( 1511 AGmGrAA 75312 AAmGrAA

613 GrAA 9914 AAmA k

n

knMMnnB 2

2

2

20

1

01

2

1

1015

Bmm

AA

03

2

3

1116

Bmm

AA

05

2

5

1217

Bmm

AA

07

2

7

1318

Bmm

AA

09

2

9

1419

Bmm

AA

191817161520 AAAAAA

Table: Variations in Skin friction, Sherwood number and Nusselt number

Sc α2 Pr γ Q1 φ M A K Gr Gm τ Sh Nu

0.22 1 0.71 1 1 1 1 1 1 5 20 14.2065 -0.7008 -0.6478

0.60 1 0.71 1 1 1 1 1 1 5 20 9.4799 -0.7267 -1.2384

0.78 1 0.71 1 1 1 1 1 1 5 20 9.5134 -0.7600 -1.4851

0.96 1 0.71 1 1 1 1 1 1 5 20 8.6703 -0.8873 -1.7218

0.22 1 0.71 1 1 1 1 1 1 5 20 13.3957 -0.7008 -0.6478

0.22 2 0.71 1 1 1 1 1 1 5 20 13.2966 -0.7472 -0.6541

0.22 3 0.71 1 1 1 1 1 1 5 20 13.2425 -0.7593 -0.6597

0.22 1 0.71 1 1 1 1 1 1 5 20 13.3957 -0.7008 -0.6478

0.22 1 1 1 1 1 1 1 1 5 20 13.2285 -0.8617 -0.6478

0.22 1 7.1 1 1 1 1 1 1 5 20 12.5174 -3.4921 -0.6478

0.22 1 0.71 5 1 1 1 1 1 5 20 11.2350 1.2377 -1.2443

0.22 1 0.71 10 1 1 1 1 1 5 20 8.6426 -0.9057 -1.6980

0.22 1 0.71 15 1 1 1 1 1 5 20 7.6589 -0.9780 -2.0473

0.22 1 0.71 20 1 1 1 1 1 5 20 6.9966 -1.0163 -2.3424

0.22 1 0.71 5 1 1 1 1 1 5 20 11.2350 1.2377 -1.2443

0.22 1 0.71 5 2 1 1 1 1 5 20 12.5446 3.7735 -1.2443

0.22 1 0.71 5 3 1 1 1 1 5 20 13.8542 6.3093 -1.2443

0.22 1 0.71 5 4 1 1 1 1 5 20 15.1638 8.8451 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 20 11.2350 1.2377 -1.2443

0.22 1 0.71 5 1 2 1 1 1 5 20 10.2886 -0.3650 -1.2443

0.22 1 0.71 5 1 3 1 1 1 5 20 9.6995 -1.6204 -1.2443

0.22 1 0.71 5 1 4 1 1 1 5 20 9.5256 -1.8933 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 20 11.2350 1.2377 -1.2443

0.22 1 0.71 5 1 1 2 1 1 5 20 9.2178 1.2377 -1.2443

0.22 1 0.71 5 1 1 3 1 1 5 20 7.8009 1.2377 -1.2443

0.22 1 0.71 5 1 1 4 1 1 5 20 6.7174 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 2 1 5 20 12.2228 3.4578 -1.2504

0.22 1 0.71 5 1 1 1 4 1 5 20 14.1984 7.8982 -1.2625

0.22 1 0.71 5 1 1 1 6 1 5 20 16.1740 12.3386 -1.2746

0.22 1 0.71 5 1 1 1 8 1 5 20 18.1496 16.7789 -1.2867

0.22 1 0.71 5 1 1 1 1 1 5 20 11.2350 1.2377 -1.2443


0.22 1 0.71 5 1 1 1 1 2 5 20 12.6597 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 3 5 20 13.2362 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 4 5 20 13.5496 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 20 4.0112 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 10 20 7.6417 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 15 20 11.2722 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 20 20 14.9022 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 5 4.0112 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 10 7.6417 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 15 11.2722 1.2377 -1.2443

0.22 1 0.71 5 1 1 1 1 1 5 20 14.9026 1.2377 -1.2443

Acknowledgement

Authors express their sincere gratitude to the editor and the reviewer for their constructive

comments which helped to strengthen the manuscript.

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