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International Journal of Energy & Technology 3 (34) (2011) 1– 6
TRANSIENT MHD NATURAL CONVECTION PAST A VERTICAL CONE
HAVING RAMPED TEMPERATURE ON THE CURVED SURFACE
Anand Kumar* and A. K. Singh°
*Department of Mathematics, Central University of Rajasthan, Ajmer, India,
aanandbhu@gmail.com
°Department of Mathematics, Banaras Hindu University, Varanasi, India,
ashok_56vns@rediffmail.com
ABSTRACT
The transient free convective flow of an electrically conducting and viscous incompressible fluid past a vertical cone
having ramped type temperature on the curved surface is considered here in the presence of a transverse magnetic field.
The non-linear partial differential equations are solved by employing implicit finite difference method. The numerical
results for the velocity, temperature, skin-friction and Nusselt number are presented by the graphs. A comparative study
of the results corresponding to the cases of ramped and isothermal temperatures on the curved surface of the cone has
revealed that the same steady state values for average skin-friction and average Nusselt number are attained almost in
the same time.
Keywords: Natural convection, Ramped temperature, Isothermal temperature, Vertical cone, Crank-Nicolson
1. INTRODUCTION
The laminar free convective flows of an incompressible
viscous fluid over a vertical cone in the presence of a magnetic
field have many industrial and technological applications such
as reactor cooling, surface coating of metals and crystal
growth etc. Merk and Prins [1, 2] have obtained the similarity
solution of laminar free convective flow over an isothermal
vertical cone. Hering and Grosh [3] have discoursed the
similarity solution for free convective flow from the vertical
cone with prescribed wall temperature being a power function
of the distance along a cone ray. Further studies have been
performed by Hering [4] and Roy [5] by considering low and
high Prandtl number respectively.
Pop and Takhar [6] have studied the compressibility
effects of the laminar free convective flow from a vertical
cone while Watanable [7] and Hossain and Paul [8] have
considered the effect of suction/injection. Pop and Na [9] have
studied the effects of suction/injection on free convective flow
from a vertical cone by taking uniform surface heat flux
condition
Yih [10] has presented the radiation effect on mixed
convection over an isothermal cone embedded in a saturated
porous media by using a numerical approach. Takhar et al.
[11] have investigated the unsteady mixed convective flow
from a rotating vertical cone by considering time-dependent
angular velocity under the transverse magnetic field.
Chamkha and Al-Mudhaf [12] have shown the numerical
solution of unsteady laminar heat and mass transfer from a
rotating vertical cone with a magnetic field by considering
heat generation or absorption effects. Ece [13, 14] have
studied the numerical solution of laminar free convective flow
over a vertical cone in the presence of a transverse magnetic
field by considering the mixed type thermal boundary layer
and a constant angular velocity solution respectively.
In this paper, the numerical solutions are obtained for the
transient free convective flow past a vertical cone under the
influence of a transverse magnetic field when the thermal
boundary condition is modified like ramped type. The
governing boundary layer equations are solved by an implicit
finite difference scheme of Crank-Nicolson type.
2. MATHEMATICAL FORMULATION
Consider the transient laminar free convective flow of a
viscous and incompressible fluid past an axi-symmetric
vertical cone. The -axis is taken along the surface of the
cone from the apex () while the -axis is taken
normal outward to it. The semi vertical angle and local radius
of the cone are considered as and respectively. Initially,
for time , the temperature of fluid as well as the cone
are same as
. At time , the temperature of the curved
surface of the cone is raised or lowered to
when
, and thereafter, for
, is
maintained at the constant temperature
By using
boundary layer and Boussinesq approximations, the
governing continuity, momentum and thermal energy
conservation equations for the considered model are derived
as follows:
(1)
International Journal of
Energy & Technology
www.journal-enertech.eu
ISSN 2035-911X
International Journal of
Energy & Technology
Anand Kumar & Singh / International Journal of Energy & Technology 3(34) (2011) 1-6
(2)
(3)
The initial and boundary conditions for the considered
problem are as follows:
(4)
By defining the non-dimensional variables
(5)
the Eqs. (1) and (3) in non-dimensional form can be now
expressed as follows:
(6)
(7)
(8)
The initial and boundary conditions in non-dimensional
form are obtained as
(9)
The non-dimensionalisation process has suggested that the
Prandtl number , magnetic parameter , the characteristic
time can be taken as
(10)
and .
3. NUMERICAL SOLUTION
The coupled non-linear nature of the partial differential
Eqs.(6)-(8) under the corresponding initial and boundary
conditions (9) clearly suggests that the solutions must be
obtained numerically and for this purpose we have used the
implicit finite difference method of Crank-Nicolson type. In
this process the obtained finite difference equations after some
algebraic manipulations are written into a system of equations
in tri-diagonal form and finally they have been solved by
Thomas algorithm.
The computational space has been restricted to finite
dimensions in the solution procedure. So the domain of
integration is considered as a rectangle with =1 and
=20 (corresponding to ) have been considered.
We have taken grid points in the numerical
computation for better results and the value of is taken as
. The mesh sizes have been fixed as ,
with time step . During any one time
step, the computed values of the previous time step have been
used for the coefficients and appearing in Eqs. (6)-
(8). At the end of each time step, first we have computed the
temperature field and then the evaluated values are employed
to obtain the velocity components in and directions,
respectively. The unsteady values of the components of
velocity and temperature field for a desired time have been
obtained by taking required number of iterations. The steady
state numerical solutions have been obtained for the
temperature and velocity fields when the following
convergence criterion is satisfied
(11)
where
stands for either the temperature or velocity field.
The superscripts denote the values of the dependent variables
after the th and th iterations respectively. Other
most important results of practical interest are the skin-friction
and Nusselt number. By using the computed values of the
velocity field, the local skin-friction and the average
skin-friction in non-dimensional form on the curved surface of
the cone are obtained as follows:
(12)
(13)
In a similar way, the local Nusselt number and the average
Nusselt number on the curved surface of the cone can be given
as follows:
(14)
(15)
4. RESULT AND DISCUSSION
In this study, we show the influence of the magnetic
parameter, semi-vertical angle of the vertical cone and time
parameter in order to clearly observe their respective effects
on the velocity profiles of the flow, temperature field, local
skin-friction, average skin-friction, local Nusselt number and
average Nusselt number.
The variations of tangential velocity distribution of the
fluid with at a cross section for different values of
the magnetic parameter and semi-vertical angle are shown in
Figs.. It can be clearly seen from these figures
that as the semi-vertical angle increases, the tangential
velocity component reduces. This causes to reduce the
momentum boundary layer thickness. Also the tangential
Anand Kumar & Singh / International Journal of Energy & Technology 3(34) (2011) 1-6
velocity of the fluid decreases with the magnetic parameter.
By increasing and , the time taken to reach to the steady
state increases. Further, the pick of the temporal maximum
velocity moves away from the surface of the cone by increase
with the time while has opposite effect. In Figs.
, we have shown the influence of and on the lateral
velocity distribution of the fluid with at a fixed cross
section . It is apparent from the figures that the lateral
velocity increases as and increase.
0 1 2 3 4 5 6
0.00
0.02
0.04
0.06
0.08
0.10
0.12
(a)
=150
=300
SS =13.1
SS = 15.9
2.0 2.0
1.0
1.0
0.5
t = 0.2
u
y
0 1 2 3 4 5 6
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
(b)
=150
=300
SS = 16.4
SS = 15.2
2.0
2.0
1.0
1.0
0.5
t = 0.2
u
y
Figure 1: Velocity profile at for different values
of when and
The derivatives involved in Eqs. (12) to (15) are obtained
using a five point finite difference formula and the integrals
are evaluated using Simpson's rule of integration.
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 0 1 2 3 4 5 6 7 8
(b)
=150
=300SS
SS
2.0
2.0
1.0
1.0 0.5 t = 0.2
y
- v
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 0 1 2 3 4 5 6
(a)
SS
SS
2.0
2.0
1.0
1.0
0.5 t = 0.2
=150
=300
y
- v
Figure 2: Velocity profile at for different values
of when and
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
(a)
=150
=300
SS SS
2.0
2.0
1.01.0
0.5 0.5
0.2 t = 0.2
x
Anand Kumar & Singh / International Journal of Energy & Technology 3(34) (2011) 1-6
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
(b)
=150
=300
SS SS
2.0
2.0
1.0
1.0
0.5
0.5
0.2
t = 0.2
x
Figure 3: Local skin-friction profile for different values of
when and
In Figs. the local skin friction are shown for
different values of and as a function of . It is observed
that the effect of over the local skin friction is to reduce it
gradually and finally reaches the steady state value. The
influence of magnetic parameter () is to decrease the value
of . Figs. shows the effects of and on
the local Nusselt number as a function of . From these
figures we can see that by increasing , the local Nusselt
number has attained maximum value and after that, by
decreasing minutely, it has attained almost constant value. At
time , the has attained its maximum value and then
by decreasing it has reached to the steady state value. The
maximum value of is decreases with and
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
(b)
=150
=300
t = 0.2
0.5
1.0
1.0
2.0
2.0
SS SS
Nu
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a)
=150
=300
SS
SS
2.0
2.0
1.0
0.5
0.5
t = 0.2
Nu
x
Figure 4: Local Nusselt number for different values of
when and
The effects of and on the average skin friction and
average Nusselt number are shown in Figs. 5 and 6
respectively. From these figures, we can see that the effect of
and are decrease the values of and . It is
noticed from Fig. 6 that the is gradually increasing with
and attaining maximum value almost at and then
by slightly decreasing it has attained the steady state value.
Fig. 7 illustrates the effect of magnetic parameter and the
semi vertical angle on the steady state values of and
. From this figure we can see that the maximum steady
state values of and occur when and
and gradually decreases with increase in the values of
and .
0 1 2 3 4 5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
=150
=300
M = 10 M = 5
M = 0
av
t
Figure 5: Average skin-friction for different values of
when and
Anand Kumar & Singh / International Journal of Energy & Technology 3(34) (2011) 1-6
0 1 2 3 4 5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
M = 10
M = 5
M = 0
=150
=300
Nuav
t
Figure 6: Average Nusselt number for different values of
when and
015 30 45 60 75
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
M = 10 M = 5
M = 0
Nuav ,
av
av
Nuav
Figure 7: Average Nusselt number, average skin-friction for
different values of when and in
the steady state.
Table 1 illustrates the comparative study of the average
skin-friction and average Nusselt number for the
cases of ramped and isothermal temperatures respectively
over the curved surface of the cone. An important observation
from the table is that initially the numerical values of the
average skin-friction in the case of ramped temperature are
less than compare to the case of constant temperature.
However, by increasing monotonically, the average
skin-friction for both cases has attained the same steady state
value in almost same time. Further, we can observe that the
average value of Nusselt number in the case of ramped
temperature has increasing tendency in time interval
and then by decreasing gradually it has attained the steady
state value while in the case of constant temperature, the value
of average Nusselt number is maximum initially and then by
decreasing gradually it has attained the same steady state
value as in the case of ramped temperature almost in the same
time.
Table 1: Comparison values of the average skin-friction ()
and average Nusselt number ( ) for ramped and
isothermal cases.
(ramp.
case)
(iso.
case)
(ramp.
case)
(iso.
case)
5
0.2
0.02218
0.11063
0.12753
0.64336
0.5
0.07258
0.14417
0.16793
0.34572
1.0
0.16113
0.16113
0.26901
0.26901
2.0
0.17680
0.17055
0.19640
0.22515
13
0.17552
0.17552
0.20105
0.20104
14
0.17553
0.17553
0.20104
0.20104
15
0.17553
0.17553
0.20104
0.20104
0.2
0.01989
0.09922
0.12750
0.64263
0.5
0.06512
0.12944
0.16738
0.34376
1.0
0.14487
0.14487
0.26564
0.26564
2.0
0.15972
0.15364
0.18861
0.21997
16
0.15867
0.15867
0.19265
0.19265
17
0.15867
0.15867
0.19265
0.19264
18
0.15867
0.15867
0.19264
0.19264
Table 2: Comparison of the steady state local skin-friction and
local Nusselt number values at with Bapuji et al.
[15].
Bapuji [15]
Present
results
Bapuji [15]
Present
results
0.03
1.2368
1.2385
0.1244
0.1259
0.10
1.0911
1.0910
0.2115
0.2144
0.70
0.8168
0.8154
0.4529
0.4621
1.00
0.7668
0.7595
0.5125
0.5139
For result validation, we have compare with the result by
Bapuji et al. [15] for the case of no magnetic field. For the
steady state values of local skin-friction () and local Nusselt
number () at for different values of are listed
in Table 2. It is observed that the results are in good
agreement.
5. CONCLUSION
The transient MHD natural convective flow past a vertical
cone having a ramped temperature along the curved surface of
the cone has been studied numerically by using implicit finite
difference method. The effects of magnetic field and semi
vertical angle on the velocity profiles are found to decrease it.
The local Nusselt number and average Nusselt number are
maximum when the non-dimensional time is equal to .
The steady state time in the reference of the velocity
components, local skin-friction, local Nusselt number,
average skin-friction and average Nusselt number increases as
magnetic parameter and semi-vertical angle increase. A
comparative study suggests that the steady state time is almost
Anand Kumar & Singh / International Journal of Energy & Technology 3(34) (2011) 1-6
same in the cases of ramped temperature and isothermal
temperature on the curved surface of the cone.
6. REFERENCES
[1] H. J. Merk and J. A. Prins, Thermal convection laminar
boundary layer-I, Applied Science Reasearch, vol. 4, pp.
11-24, 1953.
[2] H. J. Merk and J. A. Prins, Thermal convection laminar
boundary layer-II, Applied Science Reasearch, vol. 4,
pp. 195-206, 1954.
[3] R. G. Hering and R. J. Grosh, Laminar free convection
from a non-isothermal cone, International Journal of
Heat and Mass Transfer, vol. 5, pp. 1059-1068, 1962.
[4] R. G. Hering, Laminar free convection from a
non-isothermal cone at low Prandtl numbers,
International Journal of Heat and Mass Transfer, vol. 8,
pp. 1333-1337, 1965.
[5] S. Roy, Free convection from a vertical cone at high
Prandtl numbers, ASME Journal Heat Transfer, vol. 96,
pp. 115-117, 1974.
[6] I. Pop and H. S. Takhar, Compressibility effects in
laminar free convection from a vertical cone, Applied
Scientific Research, vol. 48, pp. 71-81, 1991.
[7] T. Watanable, T., Free convection boundary layer flow
with uniform suction/injection over a cone, Acta
Mechanica, vol. 87, pp. 1-9, 1991.
[8] M. A. Hossain and S. C. Paul, Free convection from a
vertical permeable circular cone with non-uniform
surface heat flux, Heat and Mass Transfer, vol. 37, pp.
167-173, 2001.
[9] I. Pop and T. Y. Na, Natural convection over a vertical
wavy frustum of a cone, International Journal of
Non-Linear Mechanics, vol. 34, pp. 925-934, 1999.
[10] K. A. Yih, Radiation effect on mixed convection over
an isothermal cone in porous media, Heat and Mass
Transfer, vol. 37, pp. 53-57, 2001.
[11] H. S. Takhar, A. J. Chamkha and G. Nath, Unsteady
Mixed convection flow from a rotating vertical cone
with magnetic field, Heat and Mass Transfer, vol. 39,
pp. 297-304, 2003.
[12] A. J. Chamkha and A. Al-Mudhaf, Unsteady heat and
Mass transfer from a rotating vertical cone with a
magnetic field and heat generation or absorption effects,
International Journal of Thermal Sciences, vol. 44, pp.
267-276, 2005.
[13] M. C. Ece, Free convection flow about a cone under
mixed thermal boundary conditions and a magnetic field,
Applied Mathematical Modelling, vol. 29, pp.
1121-1134, 2005.
[14] M. C. Ece, Free convection flow about a vertical spinning
cone under a magnetic field, Applied Mathematics and
Computation, vol. 179, pp. 231-242, 2006.
[15] P. Bapuji, K. Ekambavanan and I. Pop, Finite difference
analysis of laminar free convection flow past a
non-isothermal vertical cone, Heat Mass Transfer, vol.
44, pp. 517-526, 2008.
NOMENCLATURE
specific heat at constant pressure [J Kg-1 K-1]
acceleration due to gravity [m s-2]
constant magnetic field [T]
magnetic parameter [-]
local Nusetl number [-]
average Nusetl number [-]
Prandtl number [-]
local radius of the cone [m]
dimensionless radius of the cone [-]
Temperature [K]
dimensionless temperature [-]
wall temperature [K]
free stream temperature [K]
Time [s]
dimensionless time [-]
characteristic time [-]
velocity components along and perpendicular to
the surface of the cone from the apex [m s-1]
dimensionless velocity components along and
perpendicular to the surface of the cone from the
apex
distance along and perpendicular to surface [m]
dimensionless distance along and perpendicular
to the surface [-]
Greek
Symbols
coefficient of thermal expansion [K-1]
fluid thermal conductivity [W m-1 K-1]
semi vertical angle [radian]
kinematic viscosity of the fluid [m2 s-1]
density of the fluid [Kg m-3]
fluid electrical conductivity [s m-1]
local skin friction coefficient
average skin friction
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