Path of the Particles in Case of Unsteady State MHD Flow …€¦ · Path of the Particles in Case...

International Journal of Advances in Science and Technology,

Vol. 3, No.1, 2011

Path of the Particles in Case of Unsteady State MHD Flow in a

Porous Channel with an Exponentially Decreasing Suction

Ch. V. Ramana Murthy 1 and P. Hari Prasad 2

1Department of Applied Mathematics, Lakireddy Balireddy College of Engineering,

Mylavaram - 521 230 (INDIA)

Email id: [email protected]

2 K. L. University, Vaddeswaram - 522502 (INDIA)

Abstract

The trace of the particles in the case of unsteady state MHD flow in a porous channel with an exponentially

decreasing suction has been analysed in this paper. The expression for the path of the particles has been obtained in

the best closed form. The factors influencing the loci of all such particles have been examined with respect to

various flow entities and are illustrated graphically. As the porosity increases the paths of particles are found to be in

the gradually decreasing trend. In this situation more of backward flow is observed. It is noticed that as the porosity

increases, the path of the particles travelled is found to be decreasing.

1. Introduction

When a conductive fluid moves through a magnetic field, an ionized gas is electrically conductive, and the

fluid is influenced by the magnetic field. Natural convection and transfer of heat is of considerable interest in problems

that arises in magneto hydrodynamic (MHD) especially in the technical field due to its frequent occurrence in industrial

technology and geothermal applications. The applications are wide in variety of situations where the high - temperature

plasmas are applicable in nuclear fusion energy conversion, liquid metal fluids, and (MHD) power generation systems.

Further, in several problems related to geophysical, petroleum, chemical and biomechanical that are usually bounded by

porous medium, the problem assumes greater significance. Convective boundary layer flows are often controlled by

fluid suction or injection through a porous heated wall. This process can lead to enhancement of the heat transfer

coefficient or cooling of the system. Due to several applications in the fields of geo physics, metallurgy, petroleum

engineering, chemical engineering, composite metal engineering and heat exchanges, the problem of mass transfer and

radiation effects are unsteady MHD flows. Free convective flow embedded in a porous medium with a heat generation/

absorption assumes greater significant over the last two decades.

Generally. a porous medium usually consists of large number of interconnected pores each of which is

saturated with the fluid. The exact form of structure however is highly complicated and differs from one medium to

another medium. Flow through porous media can be considered as an ordered flow in a disordered geometry. The

transport process of fluid through a porous medium involves the porous matrix and the fluid under consideration.

Therefore, the problem is characterized by specific properties of these two matrices. In view of the complexity of the

matrices that arises out of the situation, a porous medium can be considered as statistical aggregate of large number of

solid particles containing several capillaries such as porous rock. Due to the complexity of microscopic flow in the

pores, the true path of individual fluid particle cannot be estimated analytically. The final total gross effect of the

phenomena that is being represented by a macroscopic view and applied to the masses of the fluid which are large

compared to the dimensions of the pore structure of the medium must be considered in all such complicated situations. The phenomenon of free convection arises in the fluid when temperature changes cause density variations leading to

buoyancy forces acting on the fluid elements. This can be seen in the atmospheric flow, which is quite often driven by

temperature differences. Some examples in living organisms are related to fluid transport mechanisms out of which the

blood flow in a circulatory system, air flow, in air ways circulatory systems and transfiguration of cooling systems in

heating chambers.

Berman [1] initiated the problem of flow of incompressible viscous fluid with parallel rigid porous walls when

the flow is being driven by uniform suction or injection at the walls. Subsequently, for case of high suction Reynolds

number, the work of Berman was examined by Sellars [2]. Thereafter, Yuan and Finklestein [3] studied the flow in a

porous circular pipe for obtaining solution for small suction and injection values, while Eckert [4] studied the influence

of mass transfer cooling of foreign gas injected in to the boundary layer. Later, the radiation effects on free convection

flow of a gas past a semi-infinite flat plate was examined by Soundalgekar and Takhar [5]. Thereafter, Sparrow and

July Issue Page 93 of 145 ISSN 2229 5216


Vol. 3, No.1, 2011

Cess [6] studied the effect of magnetic field on the natural convection and heat transfer. The case of uniform

suction and blowing through an isothermal vertical wall was treated by Sparrow and Cess [7]. Later, the flow in a renal

tubules as a viscous flow through a circular tube of uniform cross section and permeable boundary with permeable

radial velocity at the walls as an exponentially decreasing function of axial distance was examined by Macey [8]. It has

been pointed out in such situations that, the driving force that is necessary to move a specific volume of the fluid at a

certain velocity through a porous medium must be in equilibrium with the resistive force being generated by internal

friction between the fluid and the pore structure. The effect of electric and magnetic fields on the heat transfer to

electrically conducting fluids was investigated by Romig, [9] while, Riley [10] analyzed the MHD free convection

heat transfer. In the absence of any heat source, the MHD flow in a vertical parallel channel was discussed by Yu [11].

Terril and Thomas [12] developed the theory by considering the laminar flow in a porous pipe with uniform suction and

injection applied at the boundary wall. Later, Clarke [13] extended the range of suction or injection parameter using the

method of matched asymptotic expansions in which the analytical solution was presented as an asymptotic expression.

The problem of fully developed flow between two vertical plates taking into account, the radiation effects was

examined by Gupta and Gupta [14]. Subsequently, in a series of two papers on various aspects of blood flow in a

pulmonary channel with a view of understanding the flow and the corresponding dispersion of fluids flowing in the

lungs was presented by Fung and Tang [15]. They had considered a model in which the channel is bounded by two thin

layers of porous material with blood flow in the channel and the water moment in the porous walls. The flow pattern

was considered using physical conditions rather than Beavers and Joseph slip conditions while their investigations were

found to be independent of stability analysis. Subsequently, Merkin [16] obtained asymptotic solutions which are found

to be valid at large distances from the leading edge with suction or blowing. More recently, Lin and Yu examined [17]

the permeable wall heat transfer problem. In the present article, the effect of various parameters on the paths of the fluid

particles in the case of free convection flow of a viscous incompressible conductive fluid from a radiative vertical

porous plate with uniform surface temperature with constant rate of suction or injection, and uniform magnetic field

flux is investigated. The influence of radiation is included by using the Roseland diffusion approximation. Brady [18]

extended the study of Robinson for the fully developed flow in a porous channel as well as in a circular and elliptical

cross section while Rao et al [19] examined the MHD oscillatory flow of blood through channels of variable cross

section and presented the investigations. in a linear analysis. Numerical solution in the case of natural convective

radiation and influence of various parameters are obtained by Keller [20] using the Keller-box method.

The buoyancy driven convection in a rectangular enclosure with a transverse magnetic field was examined by Garandet

et al. [21] while, Sacheti et al. [22] found an exact solution for the transient (MHD) free convection flow with constant

surface heat flux.

Radiation heat transfer effects on free convection flow are very important in space technology and high

temperature process and very little is known about the effects of radiation on the boundary layer of a radiate-MHD fluid

past a body. The inclusion of radiation effects in the energy equation had lead to a highly nonlinear partial differential

equation. More recently, Hossain and Takhar [23] studied the effect of radiation on forced and free convection flow of

an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream velocity and

surface temperature. Subsequently, Hossain et al. [24] studied the effect of radiation on the free convection heat transfer

problem in the absent of magnetic field and viscous effects. Thereafter, the disturbances due to the sinusoidal motion of

the bounding surface, when the fluid under consideration was of visco elastic in nature was analyzed by Ramana

Murthy et al [25]. Recently, a detailed analysis of visco elastic fluid of second order type between two parallel plates

with lower plate possessing natural permeability was presented by Ramana Murthy et al [26]. In their analysis it has

been noted that the skin friction on the upper plate is almost linear with respect to the visco elasticity of the fluid.

However, the situation seems to be not stated as above at the lower plate. The reason can be attributed to the fact that

the lower plate possessing natural permeability. The case of linear analysis by considering visco elasticity of the fluid

over an inclined porous plate was studied by Ramana Murthy and Kavitha [27]. Similar such analysis, but of course

when the bounding surface is rigid has been examined by Ramana Murthy et al [28]. In a recent study by Ramana

Murthy and Hari Prasad [29] in a situation of unsteady state MHD flow in a porous channel with an exponentially

decreasing suction, it has been established that, as the porosity decreases the velocity increases. It is observed that as the

Reynolds no increases, the velocity decreases. Further, it is seen that for a constant porosity value, as the frequency of

excitation increases, the skin friction decreases. Also for a constant value of frequency of excitation, as the porosity

increases, then skin friction decreases. It is observed that as the porosity is increased, the flow rate is found to be

sinusoidal. Also it is seen that the pattern remains unchanged even when the frequency of excitation is increased. In all

above such situations, it is seen that as the visco elasticity increases, there is a decreasing trend in the velocity profiles.

And also the velocity profiles are significantly distributed and are found to be more parabolic with the inclusion of the

visco elasticity term.

In all above mentioned investigations, the nature of velocity profiles and skin friction with respect to various

flow entities have been studied. But not much of importance has been paid in case of the paths of the fluid particles and

factors effecting thereon. Therefore, an attempt has been made in this paper to study the influence of various flow

entities that influence the path of the fluid particles.



Vol. 3, No.1, 2011

2. Mathematical formulation

)(, xvvou y = a

v u

ovu y = 0

Fig1: Schematic diagram of the Problem.

We consider the steady flow of a viscous electrically conducting fluid through a long channel, with one wall

permeable and the other impermeable, under the influence of an externally applied homogeneous magnetic field. It is

assumed that the fluid is incompressible with small electrical conductivity and the electromagnetic force produced is

very small. Consider a Cartesian coordinate system (x, y) where Ox lies along the centre of the channel, y is the

distance measured in the normal section. Let u and v be velocity components in the directions of x and y increasing

respectively and *p Then, in two- dimensional, the governing equations of continuity, momentum and energy are

given in dimensional form as

0*

*

*

*

y

v

x

u (1)

*

2

0

2

*2

*

*

*

**

*

**

*

*

*

1u

B

y

u

x

p

y

uv

x

uu

t

u e

(2)

where B0= (0, B0) is the magnetic field vector.

The appropriate boundary conditions are

0* u on ay and 0* u on 0y (3)

where ''a the width of the channel is e is the electrical conductivity, the fluid density v the kinematics viscosity.

Introducing the following non-dimensional quantities:

U

aBH

v

Ua

va

tt

U

pp

a

yy

a

xx

U

vv

U

uu e

2

0

2

*

2

*****

,Re,/

,,,,, (4)

and substituting into equations (1) - (2), we obtain

0

y

v

x

u . (5)

uH

y

utxh

y

uv

x

uu

t

u 2

2

2

Re

1),(

(6)

The boundary conditions reduces to

0u on 0y and 0u on 1y (7)

Under the assumptions that the flow is laminar and has the velocity components are [u(y, t), 0 0] and the plate

is large enough always along x-axis the equation of motion is given by

0

x

u (8)

1

2

2

2

Re

1),(

k

uuH

y

utxh

t

u

(9)



Vol. 3, No.1, 2011

and the boundary conditions reduces to

0u on 0y and 00 u on 1y (10)

3. Solution of the problem

Let the solution of the Eqn (8) and Eqn (9) be of the form

tieyutyu )(),( 0 (11)

tiehtxh 0),( (12)

Substituting Eqn (11) and Eqn (12) in Eqn (9) we obtain

Re)1

(Re 00

1

2

2

0

2

huk

Hiy

u

(13)

The boundary conditions are transformed to

00 u on y =0 and 00

y

u on y = 1 (14)

The solution of Eqn (13), satisfying boundary conditions in Eqn (14) is:

tie

p

yp

p

htyu

1

cos

)1(cosRe),(

2

0 Where )

1(Re

1

2

kHip (15)

The paths of the particles is given by

tiep

yp

p

ihdttyu

1

cos

)1(cosRe),(

2

0 (16)

4. Results and conclusions

1. The trace of the particles for various Reynolds number and Porosity of the fluid bed for a constant value of time

has been illustrated in fig. 2, fig. 3, fig. 4 and fig. 5. In a situation when t = 0 and Reynolds number = 5, as the

porosity increases the paths of particle are found to be in the gradually decreasing trend. In this situation more of

backward flow is observed. Such a backward flow can be attributed to the fact that the driving force necessary for

the fluid particles to gain momentum is not that sufficient to push the fluid in the forward direction. However in a

case that when t = 15 for the same pore size the fluid bed the paths of the fluid particles are observed to be

gradually in the increasing trend. However the magnitudes of the paths of the particle are large enough in the

boundary layer region and thereafter gradually decrease as we move far away from the bounding surface. When

the situation was examined for the Reynolds number 10 and at t = 0, as the pore size increases more of backward

flow is noticed. In a situation when t =15, for the same pore size the magnitude of the path of the Particles

gradually increases and then decreases almost in the exponential order. It can be concluded that from the fig.2 and

fig.4, the paths of the particle is found to be negative while for t = 15 and Reynolds number = 10. The paths of the

particle are observed to be positive. All these illustrations indicate that the paths of the particle depend mainly on

the Reynolds number and also time t. Further, the driving force required to push the fluid the forward direction will

only plays a role only when t > 0 and is almost independent of the pore size of the fluid bed.



Vol. 3, No.1, 2011

Fig. 2: Effect of Porosity on path of the particles. (Re = 5)

Fig. 3: Effect of Porosity on path of the particles. (Re = 5)



Vol. 3, No.1, 2011

Fig. 4: Effect of Porosity on path of the particles. (Re =10)

Fig. 5: Effect of Porosity on the path of particles. (Re =10)



Vol. 3, No.1, 2011

2. Fig. 6 and Fig. 7 shows the influence of time parameter (t) for a fixed porosity of the fluid bed. In both the

illustrations it is noticed that as t increases the path of the particles travelled decreases. Such a decrease is observed

to be very significant in the boundary layer region. However, as we move far away from the plate the path of

particles remains unchanged. From fig. 5 and fig. 6, it is seen that the porosity of the fluid bed plays a significant

role on the locus of the particles traversed. It is noticed that as the porosity increases, the path of the particles

travelled is quite decreasing.

Fig. 6: Effect of time on path of the particles. (Re = 5)



Vol. 3, No.1, 2011

Fig. 7: Effect of time on path of the particles. (Re = 10)

5. References

1. .Berman A S., “Laminar flow in a channel with porous walls” ,J.Appl.Phys., 24, Pp1232-1235

2. Sellars J R., “Laminar flow in channels with porous walls at high suction Reynolds number” J.Appl. Phys., 26,

489, (1953).

3. Yunan S. and Finklestein A., “Laminar pipe flow with injection and suction through a porous wall” , Trans

ASME, 78, 719, (1956).

4 Eckert E. R, ” Mass transfer cooling of a laminar boundary by injection of a light weight foreign gas”. Jet

Propulsion 28 Pp 34–39, (1958).

5 Soundalgekar V M., Takhar H.S; and Vighnesam N V, “The combined free and forced convection flow past a

semi-infinite plate with variable surface temperature” . Nuclear Eng esign 110 Pp 95–98, (1960).

6 Sparrow E M; Cess R D, ” Free convection with blowing or suction”. J Heat Transfer 83 Pp 387–396, (1961).

7 Sparrow E M; Cess R D “Effect of magnetic field on free convection heat transfer” . Int J Heat Mass Transfer

3 Pp 267–274, (1961).

8 Macey R I., “Pressure flow patterns in a cylinder with reabsorbing wals”, Bull.Math.Biophys, 25 (1). (1963).

9 Romig M, “The influence of electric and magnetic field on heat transfer to electrically conducting fluids” .

Adv Heat Transfer 1 Pp 268–352, (1964).

10 Riley N, “Magnetohydrodynamics free convection” . J Fluid Mech 18 Pp 577–586, (1964).



Vol. 3, No.1, 2011

11 Yu C.P., ” Combined forced and free convection channel flows in magneto hydrodynamics”, AIAA Jan., 3,

1181-1186, (1965).

12 Terril R.M. and Thomas P.W., ”Laminar flow through a uniformly porous pipe” , Appl, Sc. Res. 21, Pp 37-67,

(1969).

13 Clarke J F “Transipiration and natural convection, the vertical flat plate problem” . J Fluid Mech 57 Pp 45–61,

(1973).

14 Gupta P.S. and Gupta A.S., “Radiation effect on hydromagnetic convection in a vertical channel” , Int.J.Heat

and Mass transfer, 17, Pp 182-189, (1974).

15 Fung Y. C. and Tang. “Solute distribution in the flow in a channel bounded by porous layers” . ASME

Jnl.Appl.Mec., 97 Pp 531-535, (1975).

16 Merkin H J, “The effects of blowing and suction on free convection boundary layers”. Int J Heat Mass

Transfer 18 Pp 237–244, (1975).

17 Lin H T., and Yu W S., “Free convection on a horizontal plate with blowing and suction”. Trans ASME J Heat

Transfer 110 Pp 415–431 (1976).

18 Brady J.F., ” Flow development in porous channel and tube” , J.Fluid Mech., 112, Pp 127-150, (1984).

19 Rao A.R. and Deshikachar K.S., “MHD oscillatory flow of blood through channels of variable cross-section” ,

Int.J.Engg.Sci., 24(10), Pp1615-1628 (1986).

20 Keller H B., ” Numerical methods in boundary layer theory” . Ann Rev Fluid Mech 10 Pp 793–796, (1988).

21 Garandet J P., Alboussiere T.and Moreau R , “Buoyancy driven convection in a rectangular

enclosure with a transverse magnetic field”. Int J Heat Mass Transfer 35 Pp 741–749, (1992).

22 Sacheti N C. Chamdran P and Singh A K., “An exact solution for unsteady magneto hydrodynamic free

convection flow with constant heat flux”. Int Commun Heat Mass Transfer 21 Pp 131–142. (1994).

23 Hossain M A; Takhar H S, ” Radiation effect on mixed convection along a vertical plate with uniform surface

temperature”. Heat Mass Transfer 31 Pp 243–248, (1996).

24 Hossain M A; Alim M A; Rees D A, “The effect of radiation on free convection from a porous vertical plate”.

Int J Heat Mass Transfer 42 Pp 181–191, (1999).

25 Ramana Murthy Ch.V., Kulkarni S.B. et al: “On the class of exact solutions by creating sinusoidal

disturbances” . Def. Sc. J., Vol. 56, Pp 733-741 (2006).

26 Ramana Murthy Ch.V., Kulkarni S.B. et al: “Exact solutions by creating forced oscillations on the porous

boundary” . Def. Sc .J. Vol. 57, Mar.Pp 197-209 (2007).

27 Ramana Murthy Ch.V., Kavitha K.R., “Flow of a second order fluid over an inclined porous plate. Int. J.

Physical Sciences” , Vol.21 (3), Pp 585-594 (2009).

28 Ramana Murthy Ch.V., Gowthami K. et al: “Flow of a second order fluid over an inclined rigid plane”, Int. J.

of Emerging Technologies and Applications in Engineering, Technology and Sciences (IJ-ETA-ETS), Vol.3,

Pp 13-17 (2010).

29 Ramana Murthy Ch. V. and Hari Prasad P., “Unsteady state MHD flow in a porous channel with an exponentially decreasing suction”, Ultra Scientists of Physical Sciences”, Vol.22(2), Pp 337-346 (2010).



Vol. 3, No.1, 2011

Dr. Ch. V. Ramana Murthy received his B.Sc., from Kakatiya University,

Waranagal (A.P.), and M. Sc. (Applied Mathematics) from Regional

Engineering College Warangal (A.P.) and Ph.D. (Fluid Mechanics) from

National Institute of Technology, Warangal (A.P.). He has around 25 years

of teaching experience. Presently he is working as Full time Professor of

Applied Mathematics in Lakireddy Bali Reddy College of Engineering,

Mylavaram (A.P.). He has more than 60 Research Publications in various

Journals of National and International repute. As on date 3 students had

submitted their Thesis for the Degree of Doctor of Philosophy under his

guidance

P. Hari Prasad had obtained his M.Sc. Degree from Acharya Nagarjuna

University in 1985 and M. Phil from Algappa University in 2007.He has

been associated with institutes of excellence for the last 20 years in teaching

Applied Mathematics to the students of Engineerinng/Technology at

Graduate and Post Graduate levels. To his credit he has several publications

in Journals of National and International repute. At present he is pursuing his

Ph.D. at Acharya Nagarjuna University at Guntur.

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