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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