iii radiation convective flow a vertical...
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CHAPTER - III RADIATION EFFECTS ON MHD CONVECTIVE FLOW PAST A
VERTICAL PLATE WITH VARIABLE SURFACE
TEMPERATURE IN THE PRESENCE OF HEAT SOURCE/SINK
1. INTRODUCTION
Free convection flow and heat transfa problems are of important considerations
in the ~ ~ a l design of a variety of indu&al equipment and atso in nuclear reactors and
gm~h~s ica l fluid dynamics. The transient natural convection flows over vertical bodies
has a wide range of applications in engineering ruid technology. The study of natural
c o n ~ t i o n flow past a semi-imfinite plate was first studied by Pohlhausen [I] using an
integral method, whereas the similarity method was f~ used to study this problem by
Ostmch [2] and the resulting non-linear ordinary differential equations were solved
numerically. When the surface heating is variable, the steady state free convection for a
semi-infinite vertical plate becomes quite complicated. This was studied by Sparrow and
Gregg [3] by assuming the power-law variation of the wal13;: = T' +an, where a is a
constant. Takhar et al. [4] studied transient free convection flow past a semi-infinite
vertical plate with variable surface temperature. The non-dimensional governing partial
differential equations are solved using an implicit finite-difference scheme.
Laminar free convection boundary-layer flow of an electrically conducting fluid
in the presence of magnetic field has been investigated by many researchers because of
its wide applications in industry and technology. The influence of a magnetic field on a
viscous incompressible flow of an electrically conducting fluid is of importance in many
applications such as extrusion of plastics in the manufacture of Rayon and Nylon,
purification of crude oil, magnetic materials processing, glass manufacturing control
processes, the paper industry and different geophysical systems. In many industrial
processes, the cooling of threads or sheets of some polymer materials is of importance in
the production line. Magneto convection plays an important role in various industrial
applications including magnetic control of molten iron flow in the steel industry and
liquid metal cooling in nuclear reactors.
Free convection heat transfer due to simultaneous action of buoyancy and induced
magnetic forces was investigated by Sparrow and Cess [5]. They observed that the free
convection heat transfer to liquid metals may be significantly affected by the presence of
a magnetic field, Kumari and Nath [6] studied the development of asymmetric flow of a
viscous electrically conducting fluid in the forward stagnation point repion of a two-
dimensional body and over a stretching surface with an applied magnetic field, whsn the
external stream or the stretching surface was set into an impulsive motion fmn the rest. MHD natural convection h m a non-isothermal inclined m h c e with multiple
~uc t i~n jec t ion slots embedded in a thumally mtified high-pmity medium has been
studied by' Takhar et al. [7]. The non-linear coupled parabolic partial d i h t i a l
equations are solved numerically using an implicit finite- difference scheme. Laminar
free convection boundary layer flow in the presence of a transverse magnetic field over a
heateddown pointing cone spinning with constant angular velocity about the symmetry
axis was studied by Mehmet [8] using similarity variables.
Radiative-convective heat transfer flows fmd numerous applications in glass
manufacturing, furnace technology, high temperature aerodynamics, fire dynamics and
space craft reentry. Cogley et al. [9] had shown that for an optically thin limit, the fluid
does not absorb its own emitted radiation or there is no self-absorption, but the fluid does
absorb radiation emitted by the boundaries. Grief et al. [lo] have shown that in the
optically thin limit, the physical situation can be simplified and then they derived an
exact solution to fully developed vertical channel flow for a radiative fluid. Soundalgekar
and Takhar [ll] considered the radiative free convective flow of an optically thin gray-
gas past a semi-infinite vertical plate. Radiation effects on mixed convection along an
isothermal vertical plate were studied by Hossain and Takhar [12]. Raptis and Pdik i i
[13] studied the effects of thermal radiation and fit% convection flow past a moving
vertical plate. Radiation effects on free convection flow past a vertical plate with mass
transfer are studied by Chamkha et al. [14]. The governing boundary layer equations for
these problems are reduced to non-similar form and are solved numerically by an implicit
finite difference technique. Muthucumaraswamy and Ganesan [IS] studied radiation
effects on flow past an impulsively started infinite vertical plate with variable
temperatures using the Laplace transform technique.
Many processes in new engineering areas occur at high tempembe8 and
knowledge of radiation heat transfer besides the convective heat transfer becomes very
important for the design of the pertinent equipment. Nuclear power plants, gas turbines
and the various propulsion devices for air crafts, missiles, satellites and space vehich atc
examples of such engineering areas. Accordingly, it i8 of interest to examine the e m of
magnetic field on the flow. Studying swh effect has a great importance in the application
fields, where thermal radiation and magnetic field are correlative. The process of h i o n
of metals and the process of cooling of the fust wall inside the nuclear reactor container
vessel, where the hot plasma is isolated from the wall by applying magnetic field are
examples of such fluids.
Magnetohydrodynamic flow padt a plate by the p m c e of radiation was studied
by Raptis and Massalas [16]. An analytical solution for the mean tempenrture, velocity
and the magnetic field have been anived and the effects of radiation on tern- cue
discussed. The combined effects of thermal radiation flux, thermal conductivity,
Reynolds number and non-Darcian (Forcheimmer drag and Brinkman boundary
resistance) body f m s on a steady laminar boundary layer flow along a vertical suface
in an idealized geological porous medium were investigated by Takhar et al. [17]. The
effects of radiation on flee convection flow and mass transfer past a vertical isothermal
cone surface with chemical reaction in the presence of magnetic field was investigated by
Ahmed [IS] using similarity variables. The effect of radiation on magnetohydrodynamic
unsteady fne-convection flow past a semi-infinite vertical porous plate was studied by
Abd El-Naby et al. [19]. The effects of thermal radiation and porous drag forces on the
natural convection heat and mass transfer of a viscous, incompressible, gray, absorbiig
emitting fluid past an impulsively started moving vertical plate adjacent to a non-Darcian
porous regime was studied by Anwar et al. [20].
The heat soucodsink effects in thermal convection, are significant where there
may exist a high temperature differences between the surface (e.g. space craft body) and
the ambient fluid. Heat generation is also important in the context of exothermic or
endothermic chemical reactions. Sparrow and Cess [21] provided one of the earliest
studies using a similarity approach for stagnation point flow with heat source/sink which
vary in time. Pop and Soundalgekar [22] studied unsteady free convection flow past an
infinite plate with constant suction and heat source. Much later Takhar et al. [23]
presented one of the most robust studies of thermal and concentration boundary layers
with MHD effects for the w e of a point sink. Takhar et al. [24] extended this analysis to
examine combined variable lateral mass flux (wall injection/suction), heat source effects
and hall current effeots on doublediffusive boundary layers under strong magnetic fields.
Sahoo et al. [25] studied magnetohydrodynamic unsteady free convection flow past an
infinite vertical plate with constant suction and heat sink.
However, convective flow under the influence of magnetic field and thermal
radiation past a vertical plate subject to a variable surface temperature in the presence of
heat soundsink has not received the attention of any researcher, Hence, the object of the
present investigation is to study the combined effects of magnetic field and thermal
radiation past a semi-infinite vertical plate subject to a variable surface temperature. The
set of non-dimensional govming equations are solved by an implicit finite difference
method of Crank-Nicolson type. The behavior of the velocity, temperature, skin-friction
and Nusselt number has been discussed for variations in the governing parametem.
2. MATHEMATICAL ANALYSIS
A twodimensional unsteady flow of an electrical conducting, radiating, viscous
incompressible fluid past a semi-infinite vertical plate with temperature~(x) = TA + x",
(where n is a constant) is considered. The surrounding fluid which is at nst has a
temperaturel",. The co-ordinate system is chosen such that x- axis measured along the
plate vertically upward and y- axis is taken n o d to the plate. It is assumed that the
viscous dissipation effect is negligible in the energy equation. A unifonn magnetic field
is applied in the direction perpendicular to the plate. The fluid is assumed to be of small
electrical conductivity so that the magnetic Reynolds number is much less than unity and
hence the induced magnetic field is negligible in comparison with the applied magnetic
field. All the fluid properties are considered constant except the influence of density
variation in the body force term (Boussinesq's approximation). Then, in the absence of an
input electric field, the boundary layer equations which govern the flow field m
Mass conservation
Momentum conservation
Thermal Energy conservation
where u, v are the velocity components in x- and y- directions respectively, t '- the time,
o- the kinematic viscosity, g- the acceleration due to gravity, /I- the coefficient of
volume expansion, T'- the temperature of the fluid in the boundary layer, T, - the
temperature of the fluid far away from the plate, (T- the electrical conductivity of the
fluid, Bo- the magnetic induction, p - the density of the fluid, c p - the specific heat at
constant pressure, k - the thermal conductivity, q, - the radiation heat flux and Qo- the
heat generation constant.
The radiating gas is said tc be non-gray, if the absorption co-efficient K, is
dependent on the wave length. The equation that describes the conservation of radiative
transfer in a unit volume, for all wave lengths is
A; = 1 K A ( T ~ ) [ ~ ~ * A ( T ~ ) - G A M (2.4)
where I,,is the spectral density for a block body, is the radiative heat flux and the
incident radiation G, is defined as
G, = jl*Ja)a R=4x
where R is the solid angle.
Now for an optically thin fluid exchange radiation with plate at the average I
temperature value T, and according to the equation (2.5) and Kirchoffs law, the incident
relation is given by
GA = 41d,,(C) = 4eb,(Ti) . Thus equation (2.4) reduces to
& = 4 ~ ~ L ( f ) [ e b A ( T ' ) - e ~ A ( T i ) ] d i l (2.6)
Expanding eb,(Tf) and KA(T1) in Taylor's series about Ti for small(^ - TL), then we
can rewrite the radiative flux divergence as
A&=4(Tf- ti)[^,($) d = 4 ( T 1 - T L ) T (2.7) W
where KAw = K,(TL) is the mean absorption coefficient, T = % K, ($1 and ebA is
Plank's function.
Hence for an optical thin limit for a non-gray gas near equilibrium, the following
relations hold
A; = 4(T1-T$ (2.8)
and hence
Thus the energy equation (2.3) reduces to
The initial and boundary conditions are
t f < O:u=O, v=O, T' = T: for all x and y
On introducing the following non-dimensional quantities
where Gr, , M , Ra , Pr and pj are the Grehof number, magnetic field parameter, radiation
parameter, Prandtl number and heat generation/absorption coefficient respectively,
equations (2.1), (2.2) and (2.10) reduce to the following dimensionless form
The corresponding initial and boundary conditions in a dimensionless form are
tsO:U=O, V=O, T=O for all x and y
The local nondimensional skin friction and the local Nusselt number are given by
Also, the non-dimensional average skin friction and the average Nusselt number are
given by
3. NUMERICAL TECHNIQUE
The governing equations (2.13)-(2.15) represent coupled system of non-linear
partial differential equations, which are solved numerically under the initial and boundary
conditions (2.16) using Crank-Nicolson implicit finite difference scheme. The fmite
difference equations corresponding to equations (2.13)-(2.15) are
The region of integration is considered as a rectangle with sides X, (= 1) and
Y, (=30), where Y, corresponds to Y =a, which lies very well outside the
momentum and thermal boundary layers:. The maximum of Y was chosen as 30 after
some preliminary investigations so that the last two of the boundary conditions (2.16) are
satisfied. TO obtain an economical and reliable grid system for the computations, a grid
independence is performed. Hence the grid system of 20x120 is selected for all
subsequent analysis with AX = 0.05 and A Y = 0.25. Also the time-step size dependency
is carried out, which yields At = 0.01 for reliable results. Here, the subscript i-designates
the grid point along the Xdirection, j-aiong the Y- direction and the superscript n along
the tdirection. During any one time step, the coefficients U:, and y;appearing in the
difference equations are treated as constants. The values of U, V and T are known at all
grid points at t = 0, from the initial conditions. The computations of U, V and T at time
level (n t l ) using the values at previous time level (n) are carried out as follows. The
finite difference equation (2.21) at every internal nodal point on a particular i-level
constitute a tridiagonal system of equations. Such a system of equations are solved by
Thomas algorithm as described in Camahan et a1 [26]. Thus, the values of T are found at
every nodal point for a particular i at (ntl)' time level. Using the values of T at (ntl)'
time level in the equation (2.20), the values of U at ( n t l ) ~ time level are found in a
similar manner. Thus, the values ofT and U are known on a particular i-level. Finally, the
values of V are calculated explicitly using the equation (2.19) at eveIy nodal point on a
particular i-level at (ntl)' time level. This process is repeated for various i-levels. Thus
the values of T, U and V are known, at all grid in the rectangular region at (ntl)'
time level.
Computations are carried out until the steady-state is reached. The steady-state
solution is assumed to have been reached, when the absolute difference between the
values of U, as well as temperature T at two consecutive time steps are less than 1v5 at
all grid points.
The derivatives involved in equations (2.17) and (2.18) are evaluated using a five
point approximation formula and then the integrals are evaluated using Newton-Cotes
closed integration formula.
4. STABILITY AND CONVERGENCE OF THE FINITE DIFFERENCE
SCHEMI$
The stability criterion of the finite difference scheme for constant mesh sizes are
examined using Von-Neumann technique as explained by Carnahan et a1 [27]. The
general term of the Fourier expansion for U and T at a time arbitrarily called t = 0, are
assumed to be of the form el" ei*(here i =G). At a later time t, these terms will
become
Substituting (2.22) in equations (2.20) and (2.21), under the assumption that the
coefficients U and T are constants over any one time step and denoting the values after
one time step by F' and G' . After simplification, we get
Equations (2.23) and (2.24) can be written as
where
AAer eliminating G'and H' in equation (2.25), using equation (2.26), the resultant
equation is given by
Equations (2.27) and (2.26) can be written in matrix as
Now, for the stability of the finite difference scheme, the modulus of each Eigen value of
the amplification matrix does not exceed unity. Since the matrix (2.28) is triangular, the
Eigen values are diagonal elements. The Eigen values of the amplification matrix are
(1 -A)/(l +A) and (1 -B)/(l t B ) . Assuming that, U is everywhere non-negative and V is
[ I
everywhere non-positive, we get
M A=2a sinZ (aAX/2) + 2csin2 (PAYl2) + i (a sinam-b sin BAY) + - Ar
2
1-A & - - it^ (I+AXI+B)
0 - I-B I t B
2(1+ A)(I + B )
- B )
Since the real part of A is greater than or equat to zero,l(l- A)/(I t A) 1 S1 always.
Similarly [(I - B)l(l t B) 1 51 andl(l- ~ ) l ( l + E) 1 11.
Hence, the finite difference scheme is unconditionally stable. The local truncation error is
0 (At2 t AY' t AX) and it tends to zero as4 ,AY and AX tend to zero. Hence the
scheme is compatible. Stability and compatibility ensures convergence.
5. RESULTS AND DISCUSSION
A representative set of numerical results is shown graphically in Figs.1-9, to
illustrate the influence of physical parameters viz., magnetic parameter M, Prandtl
number Pr, radiation parameter Ra, heat generation/absorption parameter4 and exponent
in the power law variation of the wall temperature n on the velocity, temperature, skin-
friction and Nusselt number. Here the value of Pr is chosen as 0.71, which corresponds to
air. The other parameters are arbitrarily chosen. In order to ascertain the accuracy of our
numerical results, the present result is compared with the work available in the literature.
The present numerical results for steady state velocity profiles at X = 1.0 with 4 = 0,
n = 0, Ra = 0, M = 0 are compared with Takhar et al. [4] which is shown in figure 1. It is
observed that the results are in good agreement with each other.
The transient velocity profiles at X = 1.0 for different values of magnetic field
parameter and exponent n are calculated numerically and are presented in Fig. 2. The
velocity of the fluid increases with time until a temporal maximum is reached and
thereafter a moderate reduction is observed until the ultimate steady state is reached. It is
observed that the time taken to reach the steady state is more for higher values of
magnetic field parameter in comparison with lower values of magnetic field parameter.
The effect of a transverse magnetic field on an electrically conducting fluid gives rise to a
resistive type force called Lorentz force. This force has a tendency to slow down the
motion of the fluid and to increase its temperature. Also we observe that the d y - s t a t e
velocity decreases as n increases at all the positions from the leading edge. T i e taken to
reach the steady state increases as n increases.
Fig. 3 shows that the velocity distribution of air for different values of radiation
parameter Ra and heat source parameter/. It is observed that the velocity of air increases
as the radiation parameter increases. More time is required to reach the steady state for
higher values of radiation parameter Ra.' Due to the presence of heat source energy, the
velocity of air is found to increase. Further, it is also noticed that the more time is
required to reach the steady state for higher values of heat source parameter(.
In Fig. 4, the effect of magnetic field parameter M and n on the transient
temperature is shown for air. It is found that the temperature reduces with the increasing
values of n. The effect of n is more near the leading edge of the plate. The temperature of
the fluid decreases as the magnetic field parameter Mdecreases.
The transient temperature profiles for air at their temporal maximum and steady
state against the coordinate Y at X = 1.0 for different values of the radiation parameter Ra
and heat source parameter / are shown in Fig. 5. It is seen that the t e m p e m increases
as the radiation parameter Ra increases, This result qualitatively agrees with expectations,
since the effect of radiation and surface temperature is to increase the rate of energy
transport to the fluid, thereby increasing the temperature of the fluid. It is noticed that the
time required to reach the steady state flow increases with the increasing value of
radiation parameter Ra, this implies that the existence of radiation helps to achieve the
steady state slowly. The presence of a heat source in the boundary layer generates energy,
which causes the temperature of the fluid to increase.
The study of the effects of the physical parameters on the local as well as average
shearing stress and the rate of heat transfer is important in the heat transfer problems. The
steady state values of local skin friction and Nusselt number for different values of
radiation parameter h, magnetic fikld parameter M, heat source ( and exponent n are
calculated numerically and are depicted graphically in Figs. 6 and 7 respectively.
The local skin-friction decreases with the increasing values of n and the effect of
n over the local skin friction is more and reduces gradually with increasing the distance
along the surface. The local wall $hear stress decreases with the increasing values of M
Further it is noticed that the local skin fiiction increases with the increasing values of
radiation parameter Ra and heat source ( .The local Nusselt number reduces with the
increasing values of exponent n along b e surface. It is observed that the heat transfer
increases with the increasing values of magnetic field parameter M. It is observed that the
local Nusselt number increases with the increasing values of radiation parameter Ra and
heat source(. Average skin friction and Nusselt number are presented in Figs. 8 and 9
for different values of Ra, n, 4 and M. The influence of n on average skin friction is
more when exponent n is reduced. It is also observed that average skin friction for air
increases with the increasing values of Ra and(. Also, it is noticed that average skin
friction decreases as M increases. At an initial time, higher values of average Nusselt
number were observed and the average Nusselt number decreases with time and reaches
steady state after some time. It was observed that for short times, the average Nusselt
number was constant at each level of various parameters. This shows that initially there is
only heat conduction. The average Nusselt number increases as M or Ra or / or n
increases.
0 0 2 4 6 8
Y
Fig. 1 Comparisonn of steady state velocity profiles at X=1.0
Fig.2 Transient Velocity Profiles at X=1.0 for different M and n (*-Steady state)
Fig.3 Transient velocity profiles at X=1.0 for different Ra and 4 (*-Steady state)
0 0 4 8 Y 12 16 Fig4 Transient temperature profiles at X=1.0 for different
M and n (*-Steady state)
49
0 4 R Y 12 16
Fig.5 Transient temperature profiles at X=1.0 for different Ra and 4 (*-Steady state)
0 0.25 0.5 0.75 1
X Fig.6 Local skin friction
50
0.5 1.0 0.02 0.5 1.0 0.5 0.02 0.5 0.5 0.5 0.04 0.5 0.5 0.5 0.02 0.5
F'r = 0.71
Fig9 Average Nusselt number
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