casson fluid flow effects on mhd unsteady heat and mass transfer … · 2021. 6. 4. · chemical...

International Journal of Scientific and Research Publications, Volume 10, Issue 11, November 2020 104

ISSN 2250-3153

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Casson Fluid Flow Effects on MHD Unsteady Heat

And Mass Transfer Free Convective Past An Infinite

Vertical Plate

1 Faruk Abdullahi, 2Ibrahim Haliru Wala and 3Muhammad Abdurrahman Sani,

1&3Department of Mathematics Waziri Ummaru Federal Polytechnic BirninKebbi, Kebbi State 2Department of Statistics Waziri

Ummaru Federal Polytechnic BirninKebbi, Kebbi State

Email: [email protected]

DOI: 10.29322/IJSRP.10.11.2020.p10712 http://dx.doi.org/10.29322/IJSRP.10.11.2020.p10712

Abstract

The study was carried out numerically to study to investigate casson fluid effects on magneto-hydrodynamics (MHD) unsteady heat

and mass transfer free convective past an infinite vertical plate. Appropriate dimensional quantities were applied to change the

dimensional governing coupled non-linear partial differential equations to non-dimensional form. Numerical solution of the

dimensionless governing coupled boundary layer partial differential equations were obtained using finite element method (FEM). The

expressions of velocity, temperature, concentration, skin friction, Nusselt number as well as Sherwood number have been obtained

and discussed using line graph. From the result, it was revealed that, increase of porosity parameter K, ratio of mass transfer

parameter N, and Eckert number Ec enhances the velocity profile and reverse is the case with the with increase of Magnetic

parameter M, Casson fluid parameter β and Prandtl number Pr. Similarly all the above parameters provided the same effects on

temperature profile except on casson fluid parameters β which provided an opposite effects. Skin friction at y = 0 and y = 1 gets

enhanced with increase in casson fluid parameter β and gets reduced with increase in porosity parameter K. Increase in casson fluid

parameter β decreases Nusselt number at y = 0 and it gets enlarged by increasing the value of Eckert number Ec. At y = 1 Nusselt

number gets enhanced with increase of casson fluid parameter β and gets reduced with increase in of Eckert number Ec. Sherwood

number at y = 0 and y=1 has no any significant by increasing casson fluid parameter and Schmidt number Sc.

1 Introduction

Normally Casson fluid displays yield stress. Casson is known with a shear thinning liquid which is believed to have an

infinite viscosity at zero rate of shear. A yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear, i.e.,

if a shear stress less than the yield stress is applied to the fluid, it behaves like a solid, whereas if a shear stress greater than yield stress

is applied, it starts to move (Ananda, Reddy and Janardhan 2017). A Casson fluid which is normally exemplified as Honey, human

blood, jelly, sauces and so on are very important. Casson fluids are very important fluid used in various fields like biological,

chemical, medical, metallurgical and engineering. Also casson fluid is applied in food processing and drilling operation. Additionally,

Casson fluid is applied in pharmaceutical products, coal in water, china clay, paints, synthetic lubricants, and biological fluids such as

synovial fluids, sewage sludge, jelly, tomato sauce, honey, soup, and blood due to its contents such as plasma, fibrinogen, and protein

(Jawad, Azizah and Zurni 2016). Many researchers carried out their research work in industrial environment on the on the flow of the

Casson fluid of the effects of various parameters.

Sheikh, Parth, Sarder and Shikdar (2019) studied mass and heat transfer behavior of viscous dissipative chemically reacted

casson fluid. Which is flowing with the impact of suction, thermal conductivity and variable viscosity is being discussed explicitly in

this paper. The investigation of analytical analysis of unsteady MHD boundary layer flow of Casson fluid past an oscillating vertical

porous surface subjecting to Newtonian heating using Laplace transform technique was carried out by Manjula and Chandra (2018).

Additionally, Kartini, Syafrina and Zahir (2018) Analyzed Casson fluid flow with variable viscosity in porous media over a heated

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stretching sheet. Rama, Prof. Moreover, Viswanatha, and Balakrishna (2018) studied An Unsteady MHD free convection flow of

casson fluid past an exponentially accelerated infinite vertical plate through porous media in the presence of thermal radiation and

heat source or sink. The analysations of Couette flow of a Casson fluid in an inclined composite duct comprising of porous and fluid

layers of different thickness was investigated by Manoj, Sreenadh, Gopi Krishna and Srinivas (2019). Veena, Vinuta and Pravin

(2017) investigated two-dimensional flow of non-Newtonian MHD flow of Casson fluid heat transfer with PST & PHF was

considered, using Nevier Stoke’s Equations of Motion the momentum and energy equations of Casson fluid were derived

Imran, Sharidan and Ilyas (2016) studied combined effects of slip condition and Newtonian heating on MHD free

convectional flow of Casson fluid over a nonlinearly stretching sheet in saturated porous medium. Also Kamran, Hussain, Sagheer

and Akmal (2017) studied the Casson nanofluid past horizontal stretching surface with magnetic effect and Joule heating under the

consideration of Slip and thermal convective boundary conditions. They discovered that, increase Casson fluid parameter the declines

the velocity profile and rises temperature profile. Ananda et., al (2017) studied the combined effects of chemical reaction, radiation,

Dufour and Soret effects on Casson MHD fluid flow over a vertical plate with heat source / sink and concluded that velocity falls

down with the increase of Casson parameter and temperature is decreased when thermal radiation is increased. The study of the

effects of non-Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet in a porous medium was conducted by Bhim,

Rachid and Hasan (2019). They revealed that increase in the value of the Casson parameter decreases velocity and increases skin

friction. Hamzeh (2018) studied the effect of MHD free convective boundary layer flow about a solid sphere in a micropolar Casson

fluid. They reported that increasing in Casson parameter increases both values of the local Nusselt number and angular velocity

profile. While the skin friction coefficient, temperature and velocity profiles decrease. Hasan and Zillur (2019) investigated the flow

of Casson fluid and heat transfer over a permeable vertical stretching surface considering the effects of magnetic field and thermal

radiation. They revealed that increasing the value of Casson parameter leads to the increase of temperature and decrease in velocity

profile. Pramanik (2014) investigated the boundary layer flow of a non-Newtonian fluid accompanied by heat transfer toward an

exponentially stretching surface in presence of suction or blowing at the surface. He concluded that Momentum boundary layer

thickness decreases with increasing Casson parameter.

The analysis of (MHD) flow of a Casson fluid over an infinite vertical oscillating plate embedded in a porous medium was

carried out by Abid, Mohd, Ilyas and Razman (2017). They revealed that fluid flow can be controlled by the increasing the values of

Prandtl number as well as by the increasing of Casson parameter. Hassan, Sajjad, Rabia, Amna .and Shamila (2017) investigated the

mixed convection radiative heat transfer of electrically conducting Casson fluids. The fluid flows past a permeable stretching sheet

lying in the porous medium. They revealed that, the velocity reduces in magnitude with increase in Casson parameter β. Furthermore,

Renuka, Ganga, Kalaivanan, and Abdul Hakeem, (2017) investigated Ohmic dissipation effects of Casson fluid in the presence of

inclined magnetic field over a stretching sheet with slip and thermal radiation. They concluded that the velocity of the non-Newtonian

fluid reduces with the increasing aligned angle of magnetic field, Casson parameter, velocity slip parameter and magnetic parameter.

The investigation of the impact of induced magnetic field on Casson fluid flow past a vertical plate was done by Parandhama, Rajua,

and Changal (2019). They revealed that, the velocity falls down with an increase of casson fluid parameter . Moreover, Hymavathi

and Sridhar (2016) investigated the effect of mass transfer of a MHD Casson fluid over a porous stretching sheet in presence of

chemical reaction using Keller box method. They discovered that increase in Casson parameter β increases the velocity profile, while

reverse is the case for temperature and concentration profile.

Prabhakar (2016) studied the effects of mass transfer on an unsteady free convection flow of viscous dissipative fluid past an

infinite vertical porous plate under the influence of a uniform magnetic field applied normal to the plate. The present research work


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tends to adopt and extend Prabhakar model through incorporating the casson fluid parameter on magneto-hydrodynamics (MHD)

unsteady heat and mass transfer free convective past an infinite vertical plate. The study employed finite element method (FEM) to

find numerical solution of governing coupled non-linear boundary layer partial differential equations. The expressions of velocity,

temperature, concentration, skin friction, Nusselt number as well as Sherwood number have been obtained and discussed using line

graph.

2 Formulation of the Problem

Consider an unsteady free convection flow of an incompressible electrically conducting viscous dissipative fluid past an

infinite vertical porous plate. Let the x* -axis be chosen along the plate in the vertically upward direction and the y* axis is chosen

normal to the plate. A uniform magnetic field of intensity H0 is applied transversely to the plate. The induced magnetic field is

neglected as the magnetic Reynolds number of the flow is taken to be very small. Initially, the temperature of the plate *T and the

fluid 𝑇𝑤∗ are assumed to be the same. The concentration of species at the plate 𝐶𝑤

∗ and 𝐶0∗ are assumed to be the same. At time t*>0,

the plate temperature is changed to 𝑇𝑤∗ , which is then maintained constant, causing convection currents to flow near the plate and mass

is supplied at a constant rate to the plate. Under these conditions the flow variables are functions of time y* and t* alone. The problem

is governed by the following equations:

2 2 ** 2 * ** * * * 0

0* *2 *

1( ) ( )

1

e H uu u vug T T g C C

t y K

(1) 2

* 2 * *

* *2 *p

T T uC k

t y y

(2) * 2 *

* *2 M

C CD

t y

(3)

The corresponding initial and boundary conditions are:

(4)

We now introduce the following non- dimensional quantities into the basic equations and initial and boundary conditions in order to

make them dimensionless

* * * * *

* * * * * * *

* * * * *

0, 0, ,

0, 0, , 0

0, ,

w w

t u T T C C for all y

t u T T C C at y

u T T C C as y


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21331

30 2 1

3

* ** *

R

* ** *

0 0

* *

0 R 0 0

2

0

m

* *

g T ( )( ) , L= T

v

t y, t= , y= ,

T

T CK, K= , = , =

vT

vPr , Sc= , ,

D

( )

(

R

w

w w

P

P

w

g TU vg T

v

T T TL

T Cuu

U T T C C

UCEc

k C T

C CN

T

2 2

0 0

* *

(5)

, )

R

W

H TM

T

On the

substitution of equations (5) into (1) - (4) the following governing equations in non-dimensional form are obtained.

2

2

1 1( )

1

u uGr N M u

t Ky

(6)

22

2

uPr Ec

t yy

(7)

2

2Sc

t y

(8)

The corresponding initial and boundary conditions are

0, 0, 0, 0

0 : 0, 1, 1 0

0, 0, 0

t u for all y

For t u at y

u at y

(9)

3 Method of the Solution

Equations (6) – (8) are coupled non-linear system of partial differential equations and finite element method (Galerkin

approach) would be used to solve them under the boundary conditions (9)

By applying Galerkin finite element method for equation (6) over the element ,e i jy y y is

2

2

1 1( ) 0

1

i

i

y

T

y

u uN u M N dy

t Ky

(10)

Equation (10) is shortened as:

2

2 120

i

i

y

T

y

u uN M M u P dy

ty

(11)


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1 2

1 1Where and and M

1M M P N

K

Applying integration by part to equation (10) yield:

2 2 1[ ] 0i i i i

i

i i i i

y y y yTT yj T T T

y

y y y y

u N u uM N M dy N dy M N udy P N dy

t y y t

(12)

Ignoring the first term of equation (12) we have:

2 1 0i i i i

i i i i

y y y yTT T T

y y y y

N u uM dy N dy M N udy P N dy

y y t

(13)

Let u(e) = uiN

i+ u

jN

jÞ u(e) = [N ][u]T

be a linear piecewise approximation solution over the two nodal element ,e

( )i jy y y where ( ) [ ]e

i ju u u , [ ]i jN N N also ui and u

j are the velocity component at the

thi and thj nodes of the

typical element ( )e ( )i jy y y moreover, and i jN N are called basis ( or shape) functions defined as follows:

j

i

j i

y yN

y y

, i

j

j i

y yN

y y

Hence equation (13) after simplifying becomes:

2M

Ni

'Ni

' Ni

'Nj

'

Ni

'Nj

' Nj

' Nj

'

é

ë

êê

ù

û

úú

yi

yi

òu

i

uj

é

ë

êê

ù

û

úú

dy +N

iN

iN

iN

j

NiN

jN

jN

j

é

ë

êê

ù

û

úú

yi

yi

òu

i

·

uj

·

é

ë

êêêê

ù

û

úúúú

dy + M1

NiN

iN

iN

j

NiN

jN

jN

j

é

ë

êê

ù

û

úú

yi

yi

òu

i

uj

é

ë

êê

ù

û

úú

dy - PN

i

Nj

é

ë

êê

ù

û

úú

yi

yi

ò dy = 0

(14)

Also simplifying equation (14) above we have:

2 11 1 2 1 2 1 1

01 1 1 2 1 2 16 6 2

i i i

j jj

u u uM M ll lP

u ulu

(15)

Where j il y y h and prime and dot denotes differentiation with respect to y

and t respectively. Assembling the equations

for the two consecutive elements 1i iy y y and i 1y iy y the following is obtained:

1

1 1

2 1

2

1 11

1 1 0 2 1 0 2 1 0 11

1 2 1 1 4 1 1 4 1 26 6 2

0 1 1 0 1 2 0 1 2 1

i

i i

ii i

i ii

uu u

M M Pu u u

lu u

u

(16)

Now if we consider the row corresponding to the node i to zero with l h , from equation (16) the difference schemes reads:

2 11 11 1 1 12

1( 2 ) ( 4 ) ( 4 )

6 6i i ii i i i i i

M Mu u u u u u u u u P

h

(17)

Using the trapezoidal rule on (17), the following system of equations in Crank-Nicolson method is obtained as:

A1u

i-1

n+1 + A2u

i

n+1 + A3u

i+1

n+1 = A4u

i-1

n + A5u

i

n + A6u

i+1

n + P*

(18)

Likewise, by solving (7) and (8) using the same method we have:

B1q

i-1

n+1 + B2q

i

n+1 + B3q

i+1

n+1 = B4q

i-1

n + B5q

i

n + B6q

i+1

n + Q*

(19)


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1 1 1

1 1 2 3 1 4 1 5 6 1

n n n n n n

i i i i i iC C C C C C

(20)

Where:

2 2 2

1 2 1 2 2 1 3 2 12 6 , 8 12 , 2 6A M r rM h A M r rM h A M r rM h

2 2 2

4 2 1 5 2 1 6 2 1A 2 6 , 8 12 4 , 2 6M r rM h A M r rrM h A M r rM h

B

1= Pr - 3r, B

2= 4Pr + 6r , B

3= Pr- 3r B

4= Pr + 3r,

B

5= 4Pr -6r, B

6= Pr + 3r

C

4= Pr + 3r, C

5= 4Pr - 6r, C

6= Pr + 3r

* 2P 12 ( )n n

i irh N , and2

* Q 6 Pru

r Ec Ry

With 2

kr

h and h and k are the mesh size along y direction and time direction respectively. Index i denotes space and j denotes to

the time. In equations (18), (19) and (20), taking 1(1)i n and using the initials and boundary conditions (9), the following system of

equations is obtained

i i iA X B 1(1)i n

Where iA matrices of are order n and and i iX B are column matrices having n components. The solution of the system of equation

are obtained using Thomas algorithm for velocity, temperature and concentration. For various parameters the results are computed

and p resented graphically.

The skin friction, Nusselt number and Sherwood number are important physical parameters for this type boundary layers flow. With

known values of velocity, temperature and concentration fields. The skin-friction at the plate is given by non-dimensional form:

0,1y

u

y

(21)

The rate of heat transfer coefficient can be obtained in the terms of Nusselt number in non-dimensional form as

0,1

u

y

Ny

(22)

The rate of mass transfer coefficient cab be obtained in terms of Sherwood number in non-dimensional form given by

0,1

(23)h

y

Sy

1 2 33 , 4 6 , Pr 3 C Pr r C Pr r C r


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4 Results and Discussion

We employed finite element method to solve equations (6) to (8) under the boundary conditions (9) in order to analyze the

effects various parameters on flow field in the boundary layer region. We studied the effects Prandtl number Pr , casson fluid

parameter , Eckert number Ec Schmidt number Sc , magnetic parameter M, porosity parameter K, Buoyancy effect parameter tr ,

ratio of mass transformation (N) on fluid velocity, temperature and concentration and they were presented graphically. Pr 0.71 ,

1 , 1Ec , 0.2Sc , 1M , 1K 1N . The values above were adopted to be default parameters values under the

present study. The velocity, temperature, and concentration profiles are presented in the following figures:

Figure 1: Effect of M on velocity profile

Figure 2: Effect Pr on velocity profile


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Figure 3: Effect of K on velocity profile

Figure 1 gives the details about the control of magnetic parameter M on velocity profile. From that figure it is observed that

the velocity profile begins to diminish at all point of the flow field by increasing the values of magnetic parameter M. This is true

since magnetic parameter produce resistive force, which acts opposite direction to the fluid motion. Similarly figure 2 gives the details

about the control Prandtl number Pr on fluid velocity profile. From that figure it is noticed that fluid velocity begins to diminish at

all point of the flow field by increasing the values of Prandtl number Pr . While Figure 3 gives the details control about porosity

parameter K on fluid velocity and it is also observed that fluid velocity begins to rise at all point of the flow field on increasing the

values porosity parameter K.

Figure 4: Effect of N on velocity profile


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Figure 5: Effect of β on velocity profile

Figure 6: Effect the different values Ec velocity profile

Figure 4 and figure 5 reveals the influence of the ratio of mass transfer parameter N and Casson fluid parameter β on the

fluid velocity respectively. It is observed that the fluid velocity gets intensified by increasing the values of both the ratio of mass

transfer parameter N and casson fluid parameter β. Figure 6 displays the effect of Eckert number Ec on the fluid velocity. It is

observed that the velocity get significant enhancement by increasing the values of Ec .


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Figure 7: Effect of M on temperature profile

Figure 8: Effect Pr on temperature profile

Figure 9: Effect and on temperature profile


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Figure 10: Effect 𝐸𝑐 on temperature profile

Figure 11: Effect of N on temperature profile

Figure 7 and 8; depict the influence of magnetic field parameter M and Prandtl number Pr on fluid temperature

respectively. From the both figures it is seen that, the fluid temperature diminishes by increasing the values of magnetic field

parameter M and Prandtl number Pr respectively. While the opposite behavior is observed in figure 9 by increasing the value of

casson fluid parameter β

Figure 10 and 11 displays the effect of Eckert number Ec and ratio of mass transfer parameter N on fluid temperature

respectively. It is observed that, from the both figures that temperature profile gets enlarged by increasing the values of Eckert number

Ec and ratio of mass transfer parameter N


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Figure 12: Effect of Sc on temperature profile

Figure 12 depicts the influence of Schmidt number Sc on temperature profile. From that figure it is seen that, the fluid

temperature get reduced by increasing the values of magnetic field

Figure 13(a) & 13 b(b): Effect and K on Skin friction

Figure 14(a) &14(b): effect and Ec on Nusselt number


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Figure 15(a) &15(b): effect and Sc on Sherwood number

Figure 13(a) and 13(b) displays the effect casson fluid parameter 𝛽 and porosity parameter K on the fluid skin friction. It is

seen that, increase in casson fluid parameter β has enhancing effect on skin friction in both figure. But increase in porosity parameter

K has diminishing effect on skin friction in both figures. Figure 14(a) and 14(b) displays the effects of casson fluid parameter β and

Eckert number Ec on Nusselt number. It is observed that in figure 14(a) increase in Eckert number magnifies the Nusselt number and

reverse is the case with increase in casson fluid parameter β. In Figure 14(b) Nusselt decreases with increasing Eckert number Ec and

no significant effect is observed with increase of casson fluid parameter β. Figure 15(a) and15 (b) displays the effect of Schmidt

number Sc and casson fluid parameter β on Sherwood number and it is clearly seen that both parameters have no any significant

effect on it.

5 Conclusion

In this paper, we have investigated casson fluid effects on magneto-hydrodynamics (MHD) unsteady heat and mass transfer

free convective past an infinite vertical plate. From the investigation, the following conclusions were drawn:

i. Increase of porosity parameter K, ratio of mass transfer parameter N, Eckert number Ec enhances the velocity while

reverse is the case with the increase of Magnetic parameter M, Casson fluid parameter β and Prandtl number Pr .

ii. Similarly increase of porosity parameter N, Eckert number Ec and Casson fluid parameter β enhances the temperature

profile and reverse is the case with the increase of Magnetic parameter M, and Prandtl number Pr

iii. Concentration profile gets reduced by increasing the value Schmidt parameter Sc

iv. Skin friction at y = 0 and y = 1 gets enhanced with increase in casson fluid parameter β and gets reduced with increase in

porosity parameter K.

v. Increase in casson fluid parameter β decreases Nusselt number at y = 0 and it increases it by the increase of Eckert number

Ec. At y = 1. Nusselt number magnifies with increase of casson fluid parameter β and reduces with increase in of Eckert

number Ec.

vi. casson fluid parameter β and Schmidt number Sc has no significant effects Sherwood number at both y = 0 and y=1


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References

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To A Stretching Sheet. International Journal of Mechanical Engineering and Technology, 8 (2), 16–26

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