4.1. INTRODUCTION:
The problem of free and forced convection resulting from the flow over
a heated inclined flat plates provide probably one of the most fundamental
problems in heat transfer and is thus of considerable theoretical and practical
interest. In spite of the fact that a greater number of analytical and numerical
results are available. It still continues to be a topic of vital importance for
many practical applications, and there by, generating a need for a full
understanding. Convection heat transfer is present in many engineering
applications, such as solar collectors, environmental engineering and
electronic packaging. A large amount of papers were published dealing with
natural convection in two dimensional enclosures, channel and plates,
considering the variation of different parameters, such as Rayleigh number,
Prandtl number, aspect ratio, radiation, conduction, variable properties and
discrete sources. Recently, a wide literature on laminar mixed convection in
both vertical or horizontal channels and tubes has been developed. The main
technical applications of these researches concern cooling systems for
electronic devices and solar energy thermal conversion. Many results
available in the literature are collected in [I]. In particular, the fully developed
mixed convection in vertical channels was studied analytically by Aung and
Worku [26], Aung [14],
The study of magneto-hydrodynamic flow for electrically conducting
fluid pan heated surface has attracted the interest of many researches in view
of its important applications in many engineering problems such as plasma
studies, petroleum industries MHD power generations, cooling of nuclear
rectors the boundary layer control in aerodynamics and crystal growth. Until
recently this study was largely concerned with flow and heat transfer
characteristics in various physical situations. Watanabe and Pop [29]
investigated the heat transfer in the thermal boundary layer of magneto-
hydrodynamic flow over a flat plate. Michiyochi et al. [17] considered natural
convection heat transfer from a horizontal cylinder to the mercury under a
magnetic field. Vajravelu and Nayfeh [28] studied hydro magnetic convection
at a cone and a wedge. The study of hydro magnetic dynamic free convection
through a viscous fluid past a semi- infinite plate is considered very essential
to understand the behaviour of the performance of the fluid motion in several
applications. It serves as the basis for understanding some of the important
phenomena occurring in heat exchange devices. MHD free convection flows
past a semi- infinite vertical plate have been studied in different physical
condition by sparrow and Cess [ 5 ] , Riley [18] aid others.
The Problems mentioned above are concerned with thermal convection
only. But in nature along with free convection currents caused by the
temperature differences, the flow is also affected by the differences in material
constitution, for example, in atmospheric flows there exist differences in H20
concentration and hence the flow is affected by such concentration difference.
In many engineering applications, the foreign gases are injected. This causes
a reduction in wall shear stress, the mass transfer conductance or the rate of
heat transfer, Usually, H20, COz etc are the foreign gases, which are injected
in the air flowing past bodies. The effects of foreign mass, also know as
diffusing species concentration were studied under different conditions by
Somers [2], Mathers et a1. [lo], and others either by integral method or by
asymptotic analysis. But the first systematic study of mass transfer effects on
free convection flow past a semi infinite vertical plate was presented by
Gebhart and Pera [13] who presented a similarity solution to this problem and
introduced a parameter N which is a measwe of relative importance of
chemical and thermal diffusion causing a density difference that drives the
flow the parameter N is positive when both effects combined to drive the flow
and it is negative when these effects are opposed. Unsteady free convective
flow on taking into account the mass transfer phenomenon past an infinite
vertical porous late with constant such on was studies by Soundalgekar and
Wavre [I 91.
Callahan and Marner [16] first considered the transient free convection
flow past a semi infinite plate by explicit finite difference method. They also
considered the presence of species concentration. However this analysis is not
applicable for other fluids whose Prandtl number is different h m unity.
Soundalgekar and Ganesan [20] analyzed transient free convective flow past a
semi infinite vertical flat plate, taking into account mass transfer by an
implicit finite difference method of Crank-Nicolson type. In their analysis
they observed that an increase in the N leads to an increase in the velocity but
a decrease in the temperature and concentration. Elbashbeshy 1321 studied
heat and mass transfer along a vertical plate with variable surface temperature
and concentration in the presence of magnetic field. Adoeldahab and
Elbarbary [37] took into account the Hall current effect on the MHD free
convection heat and mass transfer over a semi infinite vertical plate upon
which the flow subjected to a strong external magnetic field. Chen [40]
studied heat and mass transfer in MHD flow by natural convection from a
permeable inclined surface with variable temperature and concentration using
Keller box finite difference method and found that an increase in the value of
temperature exponent m leads to a decrease in the local skin friction, Nusselt
and Sherwood numbers. Takhar et al. [33] considered the unsteady Eree
convection flow over a semi infinite vertical plate. Ganesan and Rani [35]
studied the unsteady free convection on vertical cylinder with variable heat
and mass flux.
In this chapter, the fully developed and laminar free and forced
convection with MHD in an inclined channel is studied. The flow is assumed
to be steady and fully developed. The walls are kept at different uniform
temperatures. Expressions for velocity, bulk temperature and Pressure
gradient are obtained for different cases of ' 4 ' . The effects of magnetic
Gr parameter M, Ratio of Grashoff number \ and Reynolds number(-), the
Re
- pressure gradienta =A, ratio of wall temperature differences (rr) are
cix
studied on velocity and bulk temperature and the results are discussed through
graphs.
4.2. NOMENCLATURE:
P : Fluid pressure
p : Fluid density
u : Kinematic viscosity
g : Acceleration due to gravity
p : Thermal expansion co efficient
h : Spacing between two walls
G, : Grashoff number.
R, : Reynolds number.
P* : Dimensionless pressure difference.
r~ : Wall temperature difference ratio.
113
: Angle of the inclination of the channel to
horizontal
: Coefficient of viscosity
: Ambient temperature
: Velocity vector
: Temperature at the wall x=-h
: Temperature at the wall x=h/2
: Axial velocity
: Dimension less velocity =U/W
: Axial transverse coordinates
: Dimensionless axial coordinate
: Dimensionless transverse coordinate
: Dimensionless temperature
: Bulk temperature
: Pressure gradient
: Electric conductivity
: Magnetic field
: Dimensionless velocities of the plates
4.4. FORMATION OF THE PROBLEM:
The steady laminar fiee and forced convection flow between two
parallel flat plates in an inclined channel is considered. Let the inclination
of the channel to the horizontal be ' 4 '. Let the x- axis (the flow
direction) be taken mid way between the plates and the y- axis is taken
perpendicular to the plates. The plates are maintained at different uniform
temperatures. The distance between the plates is taken to be 'h' and the
walls are moving with the velocities uI&u2 respectively. A uniform
transverse magnetic field is applied perpendicular to the plate. The form of
the flow is given in the figure.
The following assumptions are made in the analysis of the problem.
(i) The fluid is Newtonian, viscous, and incompressible.
(ii) The flow is steady and the Grashoff number is small.
(iii) The velocity 'u 'in the axial direction is a function of y and #only.
(iv) The heat transfer takes place by conduction and hence the temper is
linear in 'y'.
(v) The gravitational force 'g' is taken into account and the pressure is a
function of x and y only.
(vi) The channel height is much larger than the channel spacing, so that
the velocity component u in y- direction is taken as zero in the entire
cross section of the flow.
(vii) Energy dissipation is neglected.
(viii) The magnetic Reynolds number is taken to be small enough so that
the induced magnetic field is negligible compared to the applied
magnetic field.[6]
(ix) Electric field is neglected[3]
Under these assumptions the governing equations of the problem are
a 2 T -0 -... @ 2
The boundary conditions of the problem are
p=O at x=O, at y=O
(4.6)
Introducing the non-dimensional quantities
The non-dimensional forms of the governing equations are
Where o = -*=pressure gradient dx
The boundary conditions are
4.5. SOLUTION OF THE PROBLEM:
Solving the equation (4.10) subjected to the boundary conditions
(4.12) & (4.13)
and reverting the parameters, we get the fully developed temperature
profile
. I t r T 8=(1-r , )y+(- )
2 (4.1 4)
Solving equation (4.9) using the equations (4.12), (4.13), & (4.14), we get
Solving equation (4.8) using the equations (4.12), (4.13), & (4.14), we get
u = ------ - a - Bs in@ As in4 sinh My Ice:" A42cosy]c"h"t[2M2sin h y ~ : ] (4.16)
To evaluate the parameter ' a ' an additional equation expressing the
global conservation of mass at any cross section in the channel is also
required.
2
For this )udY = 1 in the dimension less form (4.17) -I - 2
Employing equation (4.16) into equation (4.17), we have
hM hM M cos--- - 2k sin -
2 Bsinb a = M ' [ hM
t-] hM M 2
M cos - 2 sin - 2 2
Using equation (4.18) in equation (4.16), we get the following velocity
Gr distribution at any value of -, rr,
Re
4.5.1 Velocity
k, u = - hM
coshMy-- cos -
hM sin -
2 2
sin
(4.19)
4-52 Bulk Temperature Ob
).@9 -1 - 2 The bulk temperature is defined as 8, =- (4.20)
Using equations (4.14) & (4.19) in equation (4.20) we get the expression
for bulktemperature
hM 2k, A? sin -
[McoshM-2sin'M]+ Oh = ------- - hM hM 2 2
Mcos- M* sin--- 2 2
2 ~ ' sin ---
2 sin --
4.5.3 Pressure gradient (a)
From equation (4.18) we get the expression for pressure gradient is given
by
hM Mcos-- - (u , +u2)sin- - - 2
Mcos - -2sin
4.6. DEDUCTIONS:
4.6.1. Velocity distribution in the absence of pressure forces: - The first and second terms of the right hand side of the equation (4.8)
represents the Buoyancy and pressure forces respectively. These two forces
are of equal importance. The Velocity distribution for mixed convection in
the absence of pressure force is obtained by putting a = 0 in the equation
U = I - COS- 'A"- M2cos- "int~]cos~My+l".f 2~~ sin -- - -+]sinhm sin --- (4.23)
4.6.2. Flow between two horizontal plates (i.e. When4 = 0)
4.6.2(a). For the flow of horizontal flat plates, when the two walls are
moving with the velocities U,, U2,
(i) The velocity is given by
UI +u2 u=-------- (UI -4 sjnhMj,t hM
COS~MY------- 2cos
hM 2sin- - cos -
2 2
(4.24)
(ii)The expression for Bulk temperature is given by
hM Mcos- - - ( u , tu,)sin--
(iii) Pressure gradient: a = dx
(4.26)
4.6.2. (b) If one plate is moving and another plate is at stationary:
U ,=0 and U2=U:
The expressions for velocity, Bulk temperature and pressure gradient
are given by
(i) Velocity
(ii) Bulk temperature:
I + r ) hM U ( - - - ) sin - hM hM
e, = 2 2 + ---- (1 - r , ';[t 2 ~ ~ ~ 1 M 2
(iii) Pressure gradient:
4.6.2. (c) When both the plates are at stationary i.e. UI,O and U2,0
The expressions for velocity, Bulk temperature and pressure gradient
are given by
cos hM - cosh Mv (i)Velocity distribution - u = 2 ---.L
hM hM M c o s -2 s in -
2 2
(ii) Bulk temperature : 8, = 0
hM ~ ' c o s - - - dp (iii) Pressure gradient u = -- = 2
hM hM dr ~ c o s -2sin- - - 2 2
4.6.3 Flow between Wo vertical plates (i.e. when,( = ) 2
4.6.3(a) when both the plates are moving with velocities UI and U2,The
expressions for velocity, Bulk temperature and pressure gradient are given
by
hM (U-U>
coshMz-cos-- u = - ' hM '' coshMy-- sihMytl[ h f 1 +
2 c0s.--- hM
2sin- M2 cos- 2 2
(4.33)
(ii) Bulk temperature:
hM (U, t U,, ( I + rr )sin
6, = 2 - (4 - U , )(I - rT )
M 2MZ
Gr hM ( l - r l ) 2 cos- 2sin---
2 hM M
2 M 2 sin - 2 -
(iii) Pressure gradient:
4.6.3(b) When one plate is at rest and the other plate is moving with
velocity U:'
Then the expressions for velocity, Bulk temperature and pressure
gradient are given by
sinhMy-2z sin
(ii)Bulk temperature:
(iii) Pressure gradient :
4.6.3(c): when the both plates are at rest; i.e. UI=O&Uz4.
The expressions for velocity, bulk temperature and pressure gradient are given
by
(i) Velocity:
(ii)Bulk temperature:
(iii) Pressure gradient:
4.6.4 NON MAGNETIC CASE (M=O): 4.6.4.1(a) Velocity
In the absence of magnetic field the velocity distribution reduces to
y3 Asin4 (U, t U , ) 3 u =-As in# t (3U , +3U, - 6 ) y 2 - [ (U, - U , ) t - ] y - t- (4.42) 6 24 4 2
4.6.4.1(b) Flow between two horizontal plates(i.e. when (4 ), the velocity distribution is given by
(i) When one plate is at rest and the other plate is moving with velocity U
U 3 i.e. u1=0, U2=U then the Velocity is given by u = (3U -6)$ +@--+- 4 2
(4.44) (ii) When the both plates are at rest; i.e. UI=O&U2=0.then the velocity
distributionk given by
4.6.4.1 (e) .Flow between two perpendicular plrtes(i.e. when ( =?) 2
In the absence of magnetic field the velocity reduces to
lir y3 u=(l-r,)-- (I - r (U, -t.U,) 3 +(3u, +3U, - 6 ) y 2 -[(U, -U,)+-- ]Y- -~-+-
Re 6 24 Re 2
(4.46)
(i) When the lower plate is at rest and the upper plate is moving with
velocity U i.e. UI=O, U2=U then the Velocity is given by
(ii) When the both plates are at rest; j.e. uI=O&u2=0, then the velocity
is given by
4.6.4.2. Bulk temperature for non magnetic case:
The expression for bulk temperature in non magnetic case is obtained by
letting M+O in equation (4.21), we get
These results are in good agreement with Aung and Worku [I 81
4.6.4.2.(a) Flow between two Horizontal p1atesti.e. when 4 = 0)
In the absence of magnetic field and q=O, the expression for bulk
U , + U , I- tr , ) U , - U , temperature is given by 0, = (-
2 ( I - ( 1 - rT (4.50)
(i)When the lower plate is at rest and the upper plate is moving with
velocity U
i.e. U1=O; U2=U then the expression for bulk temperature is
I t ) u Oh = ( - ) U + - ( I - r r )
4 12
(ii)When the both plates are at rest; i.e. UI=O&U2=0
e, = o (4.52)
4.6.4.2(b) Flow between two vertical plates(i.e. When, 4 = E ) 2
In the absence of magnetic field the expression for bulk temperature is
given by
U , t U , I t r , ) U , - U , ( I - r , ) ' Gr 0, = (----- )(----I - (- ) ( I - r,. ) t - -
2 2 12 24 Re
(i) When the lower plate is at rest and the upper plate is moving with
velocity U i.e. UI=O , UPU then the expression for bulk temperature
is
(ii)When the both plates are at rest; i.e. UII=O&U~O then the bulk
( I - r7,12 Gr temperature reduces to 0, = -- (4.55) 24 Re
4.6.4.2(c) Pressure gradient for non magnetic case. Letting M--+O in
equation (4.22) pressure gradient reduces to
4.6.4.3. (a) Flow between two horizontal plates (i.e. when4 = 0)
For non magnetic case and 4 = 0 the expression for pressure gradient is
given by
- $ a = - = 1 2 - 6 ( U , + U , ) (4.57) dx
(i)When the lower plate is at rest and the upper plate is moving with
velocity U i.e. ul=O ,u2=U then the expression for pressure
gradient is given by
(ii)When the both plates are at rest; i.e. U1=O&U2=0 then the expression
for pressure gradient is
4.6.4.3(b) Flow between two vertical plates(i.e. When, 4 = 5 ) 2
!t In the absence of magnetic field and I$ = - the expression for pressure
2
gradient is
(i) When the lower plate is at rest and the upper plate is moving with
velocity U i.e. U1=O ,U2=U then the expression for
pressure gradient is
- dp (1 + r,.) Gr a =- = 12-6u+------ dr 2 Re
(ii) When the both plates are at rest; i.e. U1=O&U2=0 then the
expression for pressure gradient is
These results are in good agreement with G.Krishnaiah et.a1[41]
4.7, RESULTS AND DISCUSSION:
A graphical representation of stream wise velocity profiles for #=dl2
and #=O are shown in figures (1) to (lo), for fixed values of
rT=0,0.25,0.5,0.75,1 .O, UI=1,U2=1, G1-100, 300,500,750 and 1000, and
M=1,2,3,4,5. In figure (1) velocity profiles are shown with the variation of
GrRe. It is noticed that for Gr/Re=100 there is no reversal flow. For
Gr/Re=300 reversal flow is observed near the upper plate, this reversal flow
increases with the increase of GrRe. Near the lower plate y= -1 velocity
increases with the increase of Gr/Re, maximum velocity takes place at y= - 0.35.
In figure (2) velocity profiles are shown for fixed values of rT=0.5,M=l,
UI=l,Uz=land #-1~:/2, It is noticed that velocity increases with the increase of
Gr/Re near the lower plate. Reversal flow is observed for Gr/Re=300,500,750
and 1000 near the upper plate. For Gr/Re=100 no reversal flow is observed. In
figure (3) velocity profiles are shown for fixed values of r d . 2 5 , M=l,
179
#=d2, when the lower plate is at rest and the upper plate is moving with
constant velocityUt=l. It is observed that for Gr/Re=100,300,500 no reversal
flow is found. For Gr/Re=750& 1000 reversal flow is observed near the
moving plate. Near the lower plate which is at rest velocity increases with the
increase of Gr/Re.
In figure (4) velocity profiles are shown for fixed values of rp0.75,
M=l, #=n/2 when both the plates are at rest i.e. UI=O and U2=0. It is observed
that maximum velocity takes place near the centre and the influence of GrRe
is not found on velocity.In figure (5) velocity profiles are displayed with the
variation of magnetic parameter M, for fixed values of r d . 5 , ~ r l ~ e = 2 5 0 ,
#=d2, Ul=l,U.L=I. It is observed that velocity decreases with the increase of
M near the lower plate and the reversal flow is observed near the upper plate,
this reversal flow decreases with the increase of M. In figure (6), velocity
profiles are shown with the variation of M for fixed values of ryO.75,
Gr/Re=500, #=O, U1=O, U2=0. It is noticed that velocity decreases with the
increase of M near the centre of the plates and the reverse action is observed
near the plates. In figure (7) velocity profiles are shown with the variation of
r~ for fixed values of Gr/Re=500, M=2, #=d2, UI=l,Uz=l.It is noticed that
velocity decreases with the increase of rr near the lower plate and the reverse
effect is observed near the upper plate,
In figure (8) velocity profiles are shown with the variation of $ for
fixed values of M . 5 , M=2, GrlRe=500, UI=0,U2=1. It is observed that
velocity increases with the increase of # near the plate which is at rest and
reverse effect is seen near the moving plate.. In figure (9), effect of angle on
velocity is shown for fixed values of r73.0.0, GrBe=500, Ul=l, U2=1. It is
observed that velocity increases with the increase of# near the lower plate and
reverse action takes place near the upper plate. In figure (lo), velocity profiles
are shown for different values of 4 with fixed values of of rfi.25,
Gr/Re=250, U,=0,U2=0.1t is noticed that Velocity increases with the increase
of 4 near the lower plate and the reverse action is observed near the upper
plate, maximum velocity takes place at z= - 0.2.
Graphical representation of Bulk temperature profiles are shown from
figures (1 1) to (17), for fixed values of rT=l.O, M=l, 2, 3, #=d2and 0, U1=l,
Uz=l. In figure (1 1) variations in Bulk temperature are shown for different
values of Magnetic parameter M. It is noticed that Bulk temperature increases
with the increase of M. In figure (12) , Bulk temperature profiles are shown
for M=2,3 and 5, for fixed values of rT=l.O, #=lt/2, UI=0,U2=1. It is noticed
that Bulk temperature increases with the increase of M. In figure (13)
variations in Bulk temperature are shown for different values of M when
r1=1.0, #=n/2and 0, UI=l , Uz=l. It is noticed that the Magnetic parameter M
is not affected the Bulk temperature.
In figure (14) effect of M on Bulk temperature for fixed values of
rT=l.O, #=O, ul=l, u2=1 is shown. It is observed that Bulk temperature
increases with the increase of M. In figure (15) effect of M on Bulk
temperature is shown for r ~ 1 . 0 , #=O,ul=O,u2=1. It is noticed that when one
plate is at rest and another plate is moving with constant velocity 1, Bulk
temperature increases with the increase of M. In figure (16) effect of r~
showed on Bulk temperature for fixed values of d=O, M=2, U1=l,Uz=l. It is
noticed that Bulk temperature decreases with the increase of rT. In figure (17) , Bulk temperature profiles are shown for different values of r~ , it is noticed
that Bulk temperature increase when both the plates are moving with constant
velocity when ( 5 4 2 .
Graphical representation of pressure gradient a is shown fiom figures
(18) to (21) for different values of rd.25,0.5,0.75,1.0, UI=I,U2=1,
M=0.2,0.4,0.5,0,.6,1,5,10. In figure (1 8), effect of Gr/Re on pressure gradient
is shown for different values of r ~ , It is observed that pressure gradient
increases with the increase of r ~ . In figure (19) effect of on pressure
gradient for fixed values of b=O, UI=1,U2=1, M=2shown. It is noticed that
pressure gradient does not change with the variation of rl.. In figure (20),
Pressure gradient profiles are shown for different values of M, for fixed values
of #=0, U1=l,Uz=l and r ~ 1 . 0 . It is observed that pressure gradient increases
with the increase of M. In figure (21) effect of large values of M is shown for
fixed values of $=7c/2, UI=1,U2=1 and r ~ 1 . 0 . It is observed that pressure
gradient increases with the increase of M.
Fig.@) : Graph of Velocity profiles with the variation of GrIRe for fixed
values of r~ =O, M=l,UI=l,l&=l, @=lr/2.
Fig.(2) : Graph of Velocity proliles with the variation of GrIRe, for fixed
values of r . ~ 0 . 5 , , M=l, UI=l,Uz=l, 4 ~ 1 2 .
Fig.(3) : Graph of Velocity profiles with the variation of Gr/Re for fixed values of r.r=0.25,M=1, UI=0,U2=1, 4 =d2.
Fig.(4) : Graph of Velocity profiles with the variation of
Gr/Re for fixed values of r&.75,M=l, U1=0,U2t0, 4 wn.
Fig.(J) : Graph of Velocity profiles with the variation of M, for fixed values of rr=0.50,Gr/Re=250,
U,=1,U2=1, 4 312 .
Fig.(6 ): Graph of Velocity profiles with the variation of M for fixed values of r1=0.75,Gr/Re=500, Ul=O,U2=0, (64.
Fig.(7) : Graph of Velocity profiles with the variation of r~ for fixed values of Gr/Re=500,M=2, Ul=l,U2=1, p=le/2.
Fig.@) : Graph of Velocity profiles with the variation of # for fixed values of ,Gr/Re=500, rl=0.50,M=2, U14,U2=1,.
3 - - - - m - - - - . - - - - . - - - . - - - - . - - - - . - - - - 1 - - - - 1 - - - - m - - - -
0 .
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
y
Fig.(9) : Graph of Velocity profiles with the variation of # for fixed values of rT=O,M=l Gr/Re=5OO, U1=1,U2=1.
Fig.(lO) : Graph of Velocity profiles with the variation of # for fixed values of rl=0.25,M=1 Gr/Re=ZSO, U l=O,U2=0.
Fig.(ll) : Graph of Bulk Temperature profiles with the variation of M for the fixed values rT=l.O, 4 =z/2 and Ul=l, U2=1.
Fig.(l2) : Graph of Bulk Temperature profiles with the variation of M for the fixed values of r p l , 4 3 1 2 and UI=O, U2=l.
Fig.(l3) : Graph of Bulk Temperature profiles with the variation of M, for ry =1.0, q4 4 2 and U I = ~ , U ~ = O .
Fig.(l4) : Graph of Bulk Temperature profiles with the variation of M , for fixed values of vr 4.0, 4 4 and UI=1,U2=1.
Fig.(lS) : Graph of Bulk Temperature profiles with the variation of M , for fixed values of r~ =LO, # =O and U1=0,Upl.
Fig.(l6) : Graph of Bulk Temperature profiles with the variation of
r~ , For iixed values of 4 =o, U ~ = I , U t l a n d Ms2.
Fig.(l7) : Graph of Bulk Temperature profiles with the variation of r~ , For fixed values of 4 3 1 2 , U1=1,U2=land M=2
Fig.(lS) : Graph of Pressure gradient profiles with the
variation of rT for fired values of =nn, U,=1,U2=land M=2.
Fig.(l9) : Graph of Pressure gradient profiles with the variation of rT for fixed values of #=O, UI=l,Uz=land M=2
Fig.(21) : Graph of Pressure gradient profiles with the
variation of M for fired values of 4 =no , U1=1,U2=land rT=l.O
4.8. CONCLUSIONS:
In this chapter, uniform transverse magnetic field effects due to free and
forced convection of conducting fluid between two parallel flat plates in an
inclined channel is studied. The walls are kept at different uniform
temperature. Expressions for velocity, Bulk temperature and pressure gradient
are obtained for different values of#. The effects of M, GrRe and rr are
studied on the above flow quantities. The following conclusions are made
from this study.
a. Velocity increases with the increase of GrlRe near the lower plate
z= -1 and the reversal flow occurs near the upper plate z= 1.
b. Velocity decreases with the increase of magnetic parameter M at
the centre and the reverse action is observed near the plates.
c. Bulk temperature increase with the increase of M
d. Bulk temperature decreases with the increase of rl. for #=0, and
increase in the case of 4 2 .
e. Pressure gradient increases with the increase of # = d 2 and no
effect of rT is observed in the case of #=0.
f. Pressure gradient increases with the increase of M.
These results are in good agreement with results of G.Krishnaiah et.al
[41], for the case M-rO and Aung and Worku 1261 when ( 4 2 and
M-tO.
4.9. REFERENCES:
1. Aung W: Int.Xof Heat and Mass transfer Vo1.15, 1952. P. 1577-80.
2. Somers E.V: J. Appl. Mech., Vol. 23, 1956, PP.295-301.
3. Rossow, V.J: NACA-TN, 1957, P.3971.
4. Ostorle J-F and F-J Young: XFluid mechanics, 111, part 4, 196 1, 512-
518.
5. Sparrow, E.M & Cess, R.D: XAppl. Mech. Trans. Of ASME, 29, 1,
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