Page 56

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Soret-Dufour Effect on Variable Viscosity MHD Flow Through Porous

Medium

Mohammad Suleman Quraishi

1*

, Nidhi Pandya

2

1

Deparment of Mathematics, Acharya University, Karakul, Uzbekistan

2

Department of Mathematics & Astronomy, University of Lucknow, Lucknow, India

*Corresponding Author

DOI:

https://doi.org/10.51583/IJLTEMAS.2026.150500006

Received: 11 April 2026; Accepted: 16 April 2026; Published: 22 May 2026

ABSTRACT

In this paper we have explored the Soret and Dufour effects on unsteady viscous incompressible and electrically

conducting MHD flow through porous medium past an inclined infinite plate with variable viscosity fluid and

periodic suction or injection of fluid through the plate with exponential increment in temperature and

exponentially decrement in concentration. The developed or governing physical model of the flow field has been

converted into the mathematical model. Then mathematical model of system of equations is represented by set

of governing non linear partial differential equations which have been non dimensionalized by the help of non

dimensional numbers. The associated dimensionless governing equation of flow field is solved numerically by

Crank Nicolson finite difference implicit method for different values of governing flow parameters. The effects

of various responsible flow parameters have been discussed through graphs and table. The effects and their

numerical corresponding variations have been concord with physical nature of the problem. The velocity profile,

concentration profile and temperature profile are shown through graphs for different values of flow parameters.

The obtained numerical values of the skin friction, Nusselt number and Sherwood number with their variations

for different values of flow parameters have been tabulated.

Keywords: Soret-Dufour effects, variable viscosity, Periodic Suction or Injection, Crank Nicolson finite

difference implicit method.

INTRODUCTION

Soret-Dufour effect on MHD ﬂows arises in many areas of engineering and applied physics. The studies of such

ﬂow have application in MHD generators, chemical-engineering, nuclear reactors, geothermal energy, and

reservoir engineering and astrophysical studies. The assumption of the pure ﬂuid is rather impossible in nature.

The presence of foreign mass in the ﬂuid plays a vital role in flow of ﬂuid. Thermal diffusion or Soret effect is

one of the mechanisms in the transport phenomena in which molecules are transported by temperature gradient

in a multi-component mixture driven. The inverse phenomena of thermal diffusion are known as Dufour effect,

if multi component mixture were initially at the same temperature, are allowed to diffuse into each other, there

arises a difference of temperature in the system. In this article we have explored the Soret and Dufour effects on

unsteady viscous incompressible and electrically conducting MHD flow through porous medium past an inclined

infinite plate with variable viscosity fluid and periodic suction or injection of fluid through the plate with

exponential increment in temperature and exponentially decrement in concentration. Sparrow and Cess (1961)

analyzed the effect of magnetic ﬁeld on free convection heat transfer, Dursunkaya et al. (1992) studied Diffusion

thermo and thermal diffusion effect in transient and steady natural convection from vertical surface. Raptis et al.

(1999) discussed radiation and free convection ﬂow past a moving plate. A. Postelnicu (2004) analyzed the

Inﬂuence of a magnetic ﬁeld on heat and mass transfer by natural convection from vertical surface in porous

media considering Soret and Dufour effects. Alam et al. (2005) Investigated Dufour effect and Soret effect on

www.rsisinternational.org

Page 57

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

MHD free convective heat and mass transfer ﬂow past a vertical ﬂat plate embedded in porous medium. Bhavana

et al. (2013) analyzed the Soret effect on free convective unsteady MHD ﬂow over a vertical plate with heat

source. Effect of Variable Permeability and Variable Magnetic field on MHD Flow past an inclined plate with

exponential Temperature and Mass Diffusion with Chemical reaction through Porous media by N. Pandya and

M. S. Quraishi (2018).

In all of these studies mentioned above the viscosity of the fluid was assumed to be constant. However, we

known that the physical property may change significantly. Therefore to predict correctly the flow behavior with

respect to variable fluid viscosity, it is necessary to view the variation of viscosity with dependent variable and

this variation would also play a vital rule.

The effect of temperature dependent viscosity in the boundary layer flow, heat and mass transfer is analyzed

Free Convection Flow over an Uniform-Heat-Flux Surface with Temperature-Dependent Viscosity by Jang, J.

-Y. and C. -N. Lin (1988). The Effect of Variable Viscosity on Flow and Heat Transfer to a Continuous Moving

Flat Plate by Pop I, R. S. R. Gorla and M. Rashidi (1992). Effects of Variable Viscosity and Viscous Dissipation

on the Flow of Third Grade Fluid in a Pipe by Massoudi, M. and I. Christie (1995). The Effect of Temperature-

Dependent Viscosity on Free-Forced Convective Laminar Boundary Layer Flow Past a Vertical Isothermal Flat

Plate by Kafoussias, N. G. and E. W. Williams (1995). Mixed Convection Boundary Layer Flow on a Continuous

Flat Plate with Variable Viscosity by Hady, F. M., A. Y. Bakier and R. S. R. Gorla (1995). Mixed Convection

Flow from a Vertical Flat Plate with Temperature Dependent Viscosity by Hossain, Md. A. and Md. S. Munir

(2000).

Flow of Viscous Incompressible Fluid with Temperature Dependent Viscosity and Thermal Conductivity Past a

Permeable Wedge with Uniform Surface Heat Flux by Hossain, Md. A., Md. S. Munir and D. A. S. Rees (2000).

Effect of Variable Viscosity on Free Convection over a Non-Isothermal axisymmetric Body in a Porous Medium

with Internal Heat Generation by Begai S. (2004). Plume Formation in Strongly Temperature-Dependent

Viscosity Fluid over a Very Hot Surface by Ke, Y. and V. S. Solomatov (2004). The Effect of Variable Viscosity

on Mixed Convection Heat Transfer along a Vertical Moving Surface by Ali, M. E. (2006). The Influence of

Variable Viscosity and Viscous Dissipation on the Non-Newtonian Flow: Analytical Solution by Hayat, T., R.

Ellahi and S. Asghar (2007). Variable Viscosity Effects on Hydromagnetic Boundary Layer Flow along a

Continuously Moving Vertical Plate in the Presence of Radiation by Mahmoud, M. A. A. (2007).

Effect of Variable Viscosity on Combined Forced and Free Convection Boundary-Layer Flow over a Horizontal

Plate with Blowing or Suction by Mahmoud M. A. A. (2007). Effect of Variable Viscosity on the Peristatic

Transport of a Newtonian Fluid in an Symmetric Channel by Hayat T. and N. Ali (2008). Effect of Thermal

Radiation and Variable Fluid Viscosity on Free Convective Flow and Heat Transfer Past a Porous Stretching

Surface by Mukhopadhyay S. and G. C. Layek (2008).

Thermal Radiation Effect on Unsteady MHD Free

Convection Flow Past a Vertical Plate with Temperature-dependent Viscosity by Mostafa A. A. Mahmoud

(2009).

Analysis of mhd carreau fluid flow over a stretching permeable sheet with variable viscosity and thermal

conductivity by

Tariq Abbas and et al. (2020). Analytical treatment of MHD flow and chemically reactive

Casson fluid with Joule heating and variable viscosity effect by Haroon Ur Rasheed and et al. (2022). Siti

Suzilliana Putri Mohamed Isa and et al. (2023) have discussed Soret-Dufour Effects on Heat and Mass Transfer

of Newtonian Fluid Flow over the Inclined Sheet and Magnetic Field.

The object of this work is to investigate the effect of Soret and Dufour with variable fluid viscosity so we have

set the viscosity as a function of dependent variable distance from the plate so that viscosity depends upon

distance from the plate on MHD unsteady flow over an inclined infinite plate through porous medium with

periodic suction or injection of fluid through the plate. Heat and mass transfer have been taken exponentially

increasing and decreasing respectively. The dimensionless governing equation of flow field is solved

numerically by Crank Nicolson finite difference implicit technique for different values of governing flow

parameters. The velocity profile, concentration profile and temperature profile are shown through graphs for

different values of flow parameters.

Page 58

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Table 1: Nomenclature

α

1

Some

constant for viscosity parameter

u

The velocity of fluid in x direction

g

Gravitational acceleration due to gravity

β

The volumetric coefficient of thermal expansion

β

*

Coefficient of volume expansion for mass transfer

ν

Kinematic viscosity of ambient fluid

µ

∞

Ambient viscosity of the fluid

ρ

∞

Ambient ﬂuid density

B

Magnetic parameter

K

Permeability of porous medium

σ

Electrical conductivity of the ﬂuid

T

Dimensional temperature

T

∞

Temperature of ambient fluid

C

∞

Concentration of ambient fluid

D

Mass diffusion coefficient

D

m

Effective chemical molecular diffusivity,

D

T

Thermal diffusion coefficient

k

Thermal conductivity of the ﬂuid

C

P

Specific heat at constant pressure

K

T

Thermal diffusion ratio

C

Dimensional concentration.

T

w

Temperature on plate

C

w

Concentration on plate

Governing Mathematical Model

An unsteady ﬂow of a variable viscosity of incompressible electrically conducting ﬂuid initially at rest past

infinite inclined plate with constant temperature and constant mass diffusion in the presence of magnetic field

with periodic suction or injection are studied. Since viscosity may depends on flow parameters like pressure,

Temperature etc. In this work Velocity, Temperature and Mass transfer are varying here with the distance from

plate so we have taken viscosity as a function of distance which exponentially decreases with distance from

plate.

The plate is inclined at angle ‘λ’ from vertical, through porous medium. x-axis is taken along the plate and

y-axis is taken normal to it. Since the plate is electrically non conducting so induce magnetic field is neglected.

Initially the plate and ﬂuid are at the same temperature and concentration level is zero and plate is at rest.

Now at time t > 0, the plate is at rest along x-direction against gravitational ﬁeld, y-axis is normal to the plate,

the plate temperature and concentration raised exponentially increasing and decreasing respectively. A

transverse magnetic ﬁeld of uniform strength B is assumed normal to the direction of ﬂow. The transversely

applied magnetic ﬁeld and magnetic Reynolds numbers are very small and hence induced magnetic ﬁeld is

negligible, Cowling (1957). Due to inﬁnite length in x-direction, the ﬂow variables are functions of y and t only.

Under the above consideration, governing equations for this unsteady problem by the help of Navier-Stokes

equations are given by

www.rsisinternational.org

Page 59

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Momentum equation

)(0

0

constantavv

y

v





K

uuB

CCCosgTTCosg

y

u

yy

u

v

t

u









































2

*

)()(

1

Since

y

u

y

u

y

u

y 



























2

, therefore above equation becomes

K

uuB

CCCosgTTCosg

y

u

y

u

y

u

v

t

u















































2

*

2

)()(

1

(1)

Energy equation

2

y

C

c

KD

y

T

k

y

T

v

t

T

C

s

Tm

p































(2)

Equation of mass transfer

2

y

T

KD

y

C

D

y

C

v

t

C

m

Tm















(3)

Where

y

e

1











(4)

As per the modeled flow governing equations (1) to (4), Velocity, Temperature and Mass transfer are the

functions with respect to dependent variable y, that is with respect to the distance from plate. Temperature is

taken exponentially decreasing with time. Moreover variable viscosity has been considered as a function of

distance that is viscosity exponentially decreases with distance from plate. Concentration has been taken

exponentially decreasing.

Considering the suction or injection velocity

0

v

as a periodic function of time define as

AtCosv 1

0

. Where



(<<1) is the amplitude of suction or injection velocity and A is some constant.

Where α

1

is some constant for viscosity parameter, u is the velocity, g is the acceleration due to gravity, β is the

volumetric coefficient of thermal expansion, β

*

is the coefficient of volume expansion for mass transfer, ν is the

kinematic viscosity of ambient fluid, µ

∞

is ambient viscosity, ρ

∞

is the ambient ﬂuid density, B is magnetic

parameter, K is the permeability of porous medium, σ is the electrical conductivity of the ﬂuid, T is the

dimensional temperature,

T

∞

is temperature of ambient fluid, C

∞

is concentration of ambient fluid, D is mass

diffusion coefficient, D

m

is effective chemical molecular diffusivity, D

T

is thermal diffusion coefficient, k is the

thermal conductivity of the ﬂuid , C

P

is specific heat at constant pressure, K

T

is thermal diffusion ratio, C is the

dimensional concentration. T

w

and C

w

are the temperature and concentration on plate. Initial and boundary

conditions are given as:

Page 60

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026























yasCC,TT,u

yate)CC(CC,e)TT(TT,u;t

yC,TtACosv,u;t

AtAt

ww

0

000

00100

0

(5)

In order to form a non dimensional partial differential equation, introducing following non dimensional

quantities:









































































2

0

1

2

0

2

0

3

0

3

0

2

0

u

A,

)TT(cc

)CC(KD

D,

u

,,

u

B

M

,

k

C

P,

)CC(T

)TT(KD

S,K

u

K,

D

S,

u

)TT(g

G

,

u

)CC(g

G,

CC

C,

TT

,

uy

y,

ut

t,

u

wps

wTm

u

p

r

wm

wTm

rc

w

r

w

*

m

ww

(6)

G

r

, G

m

, S

r

, D

u

, S

c

, P

r

and M are Thermal Grashof number, Solutal Grashof number, Soret number, Dufour

number Schmidt number, Prandtl number and Magnetic parameter respectively. Where A is some constant.

By the substitution of above non dimensional quantities we get the non dimensional form of equations (1) to (5)

as:

K

u

uMCosCGCosG

y

u

y

u

y

u

tCos

t

u

mr















































2

)1(

(7)

2

1

)1(

y

C

D

y

Py

tCos

t

u

r



















(8)

2

1

)1(

y

S

y

C

Sy

C

tCos

t

C

r

c

















(9)

And

)( yExp





(10)

The initial and boundary condition Eq. (5) in non dimensional form as following:



















yasCu

yateCeut

yCut

tt

0,0,0

0,,0;0

0,0,0;0



(11)

For the sake of convenience dropping the bars from equations (7) to (11) then we have the system as

K

u

yuMCosCGCosG

y

u

y

u

yy

u

tCos

t

u

mr

)()1(

2















































(12)

www.rsisinternational.org

Page 61

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

2

1

)1(

y

C

D

y

Py

tCos

t

u

r



















(13)

2

1

)1(

y

S

y

C

Sy

C

tCos

t

C

r

c

















(14)

and

)( yExp





(15)

With initial and boundary conditions as



















yasCu

yateCeut

yCut

tt

0,0,0

0,,0;0

0,0,0;0



(16)

Now, it is useful to calculate physical quantities for primary interest, these are coeﬃcient of skin-friction ‘τ’

at

the wall along x-axis, Nusselt number Nu and Sherwood number Sh. Dimensionless forms of these physical

quantities are:

0

















y

u



,

0

















y

Nu



,

0

















y

C

Sh

(17)

METHOD OF SOLUTION

Equations (12) to (14) are non-linear partial diﬀerential equations, are solved using initial boundary conditions

Eq. (16), it is solved here by Crank-Nicolson implicit ﬁnite diﬀerence method for numerical solution. By this

method the transform form of Equations (12) to (14) in form of finite differences with indices are expressed as:

































































































































2

1

2

1

2

1

2

11

2

111

2

111

2

1

2

j,i

u

j,i

u

K

]y[Exp

j,i

u

j,i

u

M

j,i

C

j,i

C

Cos

m

G

j,ij,i

Cos

r

G

y

j,i

u

j,i

u

]y[Exp

)y(

j,i

u

j,i

u

j,i

u

j,i

u

j,i

u

j,i

u

]y[Exp

y

j,i

u

j,i

u

)tCos(

t

j,i

u

j,i

u



















(18)

Page 62

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026





















































































2

)(2

1,11,

2

1,1,1,

2

,1

1

)(2

1,11,

2

1,1,1,

2

,1

,,1

)1(

,1,

y

jijijijijiji

r

P

y

ji

C

ji

C

ji

C

ji

C

ji

C

ji

C

D

y

jiji

tCos

t

jiji

u





(19)























































































2

)(2

1,11,

2

1,1,1,

2

,1

)(2

1,11,

2

1,1,1,

2

,1

1

,,1

)1(

,1,

y

jijijijijiji

S

y

ji

C

ji

C

ji

C

ji

C

ji

C

ji

C

c

S

y

ji

C

ji

C

tCos

t

ji

C

ji

C

r



(20)

Corresponding boundary and initial conditions form Equation (16) are now as:

000

0

000







j,n

j,nj,n

t*j

j,

t*j

j,j,

,i,i,i

C,,u

jeC,e,u

iC,,u





(21)

Here, index ‘i’ refers to y and j to time,

jj

ttt 

1

and

ii

yyy 

1

. For known values of u, θ and C at

t, we calculate these values for (t +∆ t) as follows, after substitution of i and j =1, 2, 3… n-1, where ‘n’

corresponds to ∞. Now system of equations (19) and (20) are solved by Thomas Algorithm as discussed in

Carnahan et al. (1969). Then θ and C are known for all values of y at t+∆t.

Replacing values of θ and C in equation (18) and solved by same with initial and boundary conditions (21), we

have solutions for u till desired time t. Crank-Nicolson implicit finite difference method is second order method

(o (∆t

2

)) in time also and has no limitation for space and time steps, that is, the method is unconditionally stable.

Computation has been executed for ∆y = 0.1, ∆t = 0.002 and repeated till y (max) =4, y (max) is corresponds to

infinity.

www.rsisinternational.org

Page 63

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Fig. 1. Velocity Profile for different values of ‘M’

Fig. 2. Velocity Profile for different values of ‘S

r

’

Fig. 3. Velocity Profile for different values of ‘Gr’

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

= 0.1

y

u

M = 2, 4, 6

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

= 0.1

=

30

, t = 0.2,

K = 0.4,

= 0.2,

M=2,

S

r

0.4, 0.6, 0.8

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

M=2,

= 0.1

u

y

G

r

20, 25, 30

0

1

2

3

4

0.0

0.5

1.0

1.5

Page 64

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Fig. 4. Velocity profile for different values of ‘Du’

Fig. 5. Temperature Profile for different values of ‘Du’

Fig. 6. Concentration Profile for different values of ‘S

r

’

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

= 0.1

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

M=2

D

u

0.2, 0.4, 0.6

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

= 0.1

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

M=2

D

u

0.2, 0.4, 0.6

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01, = 0.1

=

30

, t = 0.2,

K = 0.4,

= 0.2,

M=2

S

r

0.4, 0.6, 0.8

C

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

www.rsisinternational.org

Page 65

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Fig. 7. Velocity profile for different value of ‘Gm’

Fig. 8. Velocity profile for different value of ‘t’

Fig. 9. Temperature profile for different values of ‘t’

G

r

= 20,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

M=2,

= 0.1

G

m

25, 30, 35

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, K = 0.4,

= 0.2, M=2,

= 0.1

t

0.1, 0.2, 0.3, 0.4

u

y

0

1

2

3

4

0.0

0.5

1.0

1.5

2.0

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, K = 0.4, = 0.2, M=2,

= 0.1

t

0.1, 0.2, 0.3, 0.4

y

0

1

2

3

4

0.0

0.5

1.0

1.5

2.0

Page 66

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Fig. 10. Concentration profile for different values of ‘t’

Fig. 11. Velocity profile for different positive values of ‘α’

Fig. 12. Velocity profile for different negative value of ‘α’

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, K = 0.4,

= 0.2, M=2,

= 0.1

C

y

t

0.1, 0.2, 0.3, 0.4

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4, M=2,

= 0.1

0.2, 0.7, 1.2

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4, M=2,

= 0.1

0.2, 0.7, 1.2

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

www.rsisinternational.org

Page 67

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Fig. 13. Velocity profile for different values of ‘K’

Fig. 14. Velocity profile for different value of ‘λ’

Fig. 15. Velocity profile for different value of ‘Pr’

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

= 0.2,

= 0.1

u

y

K

0.4, 0.6, 0.8

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

P

r

= 0.71,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4

= 0.1, t = 0.2

K = 0.4,

= 0.2

M=2

30

, 45

, 60

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

G

r

= 20,

G

m

= 25,

M=2,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

= 0.1

P

r

0.71, 2, 2.5

u

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Page 68

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

Fig. 16. Temperature profile for different value of ‘Pr’

RESULTS AND DISCUSSION

In this paper we studied Soret and Dufour effects on MHD flow through porous medium past an inclined infinite

plate with variable viscosity fluid and periodic suction or injection of fluid through the plate with exponential

increment in temperature and exponentially decrement in concentration. The dimensionless governing equation

of flow field is solved numerically by Crank Nicolson finite difference implicit method for different values of

governing flow parameters. The effects of various flow parameters discussed through graphs and table. The

velocity profile, concentration profile and temperature profile shown through graphs for different values of flow

parameters. The consequences of the relevant parameters break down the ﬂow ﬁeld and discussed with the help

of graphs of velocity profile, temperature profiles and concentration profiles.

Fig. 2, 3, 4, 7, 8, 11 and 13 depicts that the velocity ‘u’ increases with increase in S

r

,

Gr , Du , G

m

, t, α and K.

Fig. 1, 12, 14 and 15 show that velocity decreases with increase in M,- α, λ and P

r

.

Temperature profile θ in Fig.

5 and 9 show increment with increase in G

m

and t respectively while Temperature decreases with increase in P

r

.

Concentration profile ‘C’ increases with S

r

in Fig. 6 and decreases with ‘ t’ Fig.10

Table 2. Depicts that increase in G

r

, G

m

,α, S

r

, t , K and D

u

the Skin friction coefficient ‘τ’ increases while

increase in M, P

r

, λ and - α then Skin friction ‘τ’ decreases.

Table 2. Also shows that increase in P

r

and S

r

, Nusselt Number ‘Nu’ increases while it decreases with increase

in D

u

. Nusselt number first decreases and then increases with respect to time. Sherwood Number ‘Sh’ increases

with increase in D

u

while it decreases with increase in P

r

, S

r

and t.

CONCLUSION

This study investigated the MHD flow through a porous medium past an inclined infinite plate, accounting for

Soret and Dufour effects with variable viscosity and periodic suction/injection. The governing equations were

solved using the Crank-Nicolson implicit finite difference method. The following conclusions are drawn from

the results:

Flow Field Characteristics (Velocity Profile)

 Enhancement of Flow: The fluid velocity u increases significantly with an increase in the Soret number,

Dufour number, Grashof number Gr Solutal Grashof number Gm. This is attributed to the enhancement of

thermal and solutal buoyancy forces. Additionally, an increase in the permeability parameter K reduces

Darcian resistance, therefore the flow accelerates.

P

r

0.71, 2, 2.5

G

r

= 20,

G

m

= 25,

M= 2,

D

u

= 0.2

S

c

= 2.01,

S

r

=0 .4,

=

30

, t = 0.2,

K = 0.4,

= 0.2,

= 0.1

y

0

1

2

3

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

www.rsisinternational.org

Page 69

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

 Retardation of Flow: The velocity is found to decrease with an increase in the magnetic parameter M,

Prandtl number Pr, and inclination parameter λ. The decrease with M is due to the development of the

resistive Lorentz force, which acts as a drag against the fluid motion.

 Variable Viscosity Effect: The study reveals that the fluid velocity is highly sensitive to viscosity changes.

As the viscosity parameter α increases or decreases—representing a decrease or increase in the fluid's

internal resistance—the velocity profile shows a marked increment or decrement. This confirms that treating

viscosity as a variable rather than a constant is essential for accurately predicting the flow behavior of real

fluids properties.

Thermal and Concentration Distributions

Temperature Profile: The temperature of the fluid increases with higher values of the Dufour number diffusion-

thermo effect and time. Conversely, an increase in the Prandtl number leads to a thinning of the thermal boundary

layer, resulting in decreased fluid temperature.

Concentration Profile: The concentration increases with the Soret number due to thermo-diffusion effects and

decreases with time.

Surface Gradients (Skin Friction, Nusselt, and Sherwood Numbers)

Skin Friction: The skin friction coefficient at the plate increases with higher values of Gr, Gm, Sr, Du, and K,

reflecting the acceleration of the fluid near the boundary. It decreases with increasing magnetic field strength M

and Prandtl number Pr.

Nusselt number: The Nusselt number increases with Pr and Sr, indicating enhanced heat transfer at the surface.

However, it decreases as the Dufour number Du increases.

Sherwood number: The Sherwood number is enhanced by the Dufour effect but is suppressed by an increase

in the Prandtl number, Soret number, and time. This highlights the strong coupling between heat and mass

diffusion in the presence of cross-diffusion effects.

Conflicts of Interest: The authors declare no conflicts of interest.

We are thankful to reviewers and Editors of the journal, and software company (Mathematica) for developing

the techniques that help in the computation and editing. There is no financial support for this research from any

organization from anywhere.

REFERENCES

1. Abbas Tariq , Rehman Sajid , Ali Shah Rehan, Muhammad Idrees and Mubashir Qayyum (2020).

Analysis of mhd carreau fluid flow over a stretching permeable sheet with variable viscosity and thermal

conductivity, Physica A: Statistical Mechanics and its Applications, 551: Article ID 124225.

2. Alam M. S., Rahman M. M. ( 2005). Dufour and Soret effect on MHD free convective Heat and Mass

transfer ﬂow past a vertical ﬂat plate embedded in a porous medium. Journal of Naval Architecture and

Marine engineering, 2(1): 55-65.

3. Ali M. E. (2006). The Effect of Variable Viscosity on Mixed Convection Heat Transfer Along a Vertical

Moving Surface”, Int. J. Therm. Sci., 45: 60–69.

4. Begai S. (2004). Effect of Variable Viscosity on Free Convection over a Non-Isothermal Axisymmetric

Body in a Porous Medium with Internal Heat Generation”, Acta Mech., 169: 187–194.

5. Bhavana M., Kesaraiah Chenna D., Sudhakarraiah A. (2013). The Soret effect on free convective

unsteady MHD ﬂow over a vertical plate with heat source, Int. J. of Innovative R. in Sci. Eng. And

Tech, 2(5): 1617-1628.

Page 70

www.rsisinternational.org

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

6. Carnahan Brice, Luthor H. A., Wilkes J. O. (1969) Applied Numerical Methods. John Wiley and Sons,

New York.

7.

Cowling T. G. (1957). Magneto-Hydrodynamics. Inter Science Publishers, New York.

8. Dursunkaya Z., Worek W.M. (1992). Diffusion thermo and thermal diffusion effects in transient and

steady natural convection from vertical surface, International Journal of Heat and Mass Transfer, 35(8):

2060-2065.

9.

Hady F. M., Bakier A. Y. and Gorla R. S. R. (1996). Mixed Convection Boundary Layer Flow on a

Continuous Flat Plate with Variable Viscosity”, Heat Mass Transfer, 31: 169–172.

10. Haroon Ur Rasheed, Saeed Islam, Zeeshan, Abbas Tariq & Khan Jahangir (2022). Analytical treatment

of MHD flow and chemically reactive Casson fluid with Joule heating and variable viscosity effect,

Waves In Random And Complex Media, https://doi.org/10.1080/17455030.2022.2042622.

11. Hayat T. and Ali N. (2008). Effect of Variable Viscosity on the Peristatic Transport of a Newtonian Fluid

in an Symmetric Channel”, Appl. Math. Model, 32: 761–774.

12. Hayat T., Ellahi R. and Asghar S. (2007). The Influence of Variable Viscosity and Viscous Dissipation

on the Non-Newtonian Flow: Analytical Solution, Commun. Nonlinear Sci. Numer. Simulat, 12: 300–

313.

13. Hossain Md. A., and Munir Md. S and Rees D. A. S. (2000). Flow of Viscous Incompressible Fluid with

Temperature Dependent Viscosity and Thermal Conductivity Past a Permeable Wedge with Uniform

Surface Heat Flux, Int. J. Therm. Sci., 39: 635–644.

14. Hossain Md. A., and Munir Md. S. (2000). Mixed Convection Flow from a Vertical Flat Plate with

Temperature Dependent Viscosity”, Int. J. Therm. Sci., 39: 173–183.

15. Jang, J. -Y. and Lin C. -N. (1988). Free Convection Flow over a Uniform-Heat-Flux Surface with

Temperature-Dependent Viscosity”, Heat Mass Transfer, 23: 213–217.

16. Kafoussias N. G. and Williams E. W. (1995). The Effect of Temperature-Dependent Viscosity on Free-

Forced Convective Laminar Boundary Layer Flow Past a Vertical Isothermal Flat Plate”, Acta Mech.,

110: 123–137.

17. Ke Y. and Solomatov V. S. (2004). Plume Formation in Strongly Temperature-Dependent Viscosity

Fluid over a Very Hot Surface”, Phys. Fluids, 16: 1059–1063.

18. Mahmoud M. A. A. (2007). Effect of Variable Viscosity on Combined Forced and Free Convection

Boundary-Layer Flow over a Horizontal Plate with Blowing or Suction, J-KSIAM, 11: 57–70.

19. Mahmoud M. A. A. (2007). Variable Viscosity Effects on Hydromagnetic Boundary Layer Flow along

a Continuously Moving Vertical Plate in the Presence of Radiation, Appl. Math. Sci. (Hikari), 1: 799–

814.

20. Massoudi M. and Christie I. (1995). Effects of Variable Viscosity and Viscous Dissipation on the Flow

of Third Grade Fluid in a Pipe”, Int. J. Non-Linear Mech., 30: 687–699.

21. Mostafa A. A. Mahmoud (

2009). Thermal Radiation Effect on Unsteady MHD Free Convection Flow

Past a Vertical Plate with Temperature-dependent Viscosity,

Can. J. Chem. Eng., 87: 47–52.

22. Mukhopadhyay S. and Layek G. C. (2008). Effect of Thermal Radiation and Variable Fluid Viscosity on

Free Convective Flow and Heat Transfer Past a Porous Stretching Surface, Int. J. Heat Mass Transfer,

51: 2167–2178.

23. Pandya N. and Quraishi M. S. (2018). Effect of Variable Permeability and Variable Magnetic field on

MHD Flow past an inclined plate with exponential Temperature and Mass Diffusion with Chemical

reaction through Porous media”, International Journal of Scientific Research in Physics and Applied

Sciences, 6(5): 62-68.

24. Pop I., Gorla R. S. R. and Rashidi M. (1992). The Effect of Variable Viscosity on Flow and Heat Transfer

to a Continuous Moving Flat Plate, Int. J. Eng. Sci., 30: 1–6.

25. Postelnicu A. (2004). Inﬂuence of a magnetic ﬁeld on heat and mass transfer by natural convection from

vertical surface in porous media considering Soret and Dufour effects. International Journal of Heat and

Mass Transfer, 47(67): 1467-1472.

26. Raptis, Perdikis C. (1999). Radiation and free convection ﬂow past a moving plate, Appl. Mech. Eng.,

4(4): 817-821.

27. Siti Suzilliana Putri Mohamed Isa, Hazirah Mohd Azmi, Nanthini Balakrishnan, Norihan Md. Arifin,

Haliza Rosali (2023), Soret-Dufour Effects on Heat and Mass Transfer of Newtonian Fluid Flow over

www.rsisinternational.org

Page 71

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue V, May 2026

the Inclined Sheet and Magnetic Field, Journal of Advanced Research in Numerical Heat Transfer 14,

Issue 1 (2023) 39-48.

28. Sparrow E. M., Cess R. D. ( 1961). Effect of Magnetic ﬁeld on free convection heat transfer. Int. J Heat

Mass Transfer, 3: 267-70.

Table 2: Skin friction coefficient τ, Nusselt number Nu and Sherwood number Sh for different values of

parameters

Gr

Gm

α

M

Pr

Sr

λ

t

K

Du

τ

Nu

Sh

20

25

0.2

2

0.71

0.4

300

0.2

0.4

0.2

6.77909

1.71408

1.26437

25

0.2

2

0.71

0.4

300

0.2

0.4

0.2

7.73222

1.71408

1.26437

30

25

0.2

2

0.71

0.4

300

0.2

0.4

0.2

8.68536

1.71408

1.26437

20

30

0.2

2

0.71

0.4

300

0.2

0.4

0.2

7.37239

1.71408

1.26437

20

35

0.2

2

0.71

0.4

300

0.2

0.4

0.2

7.96570

1.71408

1.26437

20

25

0.7

2

0.71

0.4

300

0.2

0.4

0.2

7.11135

1.71408

1.26437

20

25

1.2

2

0.71

0.4

300

0.2

0.4

0.2

7.41848

1.71408

1.26437

20

25

0.2

4

0.71

0.4

300

0.2

0.4

0.2

6.34006

1.71408

1.26437

20

25

0.2

6

0.71

0.4

300

0.2

0.4

0.2

5.95834

1.71408

1.26437

20

25

0.2

2

2.00

0.4

300

0.2

0.4

0.2

6.05806

3.22211

0.424553

20

25

0.2

2

2.50

0.4

300

0.2

0.4

0.2

5.91942

3.74963

0.105401

20

25

0.2

2

0.71

0.6

300

0.2

0.4

0.2

7.07635

1.75024

0.956001

20

25

0.2

2

0.71

0.8

300

0.2

0.4

0.2

7.38356

1.78967

0.622159

20

25

0.2

2

0.71

0.4

450

0.2

0.4

0.2

5.53510

1.71408

1.26437

20

25

0.2

2

0.71

0.4

600

0.2

0.4

0.2

3.91391

1.71408

1.26437

20

25

0.2

2

0.71

0.4

300

0.1

0.4

0.2

4.63594

1.86770

2.09706

20

25

0.2

2

0.71

0.4

300

0.3

0.4

0.2

8.25896

1.75878

0.795038

20

25

0.2

2

0.71

0.4

300

0.4

0.2

9.45491

1.87330

0.453853

20

25

0.2

2

0.71

0.4

300

0.2

0.6

0.2

6.96962

1.71408

1.26437

20

25

0.2

2

0.71

0.4

300

0.2

0.8

0.2

7.06911

1.71408

1.26437

20

25

0.2

2

0.71

0.4

300

0.2

0.4

6.88755

1.56573

1.35475

20

25

0.2

2

0.71

0.4

300

0.2

0.4

0.6

7.00158

1.38527

1.46660

20

25

-0.2

2

0.71

0.4

300

0.2

0.4

0.2

6.49224

1.71408

1.26437

20

25

-0.7

2

0.71

0.4

300

0.2

0.4

0.2

6.10267

1.71408

1.26437

20

25

-1.2

2

0.71

0.4

300

0.2

0.4

0.2

5.67172

1.71408

1.26437