CHAPTER - 3 UNSTEADY MHD NATURAL CONVECTION IN

VERTICAL CHANNEL PARTIALLY FILLED WITH A POROUS MEDIUM

3.1 INTRODUCTION

Magnetohydmdpmics (MIID): Magneto - having to do with electro-

magnetic fields, hydro - having to do with fluids, dynamics - dealing with forces

and the laws of motion. Magnetohydrodynamics is the mathematical model for the

low frequency interaction between electrically conducting fluids and

electromagnetic fields.

Howevet; let us return to the MHD studies carried out before the problems

of MHD were formulated and in fact even before this scientific discipline was

named. Magnetohydrodynamic phenomena were investigated extensively during

the second and third decades of this century by a number of astmphysicists,

primarily Cowling, Chapman, and Ferraro. A great deal was done then to clarify

the general features of the generation of a magnetic field by the motion of a

conducting medium. An MHD pump is a device which converts the electrical

energy of the current supplied to it into mechanical energy of the pumped liquid.

Its opemting principle, and even its actual design, is similar to those of an electric

motor. Such a motor can be inverted to obtain a generator, which is a machine

converting mechanical energy into electric current.

Fluid flow and heat transfer characteristics at the interface region in

systems which consist of a fluid -saturated porous medium and an adjacent

horizontal fluid layer have received considerable attention. This anention stems

from the wide range of engineering applications such as electronic cooling,

transpiration cooling, drying processes, thermal insulation, porous bearing, solar

collectors, heat pipes, nuclear reactors, crude oil extraction and geothermal

engineering. The work of Beaven and Joseph [3] was one of the first attempts to

study the fluid flow boundary conditions at the interface region. They performed

experiments and detected a slip in the velocity at the interface.

This chqter wm published in J, Pure & Appl Pip's., ~01.21, No.1, Jan-March, 2009, pp.69-81.

Neale and Nader [20] presented one of the earlier attempts regarding this type of

boundary condition in porous medium. In this study, the authors proposed

continuity in both the velocity and the velocity gradient at the interface by

introducing the Brinkman term in the momentum equation for the porous side.

Vafai and Kim [34] presented and exact solution for the fluid flow at the interface

between a porous medium and a fluid layer including the inertia and boundary

effects. In this study, the shear stress in the fluid and the porous medium were

taken to be equal at the interface region. Vafai and Thiyagaraja [36] analytically

studied the fluid flow and heat transfer for three types of interfaces, namely, the

interface between two different porous media, the interface separating a porous

medium from a fluid region and the interface between a porous medium and an

impermeable medium. Continuity of shear stress and heat flux were taken into

account in their study while employing the Forchheimer-Extended Darcy equation

in their analysis. Other studies consider the same set of boundary conditions for

the fluid flow and heat transfer used in Vafg and Thiyagaraja [36] such as (Vafai

and Kim, [34]; Kim and Choi, [16]; Poulikokas and Kmierczak,[26]; Ochoa-

Tapia and Whitaker, [21] ).

Somerton and Catton [32] have obtained the stability condition for a fluid

layer superposed porous layer heated from below. Using the non-Darcy model,

Jang and Chen [13] have solved numerically the fully developed forced

convection in a horizontal channel partially filled with a porous layer. An

analytical investigation was recently presented by Kuznetsov [la] for steady fully

developed laminar fluid flow in the interface region between a porous medium

and a fluid layer in a channel partially filled with a porous material, by taking into

account the jump condition for shear stress proposd by Ochoa-Tapia and

Whitaker [22, 231. Using the Brinkman-Forchheimer model and stress jump

boundary condition at the interface, an analytical solution for the steady fully

developed fluid flow in composite region has been further presented by

Kuznetsov [18].

The we of elecfrieally conducting fluids uadn the iduence of -tic

fields io V ~ O W industries has led to a renewed inam in b v e d g a a

hydmma@c flow and heat transfer in & i t gmmetrices. For example.

Sparrow and Cess [331 considered the effect of a magnetic field on the free

convection heat transfer h m surface. Raptis and Kafoussias [28] analyzed flow

and heat transfer through a Porous medium bounded by an infinite vertical plate

under the action of a magnetic field. Garandet et al. [I 11 discussed buoyancy

driven convection in rectangular enclosure with a transverse magnetic field.

Chamkha [4] analyzed free convection effects on threedimensional flow over a

vet.rical stretching surface in the presence of a magnetic field.

Free convection flow in porous medium was studied by Cheng and pop [a] to analyze the transient free convection boundary layer flow in the porous

medium. Raptis and Takhar [29] have also studied flow formation through a

porous medium. Poulikakos and Renken [27] presented a series of numerical

simulation of forced convection in porous medium using non-Darcy model,

Laminar convection in a vertical channel filled with porous material was solved

both numerically and analytically by Chandrasekhara and Narayanan [5]. A

numerical solution was provided by Kou and Lu [17] in order to investigate the

influence of inertia effect of laminar fully developed mixed convection in a

vertical channel embedded in porous media by using non-Darcy model. Free

convection flow between fluid and porous medium in a vertical channel was

studied numerically by Chang and Chang [6], while in the case of a vertical tuba

by Chang et al, [7]. In a recent study, free convection flow between vertical walls

partially filled with porous medium was studied analytically by Paul et al. [24].

Transient natural convection in a vertical channel partially filled with a porous

medium studied Paul et al. [25].

The Brinkman-Forchheimer extended Darcy model has been used to

simulate momentum transfer within the porous medium as it has some advantages

over other models as reported by Singh and Thorpe [31] and Vafai and Kim [35]

in their studies.

Rostami [30] studied unsteady natural convection in an enclosure with

vertical wavy walls. Also, Badr et al. [2] studied turbulent natural convection flow

in vertical parallel-plate channels. A new approach on MHD natural convection

boundary layer flow from a finite flat plate of arbitrary inclination in a rotating

environment Ghosh and Pop [12]. Duwairi and Damesh [lo] studied

magnetohydrodynamic natural convection heat transfer from radiate vertical

porous surfaces. Ai-Subaie and Charnkha [I] analyzed transient natural

convection flow of a particulate suspension through a vertical channel. Mendez et

al. [19] verified asymptotic and numerical transient analysis of the free convection

cooling of a vertical plate embedded in porous medium. Numerical analysis of

free convective flows in partially open enclosure studied by Jilani et al. [14].

Chowdhury and Islam [9] provided a comprehensive theoretical analysis of a two-

dimensional unsteady free convection flow of an incompyessible, visco elastic

fluid past an infinite vertical porous plate.

In this chapter we present a numerical solution for an unsteady free

convection in the interface region between fluid and porous medium bounded by

two vertical walls under the influence of magnetic field. Here we assume that the

viscosity of the fluid is different from the effective viscoshy of the porous matrix.

Also thermal conductivity of the fluid has been assumed to be different from that

of effective thermal conductivity of the porous medium. The effect of various

emerging parameters studied on velocity field and temperature field numerically.

3.2 NOMENCLATURE

C Inertia coefficient

Da Darcy number

d' Distance of interface from the wall y' = 0

d Distance of interface in nondimensional form

Gr Grashof number

g Acceleration due to gravity

N Distance between vertical walls

K t Permeability of porous medium

4 Effective thermal conductivity

k Thermal conductivity of fluid layer

t ' Dimensional time

t Time in non-dimensional form

T' Temperature of the fluid

< Temperature of the hot wall

I;: Temperature of the cold wall

u' Velocity of the fluid

u Velocity of the fluid in non-dimensional form

M Magnetic field parameter

XI, Y ' Dimensional co-ordinates

x , Y Co-ordinates in nondimensional form

Greek Symbols:

a Ratio of thermal conductivity

P Coefficient of thermal expansion

Y Ratio of kinematics viscosity

va Effective kinematic viscosity of porous layer

v Kinematic viscosity of the fluid

0 Temwrature in non-dimensional form

Subscrlprs

I Fluid layer

P Porous layer

h Hot wall

c Cold wall

3.3 MATHEMATICAL ANALYSIS

We consider the unsteady MHD free convective flow of a viscous

incompressible fluid between two vertical walls consisting of fluid and porous

layers. The x ' - axis is taken along one of the wall while y' - axis is normal to it.

The porous substrate is deposited on the wally' =H. At timetl>O, the

temperature of the wall y o = 0 is instantaneously raised toT,', causing the

phenomenon of free-convection in the vertical channel. The walls are infinitely

long so that the flow characteristics depend bn co-ordinate y' and time t' only.

A uniform magnetic field of strength 2 = (0, B,,,o) is applied normal to the flow

direction. The governing equations for the unsteady, viscous incompressible flow

of an electrically conditioning fluid for the Brinkman-extended Darcy model (the

fluid region) arc:

while for the pmwn region the momentum and energy quations ire

I b t...\'., b ).:.:.. t:.:::::

Clear ;,2.,.\. Fluid [-::.:-: region ; ::. :* :+POIDM medium

,:.:-::, region :.:-:..: j...:.. .:.. ::..,*.* t:. . I.;.'.:. I . , *:. I . *'.

Figure3.1. The schematic of thc flow configumtion

The following assumptions are made:

(i) The Flow is unsteady, viscous and impressible

(ii) Electric field E and induced magnetic field arc neglected

(iii) The energy dissipation is neglected

(iv) Pressure tenn will be neglected

(v) All the physical properties of the fluid assumed to be constant

(vi) The Boussinesq approximations have been used.

Using the above assumptions. the governing equations for the fluid region

are

wile for the porous region the respective equations are

The boundary conditions are

u'=O, T=Oforally'andtSO

u; = 0, T; = Ti for y'= 0

ub=0, ~ ~ = ~ ~ f o r y ' = 1 a n d t : , O

The equations (3.9) to (3.12) and boundary conditions (3.13) are put in non-

dimensional form by using the following transformations

The non-dimensional governing equations for the fluid region (the momentum and

energy equations) are

while for the porous region the respective equations are

- w:3,' where M 2 - P

The boundary conditions are

Equation (3.16) is written by assuming the heat capacity ratio of porous

layer to fluid layer as unity.

In modeling a composite fluid and poro& system, the use of the two-

domain approach for fluid and porous layers required matching conditions at tho

interface. In non-dimensional form, they are obtained as follows:

In describing the matching conditions at the fluid / porous interface in

equation (3.8), continuity of velocity and shear stress are taken for the velocity

field used by Neale and Nader [20] and Vafai and Kim [34]. Similarly, continuity

of temperature and heat flux is considered for the temperature field.

3.4 NUMERICAL SOLUTION PROCEDURE

The momentum and heat transfer in the fluid and porous regions given by

equations (3.13) to (3.16) are solved numerically using implicit finite difference

method. The procedure involves discretization of the transport equations (3.13) to

(3.16) into the fmite difference equations at the grid point(i, j). They are, in order

as follows:

ep(j,,) -ep(l,,-l) - a (3.24) ~t Pr (AYY

Here the index i refers to y and j refers to t . The time derivative is

replaced by the backward difference formula, while spatial derivatives is replaced

by the central difference formula. The above equations are solved by Thomas

algorithm by manipulating into a system of linear algebraic equations in the

tridiagonal form.

In each time step, the process of numerical integration for every dependent

variable starts from the process of numerical integration for every dependent

variable starts from the first neighbowing grid point of the wall at y = 0 and

proceeds towards centre using the tridiagonal form of the finite difference

equations (3.21) and (3.22) until it reaches at immediate grid point of the

interface y = d . In the next process, the tridiagonal form of equations (3.23) and

(3.24) corresponding to the porous layer is used to advance the solution procedure

fiom the grid point located immediately from other side of the interface up to the

immediate grid point of the wall y = 1. Again the value of the dependent variables

at the interfacial grid point is obtained by the matching conditions at the interface.

In each time step first the temperature field has been solved and then the

evaluated values are used to obtain the velocity field. The process of computation

is advanced &ti1 a steady state is approached by satisfying the following

convergence criterion:

with respect to the temperature and velocity fields.

Here 4, stands for the velocity and temperature fields, N is the number

of interio~ grid points and \A/_ is the maximum absolute value of4, .

In the numerical computation special attention is needed to specify AI to

get a steady state solution as rapidly as possible, yet small enough to avoid

instabilities. It is set, which is suitable for the present computation, as

At = Stabr x (AY)' (3.23)

The parameter 'Stabr' is determined by numerical experimentation in order

to achieve convergence and stability of the solution procedure. Numerical

experiments show that the value 2 is suitable for numerical computations. In order

to confirm the validity of this numerical model, the numerical result is compared

with the classical one in extreme condition (d = 1, no porous medium). Our

results coincide with those results of Paul et al. [25], whenM -, 0.

3.5 RESULTS AND DISCUSSION

Figure 3.2 depicts the influence of d (i.e., when widths of the fluid and

porous regions are different) on the velocity profile. This shows that due to

increase in drag force of the porous medium, the fluid in the porous layer

diminishes rapidly and also results in a change of the fluid velocity at the interface

of the fluid / porous region and in the fluid layer. Demonstration of the velocity

profiles for different values of d suggests that free convective flow phenomena

in the fluid layer can be suppressed by increasing the width of the porous layer.

Here too flow ~ecomes steady state when the value of nondimensional time

approaches towards the Prandtl number of the fluid.

In order to study the effect of Magnetic field parameter' M on the velocity

profile we have plotted figure 3.3,

forPr=0.71,t=O.l,y=1,Gr=100,Da=0.001 andd=O.S.Itisobservedtht

the velocity decreases with increase the M .

Figwe3.4 shows the velocity profiles for different Darcy number Da

withPr=0.71, t=0.1, y=1, Gr=100, M = 1 and d=0.5.1tisobservedthat

velocity decreases with an increase in Da .

In figure3.5 we have plotted the velocity profiles for different values of y

withPr=0.7l,t=O.l,Gr=100,M=1, Da=0.001 and d=0.5. This figure

shows that velocity decreases with an increase in y .

Figure3.6 shows the velocity profile for different values of Prandtl number

Pr witht=O.l,y=O.1,Gr=100,M=1,Da=0.001, y-1 and dz0.5 . It is

observed that velocity decreases with an increase inPr .

Figure3.7 shows the velocity profile for different values of time t

withPr = 0.71, y = 1 ,Gr = 100, M = 1, Da = 0.001 and d = 0.5. It is observed that

velocity increases with and increase int .

Figures 3.8 to 3.11 shows the effect of d ,Pr ,aandt on temperature

profiles. Figure3.8 shows that the temperature decreases as d increases. From

figure3.9, it is observed that the temperature decreases as Pr increases. Figure3.10 shows that as the ratio of thermal conductivity increases, decreases the

temperature. Figure3.11 shows that as the time increases, the temperature

decreases.

Figure 3.2 Velocity profile for different values of d with Pr = 0.71, t = O . l , G r = l O O , M = l , Da=O.OOland y = 1

Figure 3.3 Velocity profile for different values of M with Pr = 0.71, t = O . l , y = l , G r = l O O , Da=O.OOland d=0.5

Figure 3.4 Velocity profile for different values of Da with Pr = 0.71, t=O.l,y=l,Gr=lOO, M=land d=0.5

Figure 3.5 Velocity profile for different values of y with Pr = 0.71, t=O.1,Gr=100,M=1,Da=0.001and d=0.5

Figure 3.6 Velocity profile for different values of Pr with t = 0.1, Gr=IOO,M=l,y=l, Da=O.OOland d-0 .5

Figure 3.7 Velocity profile for different values of time f with Pr = 0.71, Gr=lOO,M=l,y=l, Da=O.OOland d=0.5 .

Figure 3.8 Temperature profiles for different values of d with Pr = 0.71 t=0.1 anda=l .

Figure 3.9 Temperature profiles for different values of Pr with d=0.5, t=0.1 anda=I .

Figure 3.10 Temperature profiles for differtnt values of a with Pr =0.71,d=0.5, I =0.1

Figwe 3.11 Temperature profiles for different values of t with Pr =0.71,d=0.5, a=].

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