unsteady magneto hydrodynamic f low of casson f luid ... · [email protected] abstr act the...

Unsteady Magneto Hydrodynamic Flow of Casson Fluid

through Parallel Plate Channel with Heat Source

C.K.Kirubhashankar

Sathyabama Institute of Science and Technology, Chennai, India

[email protected]

Abstract

The intention of this paper is to examine the unsteady magneto hydrodynamic flow of Casson fluid in

presence of heat transfer through parallel plate. The examination uncovers numerous essential parts of flow

and heat transfer. By employing appropriate transformations, the governing partial differential conditions

relating to the momentum and energy equations are changed into ordinary differential conditions. The flow

highlights and warmth exchange attributes for various estimations of the representing parameters viz. Casson

parameter, heat source parameter, Hartmann number, and Prandtl number are investigated and talked about in

detail. It was discovered that heat source and magnetic field transforms the flow pattern and increment the

temperature of the fluid. It helps the community working in the field of Physiological fluid dynamics yet

additionally to the medical professionals.

Key word: Parallel Plate Channel, Boundary Layer, Heat Source, Magnetic Field, Casson fluid.

1. Introduction

The investigation of Casson flow and heat transfer in a viscous fluid is of impressive premium as a result of

their consistently expanding modern applications and vital heading on a few mechanical procedures. Amid the

most recent decades broad research work has been done on the fluid dynamics of biological fluids in the

presence of magnetic field. For various reasons, utilizations of magneto hydrodynamics in physiological flow

issues are of developing interest. The flow because of extending of a flat surface was first researched by

Crane [1]. Pavlov [2] examined the impact of exterior magnetic field on the MHD flow over an extending

sheet. Andersson [3] examined the MHD flow of viscous fluid on an extending sheet and Mukhopadhyay et

al. [4] introduced the MHD flow and heat transfer over an extending sheet with variable fluid consistency.

Examples of Casson fluid incorporate jam, tomato sauce, nectar, soup and focused natural product juices, and

so on. Human blood can likewise be dealt with as Casson fluid. Due to presence of numerous substances like,

protein, fibrinogen and globulin in a fluid base plasma, human red platelets can shape a chainlike structure,

rouleaux. Several investigations were made to look at flow over an extending/contracting sheet under various

parts of MHD, suction/injection, heat and mass transfer and so on [5– 12]. In the event that the rouleaux carry

on like a plastic solid, at that point there exists a yield pressure that can be related to the steady yield stress in

Casson's fluid [13– 15]. Adhikary and Misra [16] exhibited a exact solution of the issue of oscillatory flow of

a fluid and heat transfer along a permeable oscillating channel in presence of an exterior magnetic field.

Tzirtzilakis [17] considered a scientific model of biomagnetic fluid dynamics (BFD), reasonable for the

portrayal of the Newtonian blood flow under the activity of magnetic field. This model is reliable with the standards of ferrodynamics and magneto hydrodynamics and considers both magnetization and electrical conductivity of blood. Ramamurthy and shanker [18] contemplated magneto hydrodynamic consequences for

blood flow through a permeable channel. They considered the blood a Newtonian fluid and conducting fluid.

International Journal of Pure and Applied MathematicsVolume 119 No. 15 2018, 1185-1195ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

1185

Blood vessel MHD pulsatile flow of blood under periodic body acceleratiion has been considered by Das and

Saha [19]. The blood flow in extremely narrow vessels under the impact of transverse magnetic field has

been examined by Madhu et al. [20]. The detailed discussion on the stagnation-point flow over

extending/contracting sheet can be found in progress Bhattacharyya [21– 23].

Inspired by the previously mentioned examinations on flow of Newtonian and non-Newtonian fluids owing

to an extending sheet and its tremendous applications in several enterprises. In the present study, a numerical

model for the unsteady fluid flow through parallel plate channel with heat transfer and external transverse

magnetic field is exhibited. An endeavor is made in this paper to broaden the work by Islam M. Eldesoky

[22] for non-Newtonian Casson fluid and heat transfer under the conditions characterized in our model. The

primary target of the present work is to get analytical expressions for axial velocity, temperature distribution

and normal velocity using new boundary conditions and with converting the system of partial differential

equations into system of ordinary differential equations. The impacts of magnetic field (Hartmann number

(Ha)), Casson parameter, heat source parameter (S) and Prandtl number (Pr) on the axial velocity, temperature

distribution and normal velocity are researched and analyzed with the assistance of their graphical

representations.

2. Mathematical analysis of the flow

Consider the unsteady two-dimensional flow and heat transfer of an incompressible Casson fluid over an

exponentially extending/contracting sheet at 𝑦 = 0, with the flow being restricted in 𝑦 > 0. The fluid is

electrically conducting in the presence of a uniform magnetic field applied normal to the sheet, and the

induced magnetic field is ignored under the approximation of small Reynolds number.

The rheological equation of state for an isotropic and incompressible flow of a Casson fluid as follows:

cij

c

yB

cijy

B

ij

ep

ep

22

,22

,

where ij is the (i, j)-th component of the stress tensor, = eij eij and eij are the (i, j)-th component of the

deformation rate, is the product of the component of deformation rate with itself, c is a critical value of this

product based on the non-Newtonian model, B is plastic dynamic viscosity of the non-Newtonian fluid , and

py is the yield stress of the fluid.

So, 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.

Considering u and v as velocity components in the directions of x and y respectively (axial and normal

respectively) at time t in the flow field, we may write the two dimensional boundary layer equations in

presence of transverse magnetic field as

0

y

v

x

u (1)

)(1

11

0

20

2

2

TTguB

y

u

x

p

t

u

(2)

)( 02

2

TTC

Q

y

T

C

K

t

T

pp

(3)

where 𝜐 is the kinematic fluid viscosity, 𝜌 is the fluid density, = 𝜇𝐵√2𝜋𝑐/𝑝𝑦 is the Casson parameter, 𝜍 is the

electrical conductivity of the fluid, and 𝐻0 is the strength of magnetic field applied in the 𝑦 direction. and

International Journal of Pure and Applied Mathematics Special Issue

1186

is the density and viscosity of the blood while p∗ stands for pressure. K‟ is thermal conductivity; CP is the

specific heat at constant pressure. Q is the quantity of heat, T is the temperature and β is the volumetric

expansion parameter while θ is the temperature distribution.

The boundary conditions are taken as:

te2 , teu

2 at 1y

0 , 0u at 1y

Let us introduce the non-dimensional variables,

h

xx * ,

h

yy * ,

hm

uu

2/

* , hm

vv

2/

* /2

*

h

tt ,

32

*

2/),(

hm

dxdptxp

,

32

*

2/ hm

(4)

Substituting equation (4) into equations (1) – (3), we get

0

y

v

x

u (5)

guHay

up

t

u

2

2

211 (6)

rr P

S

yPt

2

21 (7)

where the heat source parameter, TK

QhS

2

, Prandtl number,T

p

rK

CP

3. Analytical Solution of the Problem

With the above discussions in the previous section, let us choose the solutions of the equations (5) – (7)

respectively as

teyFtyu2

)(),( (8)

teyGtyv2

)(),( (9)

teyHty2

)(),( (10)

The boundary conditions are transformed to

,1,1 FH at 1y

,0,0 FH at 1y (11)

By virtue of (8) – (10), we obtain the equations (5) – (7) respectively as

yHgpyFHayF

1)(

1

22 (12)

CG (13)

02 yHPSyH r (14)

Solution of equation (14) using the boundary condition (11) is as follows

yyyH

sinsin2

1cos

cos2

1 (15)


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where rPS 2

From (10) and (15) the temperature distribution is given by

teyyty2

sinsin2

1cos

cos2

1),(

(16)

Using the equation (16) into equation (12) we get

yygpyFHayF

sin

sin2

1cos

cos2

1

1)(

1

22 (17)

From equation (8) and (17) the velocity of the flow of the fluid parallel to the direction of the channel is

obtained as,

teycycycycctyu2

sincossincos),( 54321 (18)

where te

pp

2

*

1

, 22

1Ha

,

2

11

pc ,

cos2 222

g

c ,

sin2 223

g

c

cos2

cos221 214

ccc

,

sin2

sin21 35

cc

From equations (9) and (13), the velocity of the fluid flow perpendicular to the direction of the channel is

given by tCetyv

2

),( (19)

where C is an arbitrary constant.

Equations (16), (18) and (19) show the temperature distribution, the axial velocity and normal velocity

respectively.

3. Results and Discussions

In this section, we discuss the different physical parameters, such as heat source parameter (S), Hartmann

number (M), Prandtl number (Pr) and decay parameter () on temperature distribution, axial velocity and

normal velocity. The obtained computational results are presented graphically and the variations in velocity

and temperature are discussed.

3.1. Effects of different physical parameters on temperature fields

Figure 1 shows the performance of temperature distributions versus y at = 0.5, Pr = 1, = 0.5, and t = 1for

different values of heat source parameter (S = 1, 1.75, 2.5, 3.25, 4). We observe that the temperature field

decreases with increasing the values of S, for y 0.5, and temperature field increases for y 0.5. The

maximum effect of heat source is at y = -1.

Figure 2 emphasizes that the temperature field distribution for different values of Prandtl number (Pr = 1, 3, 5,

7, 9) at S = 1, = 0.5, = 0.2, and t = 1. The effect of Prandtl number on temperature steadily decreases with

increasing the values of Prandtl number.

It is clear from Figure 3 that temperature field distribution decreases with increasing the decay parameter up

to y 0.16 and it increases with increasing the decay parameter for y 0.16, at S = 1, Pr = 1, = 0.2, and

t = 1 for different values of decay parameter ( = 0.5, 0.75, 1, 1.25, 1.5). The maximum effect of decay

parameter on the temperature field is between -0.8 y -0.4.


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Figure 1. Temperature Field for different values of values of Heat Source Parameter (S)

Figure 2. Temperature Field for different values of values of Prandtl Numer (Pr)

Figure 3. Temperature Field for different values of values of Prandtl Numer (Pr)


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3.2. Effects of different physical parameters on velocity fields

Figure 4 exhibits the axial velocity profiles for several values of heat source parameter (S = 1, 1.75, 2.5, 3.25,

4) at = 1.5, Pr = 1, = 0.2, t = 1, = 0.2, g = 9.8, = 0.5, p = 0.5 and Ha = 1. It is observed that axial

velocity increases with increasing the heart source parameter S for y -0.3 and the effect reverses for

-0.3 y 1.

Figure 5 indicates the effect of magnetic field on the axial velocity for different values of Hartmann number

(Ha = 1, 1.25, 1.5, 1.75, 2) at S = 1, = 0.5, Pr = 1, = 0.2, t = 1, = 0.2, g = 9.8, = 0.5 and p = 0.5. It is

observed that the axial velocity decreases with increasing the magnetic field up to y -0.3 and for y -0.3

axial velocity increases with increasing the magnetic field.

The effect of Prandtl number on the distribution of the axial velocity at S = 1, = 0.5, Pr = 1, = 0.2, t = 1,

= 1, g = 9.8, = 0.5, p = 0.5 and Ha = 1is shown in Figure 6. It is observed that axial velocity decreases

with increasing the Prandtl number.

Figure 7 defines the effect of decay parameter on the axial velocity for different values of decay parameter

( = 0.5, 0.75, 1, 1.25, 1.5) at S = 1, = 0.5, Pr = 1, = 0.2, t = 1, = 1, g = 9.8, = 0.5, Ha = 2 and p = 0.5.

It is observed that the axial velocity increases with increasing the decay parameter up to y -0.3 and for

y -0.3 axial velocity decreases with increasing the decay parameter.

Effects of Casson parameter on velocity profiles for unsteady motion are clearly exhibited in Figure 8 the

behavior of velocity with increasing is noted at S = 1, = 0.5, Pr = 1, = 0.2, t = 1, = 0.2, g = 9.8, = 0.5

and p = 0.5. It is observed that the axial velocity decreases with increasing the magnetic field up to y -0.2

and for y -0.2 axial velocity increases with increasing the magnetic field.

Figure 4. Axial velocity for different values of values of Heat Source Parameter (S)

Figure 5. Axial velocity for different values of values of Magnetic Field Parameter (Ha)


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Figure 6. Axial velocity for different values of values of Prandtl Numer (Pr)

Figure 7. Axial velocity for different values of Decay Parameter (S).

Figure 8. Axial velocity for different values of Casson Parameter ().


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Figure 9. Normal velocity for different values of Decay Parameter ().

Normal velocity for different values of decay parameter is shown in Figure 9. It is depicted that normal

velocity is decreasing with increasing values of and also for increasing values of t. The normal velocity is

decreasing slowly at low values of the decay parameter ( = 1) while it decreases very fast and tends to zero

at high values of decay parameter ( = 3).

4. Conclusion

The present study provides the solution for the unsteady Casson fluid flow through the parallel plate channel

with heat source and external transverse magnetic field is presented. The present work is the effect of

magnetic field, heat source and Casson parameter seems to be significant.

The present mathematical model gives a simple form of axial velocity, temperature distribution and normal

velocity of the flow. Analytical expressions are obtained by choosing the axial velocity; temperature

distribution and the normal velocity of the flow depend on y and t only.

The temperature field decreases with increasing the heat source parameter (S), Prandtl number(Pr) and the

decay parameter(). And temperature field increases with increasing the heat source parameter (S) and decay

parameter () for y 0.5 and y 0.16 respectively. For y -0.3, the axial velocity increases with increasing

the heart source parameter (S) and decay parameter () and the effect reverses in -0.3 y 1. The axial

velocity decreases with increasing the magnetic field (Ha), Prandtl number (Pr) and Casson parameter ().

And axial velocity increases with increasing the magnetic field (Ha) and Casson parameter () for y -0.5 and

y -0.2 respectively. The effect of increasing values of the Casson parameter is to suppress the velocity field.

Prandtl number can be used to increase the rate of cooling in conducting flows. The normal velocity decreases

with increasing the decay parameter and tending to zero very fast for higher values of the decay parameter.

The results may be helpful for possible technological applications in liquid-based systems involving

stretchable materials.

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