The numerical analysis of the effect of suction, slip and ... · the boundary layer flow past a stretching plate. Andersson [5] investigated the flow of viscoelastic fluid along a

© 2019, IJSRMSS All Rights Reserved 55

International Journal of Scientific Research in ______________________________ Research Paper . Mathematical and Statistical Sciences

Vol.6, Issue.3, pp.55-78, June (2019) E-ISSN: 2348-4519

DOI: https://doi.org/10.26438/ijsrmss/v6i3.5578

The numerical analysis of the effect of suction, slip and inclination, on the

non-Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet

in a porous medium

Bhim Sen Kala

Doon University, Dehradun, 248001, Uttarakhand, India.

Corresponding author: [email protected];

Available online at: www.isroset.org

Received: 02/May/2019, Accepted: 11/Jun/2019, Online: 30/Jun/2019

Abstract--In this paper, we have studied the numerical analysis of the effect of suction, slip and inclination, on the non-

Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet in a porous medium. In the mathematical model,

using similarity variables, the momentum, energy and concentration equations are transformed to non-dimensional ordinary

differential equations. And then these are solved numerically using bvp4c method, a Matlab in- built bvp4c-programm. A

discussion for the effects of the parameters involved on the boundary layer regions and the magnitude of the velocity,

temperature and concentration and Local skin friction, Local Nusselt Number and Local Sherwood Number have been done

graphically and numerically using figures and tables.

Keywords: Casson fluid; magnetic parameter; darcy parameter; slip parameters; forchheimrer parameter; power index

parameter.inclination parameter, suction parameter.

Mathematics Subject Classification: 35A22, 35A35, 35A99, 35G20, 35G30

Nomenclature

x and y are Cartesian coordinates[m]

u velocity components along the x- axes[m/s]

velocity components along the y-axes[m/s]

B magnetic field,

0B magnetic constant

dK permeability parameter[m2]

Fs Forchheimer parameter,

density of the fluid [kg m-3

]

kinematic viscosity of the fluid [m2 s

-1]

dynamic viscosity of the fluid [kg m-1

s-1

]„

B plastic dynamic viscosity fluid

yp yield stress of fluid

),( jie ji -th component of deformation rate

pK non- dimensional Permeability parameter

bC non- dimensional Forchheimer coefficient

Grashof number,

Gc solutal Grashof number,

* buoyancy parameter,

solutal buoyancy parameter,

cK chemical reaction parameter

Su velocity slip

w wall mass transfer velocity

Vs Velocity slip parameter

n stretching surface power index parameter. )(xT

temperature at the surface

Gr

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Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 6(3), Jun 2019, ISSN: 2348-4519


product of deformation rate

T coefficient of thermal expansion

C coefficient of mass expansion

chemical reaction rate of the solute

acceleration due to gravity

T thermal diffusivity of the fluid

pC specific heat at constant pressure

k thermal conductivity of the fluid

D mass diffusion coefficient

similarity variable,

stream function [m2 s

-1]

f non- dimensional stream function

S suction( blowing) parameter

Casson fluid parameter,

n stretching power index parameter.

inclination parameter

M non- dimensional Magnetic parameter

wu surface stretching speed,

wT initial temperature of the surface

0T proportionality constant

y

T

temperature gradient term

)(xC concentration at the surface

wC initial concentration of the surface

0C proportionality constant

y

C

concentration gradient term

T ambient temperature

C ambient concentration

fC local skin-friction

fRe Reynold number.

w Shearing stress

xNu local Nusselt number

xSh local Sherwood number

I. INTRODUCTION

Fluid flow over a stretching sheet has many important

applications: in polymer sheet manufacturing, in chemical

engineering, and in metal processing in metallurgy etc.

Crane [1] first initiated the study of flow of Newtonian

viscous incompressible fluid over a linearly stretching sheet.

He investigated the flow of viscous incompressible fluid

along a stretching plate whose velocity is proportional to the

distance from the slit; such situation occurs in drawing of

plastic films. The study was extended to non-Newtonian

fluids by many researchers. Rajagopal [2] studied the flow

of viscous incompressible fluid on moving (stretching)

surface in the boundary layer region.

Ishak et al. [3] investigated the MHD flow of viscous

incompressible fluid along a moving wedge under the

condition of suction and injection.

Non –Newtonian types of flow occurs in the drawing of

plastic films and artificial fibres. The moving fibre

produces a boundary layer in the medium. Surrounding

medium of the fibre is of technical importance; in that it

governs the rate at which the fibre is cooled and this in turn

affects the final properties of the yarn. Some of the studies

on non-Newtonian fluid are as follows.

Siddappa et al. [4] investigated the flow of visco-elastic fluid

(a non-Newtonian fluid) of ‘Walters’s liquid B Model’ for

the boundary layer flow past a stretching plate. Andersson

[5] investigated the flow of viscoelastic fluid along a

stretching sheet in the presence of transverse magnetic field.

Dandapat [6] investigated the effect of transverse magnetic

field on the stability of flow of viscoelastic fluid over a

stretching sheet.

Fang [7] studied that variable transformation method can be

used to get the solution of extended Blasius equation from

original Blasius equation. Mamaloukas et al. [8] have

discussed some alike nature of free-parameter method and

separation of variable method and have found exact solution

of equation representing flow of two-dimensional visco-

elastic second grade fluid over a stretching sheet. Khidir [9]

used spectral homotopy perturbation method and successive

K

g



linearization method to solve Falker-Skan equation (A non-

linear boundary value problem).

Bataller [10] investigated the flow in the boundary layer of

the viscous incompressible fluid under two situations: one

about a moving plate in a quiescent ambient fluid (Sakiadis

flow) and another uniform free stream flow over a resting

flat-plate (Blasius flow).

Motsa, et al. [11] investigated the MHD boundary layer flow

of upper-convected Maxwell (UCM) fluid over a porous

stretching surface. Motsa et al. [12] had analysed the MHD

flow of viscous incompressible fluid over a nonlinearly

stretching sheet. Rosca [13] discussed the flow of viscous

electrically conducting fluid over a shrinking surface in the

presence of transverse magnetic field.

Nadeem et al. [14] investigated the MHD boundary layer

flow of Williamson fluid over a stretching sheet.

Mukhopadhyay [15] analysed the axis symmetric boundary

layer flow of viscous incompressible fluid along a stretching

cylinder in the presence of uniform magnetic field and under

partial slip conditions. Akbar et al. [16] investigate the MHD

boundary layer flow of Carreau fluid over a permeable

shrinking sheet.

Nadeem et al. [17] investigated the MHD boundary layer

flow of a Casson fluid over an exponentially shrinking sheet.

Biswas et al. [18] studied the effects of radiation and

chemical reaction on MHD unsteady heat and mass transfer

of Casson fluid flow past a vertical plate. Ahmmed et al.

[19] analysed the unsteady MHD free convection flow of

nanofluid through an exponentially accelerated inclined

plate embedded in a porous medium with variable thermal

conductivity in the presence of radiation. Biswas et al. [20]

investigated the effects of Hall current and chemical reaction

on MHD unsteady heat and mass transfer of Casson

nanofluid flow through a vertical plate.

Noor et al. [21] investigated the hall current and

thermophoresis effects on MHD mixed convective heat and

mass transfer thin film flow. Sharada et al. [22] studied

MHD mixed convection flow of a Casson fluid over an

exponentially stretching surface with the effects of soret,

dufour, thermal radiation and chemical reaction.

Mukhopadhyay et al. [23] investigated exact solutions for

the flow of Casson fluid over a stretching surface with

transpiration and heat transfer effects. Dahakeand etal.,[24]

have analysed effects of radiation on magnetohydrodynamic

convection flow past an impulsively started vertical plate

submersed in a porous medium with suction.

Anyanwu [25] have described slip boundary condition

effects on the rate of heat transfer in a micro channel

including viscous dissipation..

Kala [26] studied the analysis of non-Darcy MHD flow of a

Casson fluid over a non-linearly stretching sheet with partial

slip in a porous medium.

This work is the extension of the work [26] in which

analysis of non-Darcy MHD flow of a Casson fluid over a

non-linearly stretching sheet with partial slip in a porous

medium is studied.

This work deals with the numerical analysis of the effect of

suction, slip and inclination, on the non-Darcy MHD flow of

a Casson fluid over a nonlinearly stretching sheet in a porous

medium.

II. MATHEMATICAL MODELLING

We consider steady two-dimensional laminar boundary-layer

flow of viscous, incompressible, electrically conducting non-

Newtonian Casson fluid in a saturated homogeneous non-

Darcy porous medium caused by nonlinearly stretching

sheet, which is inclined with an acute angle ( ) to the

vertical , placed at the bottom of the porous medium. The x-

axis is taken along the stretching surface in the direction of

the motion while the y-axis is normal to the surface. A

Cartesian coordinate system is used. The 𝑥-axis is along the

direction of the continuous stretching surface (the sheet) and

y-axis is normal to the 𝑥-axis. The sheet is assumed to be

stretched along the 𝑥-axis, keeping the position of the origin

unaltered and stretching velocity varies nonlinearly with the

distance from the origin. A uniform magnetic field of

strength B is applied normal to the sheet.

It is assumed that the fluid is optically dense, non-

Newtonian, and without phase change. Flow region is in

non-Darcy porous medium. This integrates a linear Darcian

drag for low velocity effects (bulk impedance of the porous

matrix at low Reynolds numbers) and a quadratic (second

order) resistance, the Forchheimer drag force, for high

velocity flows, as may be come across in chemical

engineering systems operating at higher velocities.

Brinkman’s equation takes into account the boundary effects

(the viscous force).

It is assumed that the induced magnetic field, the external

electric field and the electric field due to polarization of

charges are negligible in comparison to the applied magnetic

field. So, all of the Hall effects and Joule heating effects are

neglected.

The viscosity and thermal conductivity of the fluid are

assumed to be constant. The temperature and concentration

of the stretching surface are always greater than their free

stream values.

Under these assumptions the rheological equation for

incompressible flow of Casson fluid is given by (Sharada et

al. [22], Mukhopadhyay et al. [23])



c

c

y

Bjiji

c

y

Bjiji

ifp

e

ifp

e

22

22

Where is the dynamic viscosity, B is plastic dynamic

viscosity of the non-Newtonian fluid, yp is the yield stress

of fluid, is the product of the component of deformation

rate with itself, namely, jiji ee and jie is the

),( ji -th component of the deformation rate. c is critical

value of the product based on the non- Newtonian model.

The flow configuration and the coordinate system are shown

in Figure 1.

Under these assumptions, the governing boundary layer

equations for momentum, energy and mass take the

following form:

Figure 1 Physical model and coordinate system (for

stretching sheet)

The equation of continuity:

0

y

v

x

u

(1)

The Equation of Momentum:

2

2

0

2

2

cos))()((

11

uK

Cu

Ku

B

CCTTg

y

u

y

uv

x

uu

d

b

df

CT

(2)

The Equation of Energy:

2

2

y

T

y

Tv

x

Tu

(3)

The Equation of Mass concentration:

(4)

where, the sign refers to the cases of assisting and

opposing flow (here we shall consider the case of assisting

flow which is shown by positive sign), x and y are

cartesian coordinates along the stretching sheet and normal

to it respectively, u and v are the velocity components

along the x - and y -axes, dK is the permeability of the

porous medium, B is magnetic field, bC is Forchheimer

coefficient, is angle of inclination of the surface with the

vertical, is the Casson fluid parameter, , and

are density, kinematic viscosity and dynamic viscosity of

the fluid respectively. T , C and K are the

coefficient of thermal expansion, the coefficient of mass

expansion and the chemical reaction rate of the solute,

respectively, g is acceleration due to gravity,

p

TC

k

is the thermal diffusivity of

the fluid pC is the specific heat at constant pressure, k is

the thermal conductivity of the fluid, and D is mass

diffusion coefficient.

The strength of the magnetic field is assumed to vary

spatially by 2/)1(

0)( nxBxB where 0B is constant.

The sheet is assumed to move with power law velocity, and

varies nonlinearly in spatial coordinates with some index, in

the boundary layer region, so that relevant velocity boundary

conditions for equations (1) to (4) are as follows:

CCK

y

CD

y

Cv

x

Cu

c

2

2



.,,

,0:

;)(

,)(

,)(

,

)(:0

0

0

CCTT

uyAs

y

CCCxC

y

TTTxT

xxVvv

y

uNxc

uuxUuyAt

w

w

w

n

ww

(5)

where uw is the surface velocity of the sheet with n

w xcu

where c is a constant parameter related to the surface

stretching speed, Su is the velocity slip which is assumed to

be proportional to the local shear stress as follows:

y

upN

y

uNu yBS

2/ , where N

(m2 s kg

-1) is the slip constant. 2/yB p is

dynamic viscosity. yB p/2 is Casson fluid

parameter . wv is the wall mass transfer velocity with

0wv for mass suction and 0wv for mass injection. n

is the stretching surface index parameter , 0n is for

accelerated sheet and 0n is for decelerated sheet.

)(xT is temperature at the surface at a distance x along

the surface from the origin , WT is initial temperature of

the surface , 0T is proportionality constant arising in

dealing with temperature gradient term y

T

. )(xC is

concentration at the surface at a distance x along the

surface from the origin , WC is initial concentration of the

surface, 0C is proportionality constant arising in dealing

with concentration gradient termy

C

.

T andC are

ambient temperature and concentration respectively.

Dimensional analysis: We define

CC

CC

TT

TT

fn

nf

xnc

v

fxcu

xv

yu

fxn

c

yxnc

w

w

n

n

n

n

)(

,)(

,)(')1(

)1()(

2

)1(

)('

,,

),()1(

2

,2

)1(

2

12

1

2

12

1

2

12

1

(6)

Here is similarity variable, is stream function, f is

non-dimensional stream function, u is x -component of

velocity, is y- component of velocity.

Using equations (6), equations (1) to (4) can be written as

0

))/1((

1

2

cos1

2

1

2)

11(

2

*

2

fFs

fKM

n

n

fn

nfff

p

, (7)

0Pr

1 f , (8)

01

2

1

cKn

fSc

. (9)

And boundary conditions (5) as ,



(10)

where

2

)1(2

1

02

)1(

n

xnc

Tt

,

2

)1(2

1

02

)1(

n

xnc

Cc

.

Here prime denotes differentiation with respect to .

The parameters occurring in equations (7) to (10) are defined

as follows:

fc

BM

2

0 is Magnetic parameter

)1(

n

d

p

xcKK

is Permeability parameter,

d

b

K

xCFs

√ is

Forchhiemer parameter, 2

2)(

xTTgGr wT

is

Grashof number, 2

2)(

xCCgGc wC

is solutal

Grashof number, 2

*

Re

Gr is buoyancy parameter,

2Re

Gc is solutal buoyancy parameter,

xuw

x Re is

Reynolds number, n

w xcu fluid velocity along the wall

surface, D

Sc

is Schmidt number,

T

Pr is

Prandtl number, n

c xKK 11 is reaction rate parameter,

0cK represents destructive chemical reaction, 0cK

corresponds to no reaction and 0cK , stands for

generative chemical reaction.

2

)1(

)1(

2n

w xnc

vS

is suction/injection

parameter, 0S for suction and 0S for injection or

blowing ( 0wv for mass suction and 0wv for mass

injection). 2

)1(

2

)1(

n

xnc

NVs

is velocity slip

parameter. t is thermal slip parameter. c is

concentration slip parameter.

The engineering design quantities which have physical

interest include local Skin-friction coefficient xCf local

Nusselt number xNu and local Sherwood number xSh

and Reynold number (Rex) , are given as follows:

The local Skin-friction coefficient is 2

wf

w

xu

Cf

The Shearing Stress is

0

11

y

Bwy

u

,

00 2

y

y

B

y

Wy

up

y

u

The local Reynold number is

xuw

x Re ,

)0()1

1(2

1

Re

1

2

2

0

fn

u

y

up

Cf

x

wf

y

y

B

x

(11)

Local Nusselt number

)0('2

1Re

0

n

TT

y

Tx

uN

x

w

y

x (12)

as

f

c

t

fVsf

Sf

0)(

,0)(,0)(

)0('1)0(

),0('1)0(

),0(''1)0(

,)0(



Local Sherwood number:

)0('2

1Re

0

n

TT

y

Cx

Sh

x

w

y

x

(13)

III. METHOD OF NUMERICAL SOLUTION

The numerical solutions are obtained using the above

equations for some values of the governing parameters,

namely, the Magnetic parameter (𝑀), the Permiability

porosity parameter (Kp), the Forchhemier parameter(Fs),

inclination parameter( ), Casson parameter( ), Stretching

index parameter( n ) and velocity slip parameter (Vs).

Effects of S ,Vs , t , c , , , and n , on the steady

boundary layers in fluid flow region are discussed in detail.

The numerical computation is done using the Matlab in-built

numerical solver bvp4c. In the computation we have taken

12 and axis according to the clear figure-visibility.

IV. RESULT AND ANALYSIS

In order to validate the method used in this study and to

judge the accuracy of the present analysis, comparisons with

available results of Andersson [4] ,Mahdy [16] and

Ahmed[17] ,corresponding to the skin-friction coefficient

when are presented in Table 1. As it can

be seen, there are excellent agreements between the results.

so we are confident that the present numerical method

works very efficiently.

For drawing from figures 2 to 40 and from tables 3 to 20

following common parameter values are considered:

M =1; Kp =1; Fs =1; =0.2; =0.5; n =1; Vs =0.2;

* =0.5; = 0.5; Pr =1; Sc 1; Kc =0.5;

S =0.2; t =1; c =1;

f0(3)=0.2;f0(5)=0.2;f0(7)=0.2;

Table 1 Comparison of )0(''f for various values of Vs=[0.0;0.1;0.5;1.0] with 12 , =inf, M =0, Kp =inf, Fs =0

, n =1, * =0.0, =0.0, =0.0 , Pr =0, Sc =0, Kc =0.0, S =0.0, t =0, c =0, f0(3)=0.0, f0(5)=0.0, f0(7)=0.0,

Vs )0(''f Andersson [4] )0(''f Mahdy [16] )0(''f

Ahmed[17]

)0(''f Present study

0.0 -1.0000 -1.000000 -1.0000 -1.000001136721379

0.1 -0.8721 -0.8721091 -0.87208 -0.872083949541335

0.5 -0.5912 -0.591199 -0.591195 -0.591197864801179

1.0 -0.4302 -0.4302 001 -0.430160 -0.430163666996962

Table 2 Data for values of parameters for figures 2 to 11.

Fjgure 2 Fjgure 3 Fjgure 4 Fjgure 5,6 Fjgure 7,8 Fjgure 9,10 Fjgure 11

0.5,1,2,3 0.2 0.2 0.2 0.2 0.2 0.2

0.2 0.5,1,2,3 0.2 0.2 0.2 0.2 0.2

1 1 0.5,1,2,3 1 1 1 1

1 1 1 0.5,1,2,3 1 1 1

0.2 0.2 0.2 0.2 0.5,1,2,3 0.2 0.2

0.5 0.5 0.5 0.5 0.5 0.5,1,2,3 0.5

1 1 1 1 1 1 0.5,1,2,3



Figure 2 Graph between )(' f and for different

values of S

Figure 3 Graph between )( and for different

values of S


values of S .

Figure 5 Graph between )(' f and for different values

of Vs .




values of Vs .


values of Vs .

Figure 8 Graph between )( and for different values

of Vs .


of Vs .




values of Vs .


values of Vs .


values of Vs .


of Vs .




of Vs .


values of Vs .


values of Vs .


values of t .




values of t .


values of t .


values of t .


values of t .




values of c .


values of c .


values of c .


of c .




values of c .


values of


values of .


values of




of


values of


values of


values of .




values of .


of .

. .


values of n ..


values of n ..




values of n .


of n .


values of n .


values of n .

Figure 2 is for 120 . Figure 3 show velocity

boundary layer thickness and the magnitude of the

velocity decrease and this decrease slows down as the

Suction parameter increases.

Figure 3 is for 120 . Figure 3show thermal


temperature decrease and this decrease slows down as

the Suction parameter increases.

Figure 4 is for 120 . Figure 4 show concentration


concentration decrease and this decrease slows down as

the Suction parameter increases.





velocity decrease as the velocity slip parameter

increases.

Figure 6 show velocity boundary layer thickness and the

magnitude of the velocity decreases in the range

9.22 2 increases in the range 7.33 . as the

velocity slip parameter increases.

Figure 8 for 70 is part of Figure 7 . Figure 8

show thermal boundary layer thickness and the

magnitude of the temperature increase as the velocity

slip parameter increases.

Figure 10 for 2.35.1 is part of Figure 9 . Figure

10 show concentration boundary layer thickness and the

magnitude of the concentration increase and this increase

slows down as the velocity slip parameter increases.



velocity decrease as the velocity slip parameter

increases.

Figure 12 show velocity boundary layer thickness and

the magnitude of the velocity decreases in the range

9.22 2 increases in the range 7.33 . as the

velocity slip parameter increases.

Figure 14 for 70 is part of Figure 13 . Figure 14

show thermal boundary layer thickness and the

magnitude of the temperature increase as the velocity

slip parameter increases.

Figure 16 for 2.35.1 is part of Figure 15. Figure

16 show concentration boundary layer thickness and the

magnitude of the concentration increase and this increase

slows down as the velocity slip parameter increases.

Figure 18 for 2.38.1 is part of Figure 17 for

120 . Figure 18 shows velocity boundary layer

thickness and the magnitude of the velocity decrease as

the thermal jump parameter increases.

Figure 19 is for 120 . Figure 19 show thermal


temperature decrease as the thermal jump parameter

increases.


120 . Figure 21 show concentration boundary

layer thickness and the magnitude of the concentration

increase and this increase slows down as the thermal

jump parameter increases.



thickness and the magnitude of the velocity decrease as

the concentration jump parameter increases.

Figure 25 for 4.33 is part of Figure 24 for

120 . Figure 25 shows thermal boundary layer

thickness and the magnitude of the temperature increase

and this increase slows down as the concentration jump

parameter increases.

Figure 26 for 120 , shows concentration


concentration decrease as the concentration jump (solutal

slip)parameter increases.



thickness and the magnitude of the velocity decrease and

this decrease speeds up as the inclination parameter

increases.

Here suction (blowing) parameter, velocity slip parameter,

thermal slip parameter, solutal slip parameter are non

dimensional.


120 . Figure 30 shows thermal boundary layer

thickness and the magnitude of the temperature increase

and this increase speeds up as the inclination parameter

increases.


120 . Figure 32 shows concentration boundary


increase and this increase speeds up as the inclination


Figure 33 for 2.36.1 is part of Figure for


thickness and the magnitude of the velocity decrease

and this decrease slows down as the Casson parameter

increases.



Figure 34 for 5.45.1 , shows thermal boundary

layer thickness and the magnitude of the temperature

increase and this increase slows down as the Casson


Figure 35 for 32 , shows concentration boundary


increase and this increase slows down as the Casson


Figure 37 for 52 is part of Figure 36 for

120 . Figure 37 show velocity boundary layer

thickness and the magnitude of the velocity increase and

this increase slows down as the stretching index parameter

n increases.

Figure 39 for 52 is part of Figure 38 for

120 . Figure 39( which is part of Figure 38 )

shows thermal boundary layer thickness and the

magnitude of the temperature decrease and this decrease

slows down as the stretching index parameter n increases.

Transition from n =2 to n =3 shows wavy nature of the

temperature layer at or near the surface.

Figure 41 for 5.40 is part of Figure 40 for

120 . Figure 41 show concentration boundary


increase and this increase shows no variation as the

stretching index parameter increases.

Table 3 Local skin friction )0(''f with respect to variation in and S .

Table 4 Local Nusselt Number )0('

with respect to variation in and S .

Table 5 Local Sherwood Number )0('

with respect to variation in and S .

S \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.5483 0.5439 0.5407 0.4514

1.0 0.6119 0.6083 0.6056 0.5537

2.0 0.7060 0.7039 0.7023 0.6826

3.0 0.7674 0.7662 0.7653 0.7564

Table 6 Local skin friction )0(''f with respect to variation in and Vs .

S \ )0(''f 5.0 )0(''f,

0.1 )0(''f,

0.2 )0(''f,

0.3

0.5 -0.8713 -1.0209 -1.1379 -1.1884

1.0 -0.9451 -1.1230 -1.2651 -1.3272

2.0 -1.0859 -1.3191 -1.5101 -1.5947

3.0 -1.2239 -1.5086 -1.7425 -1.8457

S \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.4741 0.4631 0.4549 0.5394

1.0 0.5680 0.5611 0.5559 0.6045

2.0 0.6887 0.6858 0.6836 0.7016

3.0 0.7594 0.7579 0.7568 0.7649

Vs \ )0(''f 5.0 )0(''f,

0.1 )0(''f,

0.2 )0(''f,

0.3

0.5 -0.6278 -0.7009 -0.7535 -0.7751

1.0 -0.4525 -0.4884 -0.5127 -0.5224

2.0 -0.2926 -0.3062 -0.3150 -0.3183

3.0 -0.2168 -0.2236 -0.2278 -0.2294




with respect to variation in and Vs .

Vs \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.3836 0.3687 0.3583 0.3542

1.0 0.3633 0.3494 0.3403 0.3368

2.0 0.3411 0.3300 0.3233 0.3209

3.0 0.3291 0.3202 0.3151 0.3132


with respect to variation in and Vs .

Table 9 Local skin friction )0(''f with respect to variation in and t

t \ )0(''f 5.0 )0(''f,

0.1 )0(''f,

0.2 )0(''f,

0.3

0.5 -0.8150 -0.9448 -1.0449 -1.0877

1.0 -0.8252 -0.9579 -1.0604 -1.1043

2.0 -0.8368 -0.9733 -1.0787 -1.1240

3.0 -0.8433 -0.9820 -1.0894 -1.1354


with respect to variation in and t .

t \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.5063 0.4848 0.4693 0.4631

1.0 0.4028 0.3885 0.3779 0.3736

2.0 0.2865 0.2787 0.2729 0.2704

3.0 0.2224 0.2176 0.2138 0.2123


with respect to variation in and t .

c \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.5053 0.5008 0.4975 0.4962

1.0 0.5049 0.5002 0.4969 0.4956

2.0 0.5045 0.4996 0.4962 0.4948

3.0 0.5042 0.4993 0.4957 0.4943

Table 12. Local skin friction )0(''f with respect to variation in and c

c \ )0(''f 5.0 )0(''f,

0.1 )0(''f,

0.2 )0(''f,

0.3

0.5 -0.8157 -0.9454 -1.0453 -1.0881

1.0 -0.8252 -0.9579 -1.0604 -1.1043

2.0 -0.8347 -0.9705 -1.0755 -1.1206

3.0 -0.8394 -0.9768 -1.0832 -1.1288


with respect to variation in and c .

c \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.4038 0.3899 0.3797 0.3755

1.0 0.4028 0.3885 0.3779 0.3736

Vs \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.4969 0.4923 0.4892 0.4880

1.0 0.4891 0.4851 0.4826 0.4816

2.0 0.4813 0.4784 0.4767 0.4760

3.0 0.4773 0.4751 0.4739 0.4734



2.0 0.4019 0.3871 0.3762 0.3717

3.0 0.4014 0.3864 0.3752 0.3707


with respect to variation in and c .

c \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.6761 0.6679 0.6621 0.6598

1.0 0.5049 0.5002 0.4969 0.4956

2.0 0.3354 0.3332 0.3317 0.3311

3.0 0.2511 0.2499 0.2490 0.2487

Table 15 Local skin friction )0(''f with respect to variation in and .

Table 16 Local Nusselt Number )0(' with respect to variation in and .

\ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.4020 0.3873 0.3764 0.3719

1.0 0.3993 0.3830 0.3707 0.3656

2.0 0.3901 0.3676 0.3484 0.3397

3.0 0.3832 0.3538 0.3226 0.3032


with respect to variation in and .

Table 18 Local skin friction )0(''f with respect to variation in and n .

Table 19 Local Nusselt Number )0(' with respect to variation in and n .

n \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.3970 0.3814 0.3699 0.3653

1.0 0.4028 0.3885 0.3779 0.3736

2.0 0.4066 0.3934 0.3838 0.3799

3.0 0.4061 0.3934 0.3842 0.3805

\ )0(''f 5.0 )0(''f,

0.1 )0(''f,

0.2 )0(''f,

0.3

0.5 -0.8324 -0.9676 -1.0722 -1.1170

1.0 -0.8564 -1.0001 -1.1118 -1.1599

2.0 -0.9276 -1.0985 -1.2344 -1.2938

3.0 -0.9733 -1.1645 -1.3217 -1.3932

\ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.5047 0.4999 0.4965 0.4951

1.0 0.5037 0.4986 0.4949 0.4934

2.0 0.5009 0.4946 0.4898 0.4878

3.0 0.4990 0.4916 0.4856 0.4828

n \ )0(''f 5.0 )0(''f,

0.1 )0(''f,

0.2 )0(''f,

0.3

0.5 -0.8673 -1.0079 -1.1166 -1.1632

1.0 -0.8252 -0.9579 -1.0604 -1.1043

2.0 -0.8111 -0.9393 -1.0377 -1.0798

3.0 -0.8331 -0.9625 -1.0616 -1.1039




with respect to variation in and n .

Table 3 shows for fix value of the suction parameter S

as the Casson parameter increases, local skin friction

)0(''f decreases and for fix value of the Casson

parameter as the suction parameter S increases skin

friction )0(''f decreases.

Table 4 shows for fix value of the suction parameter S , as

the Casson parameter increases, Local Nusselt

Number )0(' decreases and for fix value of the

Casson parameter as the suction parameter, S

increases Local Nusselt Number )0(' increases.

Table 5 shows for fix value of the suction parameter S , as

the Casson parameter increases, Local Sherwood


Casson parameter as the suction parameter, S

increases Local Sherwood Number )0(' increases.

Table 6 shows for fix value of the velocity slip parameter

Vs as the Casson parameter increases, local skin

friction )0(''f decreases and for fix value of the Casson

parameter as the velocity slip parameter, Vs increases

skin friction )0(''f increases.


Vs as the Casson parameter increases, Local Nusselt


Casson parameter as the velocity slip parameter, Vs

increases Local Nusselt Number )0(' decreases.


Vs as the Casson parameter increases, Local

Sherwood Number )0(' decreases and for fix value of

the Casson parameter as the velocity slip parameter,

Vs increases Local Sherwood Number )0('decreases.

Table 9 shows for fix value of the thermal jump (thermal

slip ) parameter t , as the Casson parameter

increases, local skin friction )0(''f decreases and for fix

value of the Casson parameter as the thermal jump

(thermal slip )parameter, t increases skin friction

)0(''f decreases.



increases, Local Nusselt Number )0(' decreases and

for fix value of the Casson parameter as the thermal

jump (thermal slip )parameter, t increases Local Nusselt

Number )0(' decreases.



increases, Local Sherwood Number )0(' decreases

and for fix value of the Casson parameter as the

thermal jump (thermal slip )parameter, t increases Local

Sherwood Number )0(' decreases.

Table 12 shows for fix value of the concentration jump

(solutal slip ) parameter c , as the Casson parameter

increases, local skin friction )0(''f decreases and for fix

value of the Casson parameter as the concentration

jump (solutal slip )parameter, c increases skin friction

)0(''f decreases.


(solutal slip ) parameter c , as the Casson parameter

increases, Local Nusselt Number )0(' decreases and

for fix value of the Casson parameter

as the

concentration jump (solutal slip )parameter, c increases

Local Nusselt Number )0(' decreases.

n \ )0(' 5.0 )0(',

0.1 )0(',

0.2 )0(',

0.3

0.5 0.5242 0.5202 0.5174 0.5163

1.0 0.5049 0.5002 0.4969 0.4956

2.0 0.4811 0.4753 0.4712 0.4696

3.0 0.4660 0.4594 0.4547 0.4529




(solutal slip ) parameter c as the Casson parameter

increases, Local Sherwood Number )0(' decreases

and for fix value of the Casson parameter as the

concentration jump (solutal slip )parameter, c increases

Local Sherwood Number )0(' decreases.

Table 15 shows for fix value of the inclination parameter

, as the Casson parameter increases, local skin

friction )0(''f decreases and for fix value of the Casson

parameter as the inclination parameter increases

skin friction )0(''f decreases.


, as the Casson parameter increases, Local Nusselt


Casson parameter as the inclination parameter,

increases Local Nusselt Number )0(' decreases.


, as the Casson parameter increases, Local

Sherwood Number )0(' decreases and for fix value of

the Casson parameter as the inclination parameter,

increases Local Sherwood Number )0(' decreases.

Table 18 shows for fix value of the stretching index

parameter n as the Casson parameter increases, local

skin friction )0(''f decreases and for fix value of the

Casson parameter as the stretching index parameter n

increases skin friction )0(''f increases.


parameter n , as the Casson parameter increases,

Local Nusselt Number )0(' decreases and for fix

value of the Casson parameter as the stretching index

parameter n increases Local Nusselt Number )0('increases.


parameter n , as the Casson parameter increases,

Local Sherwood Number )0(' decreases and for fix

value of the Casson parameter as the stretching index

parameter, increases Local Sherwood Number

)0(' decreases.

V. CONCLUSION

In this paper, we have analysed numerically the effect of

suction, slip and inclination, on the non-Darcy MHD flow of a

Casson fluid over a nonlinearly stretching sheet in a porous

medium. In the mathematical model, using similarity

variables, the momentum , energy and concentration

equations are transformed to non-dimensional ordinary

differential equations. These equations are solved numerically

using bvp4c method, a Matlab in- built bvp4c-programm. A

discussion for the effects of the parameters involved on the

boundary layer regions and the magnitude of the velocity,

temperature and concentration and local Skin friction , Local

Nusselt number and Local Sherwood number have been

done graphically and numerically using figures and tables.

From this investigation, we have drawn the following

conclusions:

From the graphs we have following conclusion:

(i) As the Suction parameter, the velocity slip parameter, the

thermal jump(thermal slip) parameter, the concentration

jump(solutal slip) parameter, the inclination parameter,

the Casson parameter increases, velocity boundary

layer thickness and magnitude of velocity decreases.

(ii) As the velocity slip parameter, the concentration

jump(solutal slip) parameter , the inclination parameter,

the Casson parameter increases thermal boundary layer

thickness and magnitude of temperature increases.

(iii) As the velocity slip parameter, the thermal jump(thermal

slip) parameter, the inclination parameter, the Casson

parameter , the stretching index parameter increases,

concentration boundary layer thickness and magnitude of

concentration increases.

(iv) As the stretching index parameter increases, velocity

boundary layer thickness and magnitude of velocity

increases.

(v) As the suction parameter , the stretching index

parameter increases, the thermal jump(thermal slip)

parameter thermal boundary layer thickness and

magnitude of temperature decreases.

(vi) As the suction parameter, the concentration jump(solutal

slip) parameter increases concentration boundary layer

thickness and magnitude of concentration decreases.

From the tables we have following conclusion:

(vii) for fix value of suction, velocity slip, thermal slip,

Solutal (concentration) slip , inclination or stretching

index parameter as Casson parameter Local skin

friction, Local Nusselt Number or Local Sherwood

Number decreases.

(viii) for fix value of Casson parameter , as suction

parameter increases Local skin friction decreases but

Local Nusselt Number , or Local Sherwood Number

increases.

(ix) for fix value of Casson parameter , as velocity slip

increases Local skin friction increases but Local Nusselt

Number or Local Sherwood Number decreases.

(x) for fix value of Casson parameter , as thermal slip,

Solutal (concentration) slip , or inclination parameter



increases ,Local skin friction, Local Nusselt Number or

Local Sherwood Number decreases.

(xi) for fix value of Casson parameter , as stretching index

parameter increases Local skin friction or Local Nusselt

Number , increases but Local Sherwood Number

decreases.

ACKNOWLEDGEMENT

Author expresses sincere thanks to the Department of

Mathematics, Doon University, Dehradun, India, for its

support and cooperation in the work and especially to Dr.

Asha Ram Gairola for his help in the Nomenclature table.

COMPETING INTERESTS

Author has declared that no competing interests exist.

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AUTHOR’S PROFILE

Dr. Bhim Sen Kala, M.Sc, PhD, working in

Fluid Mechanics through Numerical

Analysis, currently teaching under

Department of Mathematics, Doon

University, Dehradun, 248001, Uttarakhand,

India.Email: [email protected]

http://jamc.net/

http://www.sciencedirect.com/

http://www.sciencedirect.com/

The numerical analysis of the effect of suction, slip and ... · the boundary layer flow past a stretching plate. Andersson [5] investigated the flow of viscoelastic fluid along a

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