JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES
An International Open Free Access, Peer Reviewed Research Journal
www.compmath-journal.org
ISSN 0976-5727 (Print)
ISSN 2319-8133 (Online)
Abbr:J.Comp.&Math.Sci.
2014, Vol.5(5): Pg.460-474
Conjugate Effects of Temperature Dependent Thermal
Conductivity and Viscous Dissipation on MagnetoHydrodynamic Natural Convection Flow Along
a Vertical Flat Plate with Heat Conduction
Md. S. Islam1, Md. M. Alam2, Md. S. H. Mollah2 and Md. R. Haque3
1
Department of Mathematics,
Mirzapur Cadet College, Tangail, BANGLADESH.
2
Department of Mathematics,
Dhaka University of Engineering and Technology, Gazipur, BANGLADESH.
3
Model Institute of Science and Technology, Gazipur, BANGLADESH.
(Received on: October 15, 2014)
ABSTRACT
In the present paper, the effects of temperature dependent
thermal conductivity and viscous dissipation on free convection
flow along a vertical plate have been investigated. Magnetohydrodynamics and heat conduction through a wall of finite
thickness are considered in the investigation. With a goal to attain
similarity solutions of the problem, the developed equations are
made dimensionless by using suitable transformations. The nondimensional equations are then transformed into non-similar forms
by introducing non- similarity transformations. The resulting nonsimilar equations together with their corresponding boundary
conditions based on conduction and convection are solved
numerically by using the finite difference method along with
Newton’s linearization approximation. Numerically calculated
velocity profiles, temperature profiles, skin friction coefficient and
the surface temperature distributions are shown both on graphs
and tables for different values of the parameters entering into the
problem.
Keywords: Free convection, magneto-hydrodynamics, viscous
dissipation and temperature dependent thermal conductivity.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
461
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
INTRODUCTION
The study of temperature, thermal
conductivity and heat transfer is most
important because it uses in many branches
of science and engineering. Thermal
conductivity is a measure of the ability of
heat transfer. Although heat transferred by
natural convection on MHD is important for
modeling of fuel elements in nuclear
reactors cores to prevent burnout. The
viscous dissipation effect plays an important
role in natural convection over various
devices in geological processes. Viscous
dissipation on MHD free convection flow is
important from the technical point of view
and such types of problems have responded
by many researchers. Experimental and
theoretical works on MHD natural
convection flows have been done widely but
a few investigations were done on the
temperature dependent thermal conductivity
and viscous dissipation on MHD free
convection flow along a vertical flat plate.
Fluid dynamics deal with gasses and liquids
on motion. Fluid dynamics and MHD has a
wide range of applications. The application’s
field is the design of air craft, weather and
climate forecasting, the extraction of oil
from porous reservoir, in industrial process;
understanding the structure of stars and
planet, flood flow, blood flow etc. Thermal
conductivity is a measure of the ability of
heat transfer. The viscous dissipation effect
plays an important role in natural convection
over various devices in geological processes.
Dissipation means the work done for
reformatting a viscous substance which
converted into energy. The discussion and
analysis of thermal conductivity, natural
convection flows, viscous dissipation, and
natural convection flows along a vertical flat
plate are usually ignored. In the present
work considered the effect of temperature
dependent thermal conductivity and viscous
dissipation on MHD natural convection
flows along a vertical flat plate. The
temperature dependent fluid viscosity is the
phenomenon by which liquid viscosity tends
to decrease as its temperature increases.
Temperature plays a vital role in almost all
fields of science and technology. Electrical
conductivity, fluid viscosity depends on
temperature. The increase of temperature
leads reduces the viscosity across the
boundary layer. That is why heat transfer
rate at flat plate is also affected. Therefore
to predicts the flow behavior accurately it is
necessary the viscosity variation for
incompressible fluids. The temperature of a
body is a quantity which indicates how hot
or cold exist in the body. It is a measure of
the thermal energy per particle of matter.
Temperature is an intensive property which
means it is independent of the amount of
material present; in contrast to energy or
heat. It is an extensive property which is
proportional to the amount of material in the
system. For example, a lightening tube light
can heat a small portion of the atmosphere
hotter than the surface of the sun. Flow of
electrically conducting fluid in presence of
magnetic field and the effect of temperature
dependent thermal conductivity on MHD
flow and heat conduction problems are
important from the technical point of view
and such types of problems have received
much attention by many researchers.
Many of the researchers have
research on these fields. For example,
combined effects of transverse magnetic
field and heat generation on the coupling of
conduction inside and the laminar natural
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
convection along a flat plate in present of
viscous dissipation was investigated by
Mamun et al. (2008). Effects of viscous
dissipation on natural convection flow over a
sphere with temperature dependent thermal
conductivity is studied by Raihanul Haque
et.al.(2014). Effect of pressure stress work
and viscous dissipation in natural convection
flow along a vertical flat plate with heat
conduction was analyzed by Alam
et.al.(2006). Joule heating effects on the
coupling of conduction with magnetohydrodynamic free convection flow from a
vertical flat plate was investigated by Alim
et.al.(2007). Alam et.al.(2007) studied on
stress work effect on natural convection flow
along a vertical flat plate with joule heating
and heat conduction. Combined effect of
viscous dissipation and Joule heating on the
coupling of conduction and free convection
along a vertical flat plate was investigated
by Alim et.al. (2008). Natural convectionradiation interaction on boundary layer flow
along a Thin cylinder has been investigated
by Hossain et.al. (1997). The coupling of
conduction with laminar natural convection
along a flat plate is studied by Pozzi and
Lupo(1998). Effect of pressure stress work
and viscous dissipation in some natural
convection flows was investigated by Joshi
and Gebhart (1981).
To the best of our knowledge in the
light of the literatures some have considered
heat generation and heating on MHD free
convection flow, viscous dissipation on
magneto hydrodynamic (MHD) free
convection flow along a vertical flat plate
with heat conduction. But no work has been
considered Temperature dependent thermal
conductivity and viscous dissipation on
MHD natural convection flows along a
462
vertical flat plate. So the analysis of the
problem may be demonstrates.
Nomenclature
b
plate thickness
cp
specific heat
d
(Tb - T∞)/ T∞
f
dimensionless stream function
g
acceleration due to gravity
h
dimensionless temperature
Joule heating parameter
J
L
reference length, ν2/3 /g1/3
l
length of the plate
viscous dissipation parameter
Vd
p
coupling parameter,
p = (κf /κs)(b/L)d1/4
Prandtl number
Pr
temperature
T
Tb
temperature at outer surface of
the plate
Ts & Tf solid and fluid temperature
fluid asymptotic temperature
T∞
u,v
velocity components
u, v
dimensionless
velocity
components
x, y
Cartesian coordinates
x, y
dimensionless
Cartesian
coordinates
Greek Symboles
co-efficient of thermal expansion
β
γ
temperature dependent thermal
conductivity
dimensionless similarity variable
η
and
solid
thermal
κf, κs fluid
conductivities
and
kinematic
µ , ν dynamic
viscosities of the fluid
dimensionless temperature
θ
density of the fluid
ρ
electrical conductivity
σ
stream function
ψ
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
463
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
Governing equations of the flow
Consider a steady, two-dimensional
free convection flow of an electrically
conducting, viscous and incompressible
fluid with variable viscosity and thermal
conductivity along a vertical flat plate.
x
Tα
Upper surface
ks
Interfaces
u
H0
l
b
g
T=Tf
v
Lower surface
y
Fig. A Physical model and coordinate system.
Thickness of the plate be band
length of the vertical plate be l in the figure.
It is assumed that heat is transferred from the
outside surface of the plate, which is
maintained at a constant temperature, where
temperature of the ambient fluid. A uniform
magnetic field of strength is imposed along
the -axis i.e. normal direction to the surface
and - axis is taken along the flat plate. The
governing equation of laminar flow with
heat generation and thermal conductivity
variable along a vertical flat plate under
Boussinesq approximations for the present
problem, the continuity, momentum and the
energy equations are respectively. In this
chapter we consider thermal conductivity,
viscous dissipation in energy equation.
∂u ∂ v
+
=0
∂x ∂ y
u
ν
∂u
∂u
=
+v
∂x
∂y
∂2u
∂ y2
u
(1)
∂ Tf
∂y
2
+ g β (T f − Tα ) −
+v
∂T f
σH 0 u
ρ
(2)
=
∂y
∂T f
1 ∂
ν ∂u 2
(K f
)+
( )
ρC p ∂ y
Cp ∂ y
∂y
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
(3)
464
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
Where β
the
coefficient
expansion Here, β = −
of
thermal
1 ∂ρ
(
)p
ρ ∂T f
Here we shall consider the form of the
temperature dependent thermal conductivity
which is proposed by Charraudeau (1975),
as bellows.
γ is the temperature dependent thermal
conductivity.
(4)
When γ = Kα {1 + δ (T f − Tα )}
Where, Kα is the thermal conductivity of
ambient fluid and δ =
1 ∂k
( ) f is constant
γ ∂T
Here u, v are velocity components
associated with the direction of increase x axes and y -axes measured along normal to
the vertical plate respectively. T is the
temperature of the fluid in the boundary
layer, g is the acceleration due to gravity, β
is the coefficient of thermal expansion, k is
the thermal conductivity, ρ is the density of
the fluid, C p is the specific heat at constant
Tα Is the temperature of the
ambient fluid, k f the kinematics viscosity of
pressure.
the fluid and σ is the equivalent thermal
diffusivity.
The approximate boundary conditions are
u = v = 0 at y = 0, x 〉 0
T f = T ( x , 0),
∂T f
∂y
=
on y = 0, x 〉 0
Ks
(T f − Tb )
bK f
u = 0, T f → T∞ as y → ∞, x 〉 0
ν = δ (Tb − T∞ ) and ν = δ
for Tb − T∞ = 1
Transformation
equations
of
the
(5)
governing
The non-dimensional governing equation
and boundary conditions can be obtained
from equation (1)-(3) using the following
dimensionless quantities are
x=
x
y 1
, y = Gr 4 ,
l
l
2
−1
ν3
u
u = lGr 2 and l = 1 ,
ν
g3
T f − T∞
gβ l 3
θ=
, Gr = 2 (Tb − T∞ )
Tb − T∞
ν
(6)
Where l is the length of the flat plate, G r is
the Grashof’s number, when Tb − T∞ = 1 θ
is the dimensionless temperature. Now from
the equation (1) – (3) we get the following
dimensionless equations.
∂u ∂v
+
=0
∂x ∂y
u
∂u
∂u
∂ 2u
+v
+ Mu = 2 + θ
∂x
∂y
∂y
u
∂θ
∂θ 1
∂ 2θ
+v
= (1 + γ θ ) 2
∂x
∂y Pr
∂y
(7)
(8)
(9)
∂u
ν ∂θ
+ Vd ( ) 2 + ( ) 2
∂y
Pr ∂y
σH 0 l 2
Where, M =
is the dimensionless
1
2
µG
magnetic parameter.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
465
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
Pr =
µC p
4
5
is the Prandtl number,
K∞
ν = δ (Tb − T∞ ) is the non-dimensional
ψ = x (1 + x)
ν 2d
(Tb − T∞ )
l 2C p
viscous
is the dimensionless
dissipation
parameter,
d = β (Tb − T∞ )
µ is kinematics viscosity.
When, ν =
ρ
The corresponding boundary condition (5)
can be written in the following form
η = yx (1 + x)
1
5
∂θ
∂y
x 〉 0 u → 0,θ → 0 y → ∞
and if v → 0, then θ = T f
Here
p=(
at y = 0 ,
as
x 〉0
(10)
1
K ∞b
)Gr 4 is the conjugate
K sl
conduction parameter. In this situation we
consider p=1.
Let us consider η is the similarity variables
and ψ is the non-dimensional stream
function which satisfy the continuity
equation and is related to velocity
components are
u=
∂ψ
∂ψ
, v=−
∂x
∂y
To solve the equation (8) and (9) the subject
to the boundary condition(10), the following
transformation were introduced for the flow
region
starting
from
upstream
to
downstream,
f ( x,η )
1
−
20
(11)
1
−
5
θ = x (1 + x) h( x,η )
Where h( x,η ) , represents the dimensionless
temperature.
The momentum and energy equation
transformed for new coordinate system as
f ′′′ +
16 + 15x
6 + 5x
f f ′′ −
f ′2
20(1 + x)
10(i + x)
2
u = v = 0 , θ − 1 = (1 + γ θ )
1
20
1
−
5
thermal conductivity variable parameter
Vd =
−
1
− Mx 5 (1 + x)10 f ′ + h
∂f ′
∂f
= x( f ′
− f ′′ )
∂x
∂x
(12)
1
1
h′′ γ
x 5
γ
x 5 2
) h h′′ + (
) h′
+ (
Pr Pr 1 + x
Pr 1 + x
16 + 15 x
f h′ + Vd x f ′′ 2
20(1 + x)
x f ′h
∂f
∂h
−
= x( h′ − f ′ )
5(1 + x)
∂x
∂x
(13)
+
Where, Prime denotes the partial
differentiation with respect toη .
The boundary condition in equation (10)
then takes the form
f ( x,0) = f ′( x,0) = 0
1
h′( x,0) =
−1
x 5 (1 + x ) 5 h( x,0) − 1
−1
4
1
5
(14)
−9
20
(1 + x) + γ x (1 + x ) h( x,0)
f ′( x, ∞) → 0, h( x, ∞) → 0
The set of equation (12) and (13)
together with the boundary condition (14)
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
are solved numerically by applying implicit
finite difference method. Considering the
practical field it is important to calculate the
value of the Skin friction coefficient in term
of surface shearing stress.
Skin friction coefficient and temperature
profile on the surface
To find the numerical value of skin
friction coefficient it is important to
calculate the value of the surface shearing
stress. The non - dimensional form of this
denoted by C f x
−3
Gr 4 l 2
τw
(15)
Therefore, C f x =
µν
∂u
Where τ w = µ ( ) y = 0 is the shearing
∂y
stress.
Now using the transformation describe in
(10) we get the local skin friction coefficient
below
2
−3
C f x = x 5 (1 + x ) 20 f ′′( x,0)
(16)
We obtained the numerical value of
temperature profile of the surface from the
relation
1
−1
θ ( x,0) = x 5 (1 + x) 5 h( x,0)
(17)
Here we have discussed all about the
present investigation that are the velocity
profile, temperature distribution for various
values for Prandtl number, magnetic
parameter, viscous dissipation and heat
generation.
Method of solution
The numerical method used is finite
difference method together with Keller box
466
Scheme (1971). To begin with, the partial
differential equations (12)-(13) are first
converted into a system of first order
differential equations. Then these equations
are expressed in finite difference forms by
approximating the functions and their
derivatives in terms of the center difference.
Denoting the mesh points in the ( x, η ) plane by xi and η j where i = 1, 2, . . . , M
and j = 1, 2, . . . , N, central difference
approximations are made, such that those
equations involving x explicitly are centered
at (xi −1 / 2 , η j −1 / 2 ) and the remainder at
(x , η
)
where η j −1/ 2 = 12 (η j + η j −1 ) etc.
The above central difference approximations
reduces the system of first order differential
equations to a set of non-linear difference
equations for the unknown at xi in terms of
i
j −1 / 2
their values at xi −1 . The resulting set of
nonlinear difference equations are solved by
using the Newton’s quasi-linearization
method. The Jacobian matrix has a blocktridiagonal structure and the difference
equations are solved using a block-matrix
version of the Thomas algorithm; the details
of the computational procedure have been
discussed further by in the book by Cebecci
and Bradshow (1984) and widely used by
Hossain and Alim (1998, 1997).
RESULT AND DISCUSSION
Here we have investigated the
problem of the steady two dimensional
laminar free convection boundary layer flow
of a viscous incompressible and electrically
conducting fluid with pressure work along a
side of a vertical flat plate of thickness ‘b’
insulated on the edges with temperature Tb
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
467
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
maintained on the other side in the presence
of a uniformly distributed transverse
magnetic field. Solutions are obtained for
the fluid having Prandtl number Pr = 0.01,
0.10, 0.50, 0.73, 1.00 and for a wide range of
the values of the viscous dissipation
parameter Vd = 0.25, 0.45, 0.65, 0.95 and
the magnetic parameter or Hartmann
Number M = 1.10, 1.40, 1.60, 1.80, 2.10 and
also the temperature dependent thermal
conductivity parameter γ = 0.10, 0.30, 0.40,
0.50. If we know the values of the functions
f ( x,η ) , h ( x ,η ) and their derivatives for
different values of the Prandtl number Pr,
the magnetic parameter or Hartmann
Number M, the viscous dissipation
parameter Vd and the dependent thermal
conductivity parameter γ , we may calculate
the numerical values of the surface
temperature θ ( x, 0 ) and the shear stress f
″(x, 0) at the surface that are important from
the physical point of view.
Figure 2.2(a) and figure 2.2(b) deal
with the effect of the viscous dissipation
parameter Vd (= 0.25, 0.45, 0.65, 0.95) for
different values of the controlling
parameters Pr = 0.73, M = 1.10 and γ =
0.10 on the velocity profiles f ′ ( x,η ) and the
temperature profiles h ( x , 0 ) . From figure
3.2(a), it is revealed that the velocity profiles
f ′ ( x,η ) increase slightly with the increase
of the viscous dissipation parameter Vd that
indicates the viscous dissipation accelerates
the fluid motion slowly. From figure 3.2(b),
it is shown that the temperature profiles
h ( x , 0 ) increase for increasing values of
Vd.
From figure 2.3(a) and figure 2.3(b),
it is seen that both the velocity profiles
f ′ ( x,η ) and the temperature profiles
h (x, 0 ) increase with the increasing values
of the variation dependent thermal
conductivity parameter γ (= 0.25, 0.45,
0.65, 0.95) with others controlling
parameters Pr = 0.73, M =1.10, Vd = 0.25.
Figure 3.4(a) and figure 3.4(b) represent the
velocity profiles f ′ ( x,η ) and temperature
profiles h (x , 0 ) for the different values of
magnetic parameter or Hartmann Number M
(= 1.10, 1.40, 1.60, 1.80, 2.10) with other
controlling parameters Pr = 0.73, Vd = 0.25
and γ = 0.10. From figure 3.4(a), it can be
seen that the velocity profiles decrease with
the increasing values of magnetic parameter.
On the other hand from figure 3.4 (b), it is
observed that the temperature profiles
increase due to the increase of magnetic
parameter or Hartmann Number M.
Figure 3.5(a) depicts the velocity
profiles for different values of the Prandtl
number Pr ( = 0.01, 0.10, 0.50, 0.73, 1.00 )
while the other controlling parameters are M
= 1.10, Vd = 0.25 and γ = 0.10.
Corresponding
distribution
of
the
temperature profiles h ( x, 0 ) in the fluids is
shown in figure 3.5(b). From figure 3.5(a), it
can be seen that if the Prandtl number
increases, the velocity distribution decreases.
On the other hand, from figure 3.5(b) it can
be observed that the temperature profile
decreases within the boundary layer due to
increase of the Prandtl number Pr.
From figure 3.6(a), it can be
observed that increase in the value of the
viscous dissipation parameter Vd leads to
increase the value of the skin friction
coefficient f ′′ (x,0 ) . Again figure 3.6(b)
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
468
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
shows that the increase the viscous
dissipation parameter Vd leads to increase
the surface temperature θ (x, 0). Similar
results hold in the skin friction coefficient
f ′′ ( x,0 ) and the surface temperature
=0.73, the viscous dissipation parameter Vd
= 0.25 and the dependent thermal
conductivity parameter γ = 0.10. Here it
distribution θ (x, 0) shown in figure 3.7(a)
and figure 3.7(b) respectively for different
values of the parameter such as γ (= 0.10,
temperature distributions θ (x, 0) both are
decreasing with the increasing values of M.
0.30, 0.40, 0.50) and other controlling
parameter are fixed (Pr =0.73, M = 1.10, Vd
= 0.25).
Figure 3.8(a) and figure 3.8(b)
represent the results of the skin friction
coefficient and the surface temperature
distribution respectively for different values
of magnetic parameters or Hartmann
Number M (= 1.10, 1.40, 1.60, 1.80, 2.10)
when the value of the Prandtl numbers Pr
From figure 3.9(a), it is shown that
skin friction coefficient f ′′ ( x ,0 )
can be observed that the skin friction
coefficient f ′′ ( x ,0 ) and the surface
the
decreases monotonically with the increase of
the Prandtl number Pr (=0.01, 0.10, 0.50,
0.73, 1.00) and from the figure 3.9(b), the
same result is observed on the surface
temperature distribution due to increase of
the value of the Prandtl number when other
values of the controlling parameters are M =
1.10, Vd = 0.25 and γ = 0.10.
0.4
γ = 0.50
γ = 0.40
γ = 0.30
γ = 0.10
0.8
Temperature profiles
Velocity profiles
γ = 0.50
γ = 0.40
γ = 0.30
γ = 0.10
0.4
(a)
(b)
0
0
0
2
η
4
6
0
2
η
4
6
Fig, 2(a) and 2(b) displayed the velocity and temperature profiles for different values of thermal dependent
conductivity γ against η with fixed values of Pr = 0.73, M = 1.10 and Vd = 0.25.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
469
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
0.4
1
M = 2.10
M = 1.80
M = 1.60
M = 1.40
M = 1.10
Temperature profiles
Velocity profiles
0.3
M = 2.10
M = 1.80
M = 1.60
M = 1.40
M = 1.10
0.8
0.2
0.1
0.6
0.4
0.2
(a)
0
0
(b)
2
4
0
6
0
2
η
η
4
6
Fig, 3(a) and 3(b) displayed the velocity and temperature profiles for different values of magnetic
parameter M against η with fixed values of Pr = 0.73, γ = 0.10 and Vd = 0.25.
0.8
Pr = 1.00
Pr = 0.73
Pr = 0.50
Pr = 0.10
Pr = 0.01
Temperature profiles
Velocity profiles
0.6
Pr = 1.00
Pr = 0.73
Pr = 0.50
Pr = 0.10
Pr = 0.01
1
0.4
0.2
0.8
0.6
0.4
0.2
(a)
0
0
3
6
9
(b)
η
12
15
0
18
0
3
6
9
η
12
15
18
Fig,4(a) and 4(b) displayed the velocity and temperature profiles for different values of Prandtl's number
Pr against η with fixed values of M = 1.10, γ = 0.10 and Vd = 0.25.
1
0.4
Vd = 0.95
Vd = 0.65
Vd = 0.45
Vd = 0.25
Temperature profiles
Velocity profiles
0.3
Vd = 0.95
Vd = 0.65
Vd = 0.45
Vd = 0.25
0.8
0.2
0.1
0.6
0.4
0.2
(a)
0
(b)
0
0
2
4
η
6
8
0
2
4
6
8
η
Fig,5(a) and 5(b) displayed the velocity and temperature profiles for different values of Viscous dissipation
parameter Vd against η with fixed values of M = 1.10, γ = 0.10 and Pr = 0.73.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
470
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
2
2
γ = 0.50
γ = 0.40
γ = 0.30
γ = 0.10
(b)
1.6
Surface temperature
Skin friction
1.5
1
1.2
0.8
γ = 0.50
γ = 0.40
γ = 0.30
γ = 0.10
0.5
0.4
(a)
0
0
5
10
x
15
20
0
25
0
5
10
15
20
25
x
Fig.6(a) and 6(b) displayed the skin-friction and surface temperature coefficient for different values of
dependent thermal conductivity γ against η with fixed values of M = 1.10, Vd = 0.25 and Pr = 0.73.
1.6
1.6
M = 2.10
M = 1.80
M = 1.60
M = 1.40
M = 1.10
1.2
Surface Temperature
Skin friction
1.2
M = 2.10
M = 1.80
M = 1.60
M = 1.40
M = 1.10
0.8
0.4
0.8
0.4
(b)
(a)
0
0
5
10
x
15
20
0
25
0
5
10
15
20
25
x
Fig.7(a) and 7(b) displayed the skin-friction and surface temperature coefficient for different values of
magnetic parameter M against η with fixed values of γ = 0.10, Vd = 0.25 and Pr = 0.73.
1.6
Pr = 1.00
Pr = 0.73
Pr = 0.50
Pr = 0.10
Pr = 0.01
1.4
surface temperature
Skin friction
1.2
1
0.8
0.6
Pr = 1.00
Pr = 0.73
Pr = 0.50
Pr = 0.10
Pr = 0.01
1.6
1.2
0.8
0.4
0.4
0.2
0
(b)
(a)
0
5
10
15
x
20
0
0
5
10
15
20
25
x
Fig.8(a) and 8(b) displayed the skin-friction and surface temperature coefficient for different values of
Prandtl's number parameter Pr against η with fixed values of γ = 0.10, Vd = 0.25 and M =1.10.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
471
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
2
Vd = 0.95
Vd = 0.65
Vd = 0.45
Vd = 0.25
1.6
Surface temperature
1.6
Skin friction
2
Vd = 0.95
Vd = 0.65
Vd = 0.45
Vd = 0.25
1.2
0.8
0.4
1.2
0.8
0.4
(b)
(a)
0
0
5
10
15
20
0
25
0
5
10
x
15
20
x
Fig.9(a) and 9(b) displayed the skin-friction and surface temperature coefficient for different values of
Viscous dissipation parameter Vd against η with fixed values of γ =0.10, Pr = 0.73 and M =1.10.
Table 2.1: Skin friction coefficient and surface temperature distribution for different values of
viscous dissipation parameter Vd against x with other controlling parameters
M = 1.10, γ = 0.10, Pr =0.73.
Vd = 0.25
x
f ′′ ( x, 0)
θ ( x, 0)
Vd = 0.45
f ′′ ( x, 0 ) θ ( x, 0 )
Vd = 0.65
f ′′ ( x, 0)
θ ( x, 0 )
Vd = 0.95
f ′′ (x, 0)
θ ( x, 0 )
0.0000
0.0154
0.2044
0.0154
0.2044
0.0154
0.2044
0.0154
0.2044
0.3150
0.4572
0.7086
0.4598
0.7128
0.4624
0.7172
0.4664
0.7239
0.7090
0.5557
0.7630
0.5617
0.7711
0.5678
0.7795
0.5773
0.7927
1.0409
0.6048
0.7854
0.6135
0.7964
0.6225
0.8078
0.6367
0.8259
2.0369
0.6930
0.8294
0.7092
0.8472
0.7264
0.8663
0.7544
0.8980
3.1340
0.7500
0.8566
0.7735
0.8804
0.7991
0.9069
0.8423
0.9525
4.0635
0.7841
0.8720
0.8132
0.9003
0.8456
0.9325
0.9018
0.9897
4.9876
0.8108
0.8848
0.8450
0.9171
0.8839
0.9546
0.9532
1.0232
6.1118
0.8369
0.8973
0.8770
0.9340
0.9236
0.9776
1.0090
1.0604
7.1132
0.8562
0.9072
0.9012
0.9475
0.9544
0.9963
1.0546
1.0918
9.1512
0.8876
0.9216
0.9418
0.9687
1.0080
1.0277
1.1399
1.1503
10.1191
0.8999
0.9275
0.9582
0.9775
1.0304
1.0412
1.1781
1.1773
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
472
Table2.2: Skin friction coefficient and surface temperature distribution for different values of
thermal depentdent variation parameter γ against x with other controlling parameters
M = 1.10, Vd = 0.25, Pr =0.73.
γ
x
0.0000
0.3150
0.7090
1.0409
2.0369
3.1340
4.0635
4.9876
6.1118
7.1132
9.1512
10.1191
= 0.10
f ′′ ( x, 0 ) θ ( x, 0 )
0.0154
0.4603
0.5680
0.6276
0.7574
0.8756
0.8743
1.0781
1.2181
1.3602
1.7166
1.9251
0.2044
0.7111
0.7720
0.8014
0.8709
0.9343
0.9879
1.0464
1.1273
1.2121
1.4313
1.5647
γ
= 0.30
γ
f ′′ ( x, 0) θ ( x, 0)
0.0154
0.4612
0.5679
0.6255
0.7446
0.8430
0.9179
0.9906
1.0808
1.1656
1.3592
1.4646
0.2044
0.7135
0.7750
0.8040
0.8691
0.9229
0.9640
1.0054
1.0580
1.1090
1.2273
1.2941
In Table 2.1 are given the tabular
values of the local skin friction coefficient
f ′′ ( x, 0) and the surface temperature
θ ( x, 0 ) for different values of viscous
dissipation parameter Vd while Pr =0.73, M
= 1.10 and γ = 0.10. Here it is found that the
values of local skin friction coefficient
f ′′ ( x, 0) increase at different position of x
for viscous dissipation parameter Vd
=0.25,.0.45.0, 0.65, 0.95. The rate of local
skin friction coefficient f ′′ ( x, 0) is increase
by 15.0108% as the viscous dissipation
parameter Vd changes from 0.25 to 0.95 and
x = 4.0635. Furthermore it is seen that the
numerical values of the surface temperature
θ ( x, 0 ) decrease for increasing values of
viscous dissipation parameter Vd. The rate
= 0.40
γ
= 0.50
f ′′ ( x, 0) θ (x, 0 )
0.0154
0.4651
0.5774
0.6397
0.7738
0.8911
0.9845
1.0786
1.2001
1.3186
1.6036
1.7661
0.2044
0.7201
0.7881
0.8220
0.9020
0.9733
1.0315
1.0928
1.1744
1.2572
1.4643
1.5884
f ′′ (x, 0) θ ( x, 0)
0.0154
0.4692
0.5872
0.6547
0.8061
0.9464
1.0639
1.1874
1.3538
1.5233
1.9567
2.2181
0.2044
0.7269
0.8018
0.8414
0.9391
1.0334
1.1157
1.2065
1.3343
1.4710
1.8432
2.0834
of increase the local surface temperature is
13.4977% at position x = 4.0635 as the
viscous dissipation parameter Vd changes
from 0.25 to 0.95.Numerical values of local
surface temperature θ ( x, 0 ) are calculated
from equation (2.14) for the surface of the
sphere from lower stagnation point to upper
stagnation point.
In Table 2.2 the tabular values of the
local skin friction coefficient f ′′ (x, 0) and
surface temperature distribution θ ( x, 0) for
different values of thermal conductivity
variation parameter γ while Pr = 1.0 and
Vd = 0.25 are given in table 2.2. Here we see
that the values of local skin friction
coefficient f ′′ ( x, 0 ) increases at different
position of x for thermal conductivity
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
473
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
variation parameter γ = 0.10, 0.30, 0.40 and
0.50. The rate of the local skin friction
coefficient f ′′ ( x, 0) increases by 21.1537%
as the thermal conductivity variation
parameter γ changes from 0.10 to 0.50 and
x = 4.0635. Furthermore, it is seen that the
numerical values of the surface temperature
increases
as
the
thermal
θ ( x, 0)
conductivity
variation
parameter
γ
increases. The rate of increases is 12.9365%
at position x = 4.0635 as the thermal
conductivity variation parameter γ changes
from 0.10 to 0.50.
CONCLUSION
The effect of viscous dissipation
parameter Vd, magnetic parameter or
Hartmann Number M, the temperature
dependent thermal conductivity variation
parameter γ and the Prandtl number Pr on
the
magneto-hydrodynamic
natural
convection boundary layer flow along a
vertical flat plate has been studied
introducing a new class of transformations.
The transformed non-similar boundary layer
equations governing the flow together with
the boundary conditions based on
conduction and convection were solved
numerically using the very efficient implicit
finite difference method together with Keller
box scheme. The coupled effect of natural
convection and conduction required the
temperature and the heat flux is continuous
at the interface. . From the present
investigation, the following conclusions may
be drawn:
Both the skin friction coefficient and the
velocity distribution increase for
increasing values of the viscous
dissipation
parameter
Vd
and
temperature
dependent
thermal
conductivity variation parameter γ .
Increased values of the viscous
dissipation parameter Vd leads to
increase the surface temperature
distribution as well as the temperature
distribution.
Increased values of the temperature
dependent
thermal
conductivity
variation parameter γ lead to increase
the surface temperature distribution as
well as the temperature distribution.
It has been observed that the skin
friction
coefficient,
the
surface
temperature distribution, the temperature
profiles and the velocity distribution
decrease over the whole boundary layer
with the increase of the Prandtl number
Pr.
For the effect of magnetic parameter or
Hartmann Number M, the skin friction
coefficient, the surface temperature
distribution and the velocity distribution
over the whole boundary layer
decreases,
but
the
temperature
distribution increases.
REFERENCES
1. Haque, Md. Raihanul, Ali, M. M., Alam,
Md. M. and Alim, M. M., “Effects of
viscous
dissipation
on
natural
convection flow over a sphere with
temperature
dependent
thermal
conductivity” J. Comp. & Math. Sci.
Vol. 5(1), pp. 5-14, (2014).
2. Alam, Md. M., Alim, M. M., and
Chowdhury, Md. M. K., “Effect of
pressure stress work and viscous
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
Md. S. Islam, et al., J. Comp. & Math. Sci. Vol.5 (5), 460-474 (2014)
dissipation in natural convection flow
along a vertical flat plate with heat
conduction”,
Jounal
of
Naval
Architecture and Marine Engineering,
3(2), pp. 69-76, (2006).
3. Alim, M. M., Alam, Md. M. and
Abdullah Al-momun , “ Joule heating
effects on the coupling of conduction
with
magnetohydrodynamic
free
convection flow from a vertical flat
plate”, Nonlinear analysis; Modeling
and Control, Vol.12,No-3, pp. 307-316,
(2007).
4. Alam, Md. M., Alim,M. A. and
Chowdhury, M. K., “Stress work effect
on natural convection flow along a
vertical flat plate with joule heating and
heat conduction” Journal Mechanical
Engineering, Vol. ME38, pp. 18-24,
(2007).
5. Alim, M. M., Alam, Md. M., Abdullah
Al-momun and Hossain, Md. Belal,
“Combined effect of viscous dissipation
6.
7.
8.
9.
474
and Joule heating on the coupling of
conduction and free convection along a
vertical flat plate”, International
communication of heat and mass
transfer, Vol.35, pp. 338-346, (2008).
Hossain, M. A. and Alim, M. A,
“Natural convection-radiation interaction
on boundary layer flow along a Thin
cylinder”, J. Heat and Mass Transfer 32,
pp.515-520, (1997).
Pozzi, A. and Lupo, M., “The coupling
of conduction with laminar natural
convection along a flat plate”, Int. J.
Heat Mass Transfer 31(9), pp. 18071814, (1988).
Cebeci and Bradshaw, P, “Physical and
Computational Aspects of Convective
Heat Transfer”, Springer, N. Y. (1984).
Joshi,Y. and Gebhart, B, “Effect of
pressure stress work and viscous
dissipation in some natural convection
flows”, Int. J. Heat Mass Transfer,
24(10), pp. 1377-1388, (1981).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481)
© Copyright 2024 ExpyDoc
