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CHAPTER-7
MHD Natural Convection Flow Past an Impulsively Moving Vertical Plate with Ramped Temperature in the Presence of Hall Current and Thermal Radiation*
7.1 Introduction
Natural convection flows are frequently encountered in science and technological problems
such as chemical catalytic reactors, nuclear waste repositories, petroleum reservoirs, fiber
and granular insulation, geothermal systems etc. Natural convection flows from bodies,
with different geometries are extensively investigated as it is evident from the several
review articles and books published so far (Ede, 1967; Gebhart et al., 1989; Gebhart, 1973;
Jaluria, 1980; Raithby and Hollands, 1985). Convective heat transfer flow from bodies with
different geometries embedded in a porous medium is of significant importance due to its
varied and wide applications in many areas of science and technology, namely, drying of
porous solids, thermal insulation, enhanced oil recovery, cooling of nuclear reactors,
underground energy transport etc. Keeping in view importance of such fluid flow problems,
*Communicated to International Journal of Applied Mechanics and Engineering
151
a number of investigations on natural convection flow near a vertical plate embedded in a
porous medium are carried out. Mention may be made of research studies of Cheng and
Minkowycz (1977), Ranganathan and Viskanta (1984), Nakayama and Koyama (1987), Lai
and Kulacki (1991) and Hsieh et al. (1993). Comprehensive reviews of free convection
flows with heat and mass transfer in porous media are well presented by Pop and Ingham
(2002), Vafai (2005) and Nield and Bejan (2006). Investigation of hydromagnetic free
convection flow in porous and non porous media under different conditions is carried out
by several researchers due to significant effect of magnetic field on the boundary layer
control and on the performance of so many engineering devices using electrically
conducting fluids viz. MHD energy generators, MHD pumps, MHD accelerators, MHD
flow-meters, nuclear reactors using liquid metal coolants etc. This type of fluid flow
problem also find application in plasma studies, geothermal energy extraction, metallurgy,
petroleum and chemical engineering etc. Unsteady magnetohydrodynamic natural
convection flow past an infinite vertical plate in non porous medium is investigated by a
number of researchers considering different sets of thermal conditions at the bounding
plate. Mention may be made of the research studies of Georgantopoulos (1979), Raptis and
Singh (1983), Sacheti et al. (1994) and Chandran et al. (2001). Raptis and Kafousias (1982)
studied steady free convection flow past an infinite vertical porous plate through a porous
medium in the presence of magnetic field. Raptis (1986) investigated unsteady two-
dimensional natural convection flow past an infinite vertical porous plate embedded in a
porous medium in the presence of magnetic field. Chamkha (1997b) studied transient MHD
free convection flow through a porous medium supported by a surface. Chamkha (1997a)
also investigated hydromagnetic natural convection flow from an isothermal inclined
surface adjacent to a thermally stratified porous medium. Jha (1991) analyzed unsteady
hydromagnetic free convection and mass transfer flow past a uniformly accelerated moving
vertical plate through a porous medium. Aldoss et al. (1995) considered combined free and
forced convection flow from a vertical plate embedded in a porous medium in the presence
of a magnetic field. Takhar and Ram (1994) analyzed MHD free convection flow of water
at 4oC through porous medium. Kim (2000) studied MHD natural convection flow past a
moving vertical plate embedded in a porous medium. Ibrahim et al. (2004) studied
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unsteady hydromagnetic free convection flow of micro-polar fluid and heat transfer past a
vertical porous plate through a porous medium in the presence of thermal and mass
diffusion with a constant heat source. Makinde and Sibanda (2008) investigated steady
laminar hydromagnetic heat transfer by mixed convective flow over a vertical plate
embedded in a uniform porous medium in the presence of uniform normal magnetic field.
Makinde (2009) investigated MHD mixed convection flow and mass transfer past a vertical
porous plate with constant heat flux embedded in a porous medium.
In all these investigations the effects of radiation is not taken into account. Radiative
heat transfer along with free convection is important in many areas of science and
engineering viz. glass production, furnace design, electric power generation, thermo-
nuclear fusion, casting and levitation, high temperature aerodynamics, propulsion systems,
plasma physics, cosmical flight, solar power technology and spacecraft re-entry
aerothermodynamics. In many practical applications depending on the surface properties
and configuration, radiative heat transfer is often comparable with that of convective heat
transfer. It is worth to record that unlike convection/ conduction the governing equations
taking into account radiative heat transfer become quite complicated and hence many
difficulties arise while solving such equations. However, some reasonable approximations
are proposed to solve the governing equations with radiative heat transfer. The text book by
Sparrow and Cess (1970) describes the essential features of radiative heat transfer. Chang
et al. (1983) investigated natural convection flow with radiative heat transfer in two-
dimensional complex enclosures. Cess (1966) studied laminar free convection along a
vertical isothermal plate with thermal radiation using Rosseland diffusion approximation.
Hossain and Takhar (1996) considered radiation effects on mixed convection boundary
layer flow along a vertical plate with uniform surface temperature using Rosseland flux
model. Chamkha (1997c) analyzed solar radiation assisted natural convection in a uniform
porous medium supported by a vertical flat plate. Takhar et al. (1998) investigated coupled
radiation-convection dissipative non-gray gas flow in a non-Darcy porous medium using
Keller-box implicit difference scheme. Chamkha et al. (2001) studied laminar free
convection flow of air past a semi-infinite vertical plate in the presence of chemical species
concentration and thermal radiation. Muthucumaraswamy and Ganesan (2003) investigated
153
radiation effects on flow past an impulsively started vertical plate with variable
temperature. Ghosh and Bég (2008) discussed the effects of radiation on transient free
convection flow past an infinite hot vertical impulsively moving plate in a porous medium.
Chamkha (2000b) discussed thermal radiation and buoyancy effects on hydromagnetic flow
over an accelerating permeable surface with heat source or sink. Raptis and Massalas
(1998) studied oscillatory magnetohydrodynamic flow of a gray, absorbing-emitting fluid
with non-scattering medium past a flat plate in the presence of radiation assuming
Rosseland approximation. Azzam (2002) considered radiation effects on MHD mixed
convection flow past a semi-infinite moving vertical plate for high temperature differences.
Cookey et al. (2003) analyzed the influence of viscous dissipations and radiation past an
infinite heated vertical plate in a porous medium with time-dependent suction. Mahmoud
Mostafa (2009) discussed thermal radiation effects on unsteady MHD free convection flow
past an infinite vertical porous plate taking into account the effects of viscous dissipation.
Ogulu and Makinde (2009) investigated unsteady hydromagnetic free convection flow of a
dissipative and radiative fluid past a vertical plate with constant heat flux.
In all these investigations, analytical or numerical solution is obtained assuming
conditions for fluid velocity and temperature at the plate as continuous and well defined.
However, there exist several practical problems which may require non-uniform or
arbitrary wall conditions. Keeping in view this fact, Hayday et al. (1967), Kao (1975),
Kelleher (1971) and Lee and Yovanovich (1991) investigated free convection flow from a
vertical plate with step discontinuities in the surface temperature. Recently, Seth et al.
(2011e) studied MHD natural convection flow with radiative heat transfer past an
impulsively moving vertical plate with ramped temperature in a porous medium taking into
account the effects of thermal diffusion. It is well known that when density of an
electrically conducting fluid is low and/or applied magnetic field is strong, effects of Hall
current become significant. Hall current plays an important role in determining flow-
features of the problem because it induces secondary flow in the fluid. Therefore, it is
appropriate to study the effects of Hall current on MHD natural convection flow with
radiative heat transfer past a moving vertical plate with ramped temperature.
154
Aim of the present chapter is to study the effects of Hall current on unsteady natural
convection transient flow of a viscous, incompressible and electrically conducting fluid
with radiative heat transfer past an impulsively moving vertical plate embedded in a fluid
saturated porous medium taking into account the effects of thermal diffusion when the
temperature of the plate has a temporarily ramped profile. Hydromagnetic natural
convection flow with radiative heat transfer resulting from such a plate temperature profile
may have bearing on several engineering problems especially where initial temperature
profiles assume much significance in designing of electromagnetic devices and several
natural convection phenomena which occur due to such initial temperature profiles. Exact
solution of momentum and energy equations, under Boussinesq and Rosseland
approximations, is obtained in closed form by Laplace transform technique for both ramped
temperature and isothermal plates. Expressions for skin friction and Nusselt number for
both ramped temperature and isothermal plates are also derived. The numerical values of
fluid velocity and fluid temperature are displayed graphically versus boundary layer
coordinate y for various values of pertinent flow parameters for both ramped temperature
and isothermal plates. The numerical values of skin friction due to primary flow, skin
friction due to secondary flow and Nusselt number are presented in tabular form for various
values of pertinent flow parameters.
7.2 Formulation of the Problem and its Solution
Consider unsteady flow of a viscous, incompressible, electrically conducting and optically
thick fluid past an infinite vertical plate embedded in a uniform porous medium. Coordinate
system is chosen in such a way that x - axis is considered along the plate in upward
direction and y - axis normal to plane of the plate in the fluid. A uniform transverse
magnetic field 0B is applied in a direction which is parallel to y - axis. Initially i.e. at time
0t , both the fluid and plate are at rest and have a uniform temperature T . At time
0t , plate starts moving in x - direction with uniform velocity 0U . Temperature of the
plate is raised or lowered to 0wT T T t t when 0t t , and it is maintained at
155
uniform temperature wT when 0t t ( 0t being the characteristic time). Geometry of the
problem is same as that in figure 6.1. Since plate is of infinite length in x and z
directions and is electrically non-conducting, all physical quantities except pressure,
depend on y and t only. The induced magnetic field generated by fluid motion is
neglected in comparison to the applied one i.e. magnetic field 0(0, ,0)B B
. This
assumption is valid because magnetic Reynolds number is very small for liquid metals and
partially ionized fluids (Cramer and Pai, 1973). Also no applied or polarized voltages exist
so the effect of polarization of fluid is negligible (Meyer, 1958) i.e. electric field
(0,0,0)E
. This corresponds to the case where no energy is added or extracted from the
fluid by electrical means.
Keeping in view the assumptions made above, governing equations for natural
convection flow of a viscous, incompressible and electrically conducting fluid within a
uniform porous medium with radiative heat transfer taking Hall current into account, under
Boussinesq approximation, are given by
22
02 2
11
Bu u u mw u g T Tt y m K
, (7.1)
22
02 2
11
Bw w mu w wt y m K
, (7.2)
21
2
1 r
p p
k qT Tt c y c y
, (7.3)
where 1 1, , , , , , , , , , , , ,e e e e pu w K g T m k c and rq are, respectively, fluid
velocity in x - direction, fluid velocity in z - direction, kinematic coefficient of viscosity,
electrical conductivity, density, permeability of porous medium, acceleration due to
gravity, coefficient of thermal expansion, fluid temperature, Hall current parameter,
cyclotron frequency, electron collision time, thermal conductivity, specific heat at constant
pressure and radiative flux vector.
156
Initial and boundary conditions for the fluid-flow problem are
0, for 0 and 0u w T T y t , (7.4a)
0, 0 at 0 for 0u U w y t , (7.4b)
0 0 at 0 for 0wT T T T t t y t t , (7.4c)
0 at 0 for wT T y t t , (7.4d)
0, 0, as for 0u w T T y t . (7.4e)
For optically thick fluid, in addition to emission there is also self-absorption and usually the
absorption coefficient is wavelength dependent and large (Bestman, 1985) so we can adopt
the Rosseland approximation for radiative flux vector rq (Azzam, 2002). Thus rq is given
by
* 4
*
43r
Tqk y
, (7.5)
where *k is the mean absorption coefficient and * is the Stefan-Boltzmann constant. It is
assumed that there is small temperature difference between fluid temperature T and free
stream temperature T . Equation (7.5) is linearized by expanding 4T in Taylor series
about a free stream temperature T . Neglecting second and higher order terms in T T ,
4T is expressed in the following form
4 3 44 3T T T T . (7.6)
Making use of equations (7.5) and (7.6) in equation (7.3), we obtain
* 32 21
2 * 2
1613p p
k TT T Tt c y c k y
. (7.7)
Equations (7.1), (7.2) and (7.7), in non-dimensional form, become
2 2
2 211 r
u u M uu mw G Tt y Km
, (7.8)
157
2 2
2 211
w w M wmu wt y Km
, (7.9)
2
2
1
r
NT Tt P y
, (7.10)
where
2 2 20 0 0 0 0 0 0, , , , , ,wy y U t u u U w w U t t t T T T T T M B U
2 2 31 1 0 0 1, ,r w r pK K U G g T T U P c k
and * 3 *116 3N T k k
. (7.11)
21, , , , and r rT M K G P N are, respectively, non-dimensional fluid temperature, magnetic
parameter, permeability parameter, Grashof number, Prandtl number and non-dimensional
radiation parameter.
It is appropriate to mention here that characteristic time 0t is defined, according to the non-
dimensional process mentioned above, as
20 0t U . (7.12)
Initial and boundary conditions (7.4a) to (7.4e), in non-dimensional form, become
0, 0 for 0 and 0u w T y t , (7.13a)
1, 0 at 0 for 0u w y t , (7.13b)
at 0 for 0 1T t y t , (7.13c)
1 at 0 for 1T y t , (7.13d)
0, 0; 0 as for 0u w T y t . (7.13e)
Combining equations (7.8) and (7.9), we obtain
2*
21
1r
F F N F F G Tt y K
, (7.14)
where 2*
2
1and
1M im
F u iw Nm
.
158
Initial and boundary conditions (7.13a) to (7.13e), in compact form, are given by
0, 0 for 0 and 0F T y t , (7.15a)
1 at 0 for 0F y t , (7.15b)
at 0 for 0 1T t y t , (7.15c)
1 at 0 for 1T y t , (7.15d)
0, 0 as for 0F T y t . (7.15e)
Equations (7.10) and (7.14) with the use of Laplace transform and initial conditions (7.15a)
reduce to
2
2 0d T saTdy
, (7.16)
2*
21
1 0rd F s N F G Tdy K
, (7.17)
where
0 0
, , , and , , , 01
st strPa F y s F y t e dt T y s T y t e dt sN
( s being
Laplace transform parameter).
Boundary conditions (7.15b) to (7.15e), after taking Laplace transform, become
2 1 , 1 at 0sF s T e s y , (7.18a)
0, 0 as F T y . (7.18b)
Equations (7.16) and (7.17), subject to the boundary conditions (7.18a) and (7.18b), are
solved and the solution for ,T y s and ,F y s is given by
2
1,
sy as
eT y s e
s
, (7.19)
2
11,s
y s y s y ase
F y s e e es s s
, (7.20)
159
where *11 , 1 and 1rG a N K a .
Exact solution for the fluid temperature ,T y t and fluid velocity ,F y t is obtained by
taking inverse Laplace transform of (7.19) and (7.20) which is presented in the following
form after simplification (Abramowitz and Stegun, 1972).
, , 1 , 1T y t G y t H t G y t , (7.21)
1,2 2 2
y yy yF y t e erfc t e erfc tt t
* *, 1 , 1F y t H t F y t , (7.22)
where
22
4,2 2
aytay y a atG y t t erfc ye
t
,
*2
1,2 2 2
2 2
1 1 1 2 2 2
ty y
y a y a
y
e y yF y t e erfc t e erfc tt t
y a y ae erfc t e e rfc tt t
y y yt e erfc t tt
224
2
1 2 2 .2 2
y
ayt
ye erfc tt
ay y a att erfc yet
1H t and erfc x are, respectively, the unit step function and complementary error
function.
7.2.1 Solution in case of Isothermal Plate
Solutions (7.21) and (7.22) represent the analytical solution for fluid temperature and
velocity for the flow of a viscous, incompressible, electrically conducting and optically
thick fluid past an impulsively moving vertical plate with ramped temperature taking Hall
160
current into account. In order to highlight the influence of ramped temperature distribution
within the plate on the fluid flow, it is appropriate to compare such a flow with the one past
an impulsively moving vertical plate with uniform temperature. Keeping in view the
assumptions made in present study, solution for fluid temperature and velocity for fluid-
flow past an impulsively moving isothermal vertical plate is obtained and is expressed in
the following form
,2y aT y t erfc
t
, (7.23)
*
*
1,
2 2 2
2 2 2
y y
ty y
d y yF y t e erfc t e erfc tt t
d e y ye erfc t e erfc tt t
*
2 2 2y a y ay a y a y ae erfc t e erfc t d erfc
t t t
, (7.24)
where *d .
7.3 Skin Friction and Nusselt Number
The expressions for primary skin friction x , secondary skin friction z and Nusselt
number Nu , which are measures of shear stress at the plate due to primary flow, shear
stress at the plate due to secondary flow and rate of heat transfer at the plate respectively,
are presented in the following form for the ramped temperature and isothermal plates.
(i) For ramped temperature plate:
, (7.25)
, (7.26)
1 111 , 1 , 1t
x zi erfc t e F y t H t F y tt
2 1 1aNu t t H t
161
where
(ii) For isothermal plate:
, (7.27)
. (7.28)
It is evident from the expressions (7.26) and (7.28) that, for a given time, Nusselt number
is proportional to in both the cases i.e. Nusselt number
increases on increasing Prandtl number while it decreases on increasing radiation
parameter N. Since expresses relative strength of viscosity to thermal diffusivity of
fluid, decreases on increasing thermal diffusivity of fluid. This implies that thermal
diffusion and radiation tend to reduce rate of heat transfer at both ramped temperature and
isothermal plates. Also it is noticed from (7.26) and (7.28) that increases for ramped
temperature plate whereas it decreases for isothermal plate on increasing time . This
implies that as time progresses, rate of heat transfer at ramped temperature plate is
enhanced whereas it is reduced at isothermal plate.
1 2
1, 1 1
1 1 11
1 11 2 .2
tt
t t
eF y t erfc t e a erfc tt
a e t erfc t et t
aerfc t tt
* *11 1 1t tx zi d erfc t e d e erfc t
t
*1 1t a ae a erfc t dt tt
aNut
Nu1
rPaN
Nu
rP
rP
rP
Nu
t
162
7.4 Results and Discussion
In order to highlight the influence of various physical quantities, namely, Hall current,
thermal buoyancy force, permeability of medium, radiation and time on the flow-field in
the boundary layer region, the numerical values of fluid velocity, computed from the
analytical solutions (7.22) and (7.24), are depicted graphically versus boundary layer
coordinate y in figures 7.1 to 7.10 for various values of Hall current parameter , Grashof
number , permeability parameter , radiation parameter and time t taking
magnetic parameter and Prandtl number . It is noticed from figures 7.1
to 7.10 that, for both ramped temperature and isothermal plates, primary velocity and
secondary velocity attain a distinctive maximum value in the vicinity of surface of the
plate and then decrease properly on increasing boundary layer coordinate to approach
free stream value. Also primary and secondary fluid velocities are faster in case of
isothermal plate than that of ramped temperature plate. It is evident from figures 7.1 to 7.10
that, for both ramped temperature and isothermal plates, primary velocity and secondary
velocity increase on increasing Hall current parameter , Grashof number ,
permeability parameter , radiation parameter and time t. This implies that, for both
ramped temperature and isothermal plates, Hall current, thermal buoyancy force,
permeability of medium and radiation tend to accelerate fluid flow in the primary and
secondary flow directions in boundary layer region. As time progresses, for both ramped
temperature and isothermal plates, primary and secondary fluid velocities are getting
accelerated in the boundary layer region.
In order to study the influence of radiation, thermal diffusion and time on temperature
field, the numerical values of fluid temperature T, computed from the analytical solutions
(7.21) and (7.23), are displayed graphically versus boundary layer coordinate in figures
7.11 to 7.13 for various values of , and t for both ramped temperature and isothermal
plates. It is revealed from figures 7.11 to 7.13 that, for both ramped temperature and
isothermal plates, fluid temperature T increases on increasing either radiation parameter
m
rG 1K N2 15M 0.71rP
u
w
y
u
w m rG
1K N
y
N rP
N
163
or time t whereas it increases on decreasing Prandtl number . This implies that radiation
and thermal diffusion tend to enhance fluid temperature in the boundary layer region for
both ramped temperature and isothermal plates. As time progresses, for both ramped
temperature and isothermal plates, there is an enhancement in fluid temperature in the
boundary layer region. It is noticed from figures 7.11 to 7.13 that, fluid temperature is
maximum at the surface of the plate for both ramped temperature and isothermal plates and
it decreases properly on increasing boundary layer coordinate y to approach free stream
value. Also fluid temperature is lower for ramped temperature plate than that of isothermal
plate.
The numerical values of non-dimensional primary skin friction and secondary skin
friction for both ramped temperature and isothermal plates, computed from the
analytical expressions (7.25) and (7.27), are presented in tabular form in tables 7.1 to 7.6
for various values of taking and whereas those of
Nusselt number for both ramped temperature and isothermal plates, computed from
expressions (7.26) and (7.28), are displayed in tables 7.7 and 7.8 for different values of
. It is evident from tables 7.1 to 7.6 that, for both ramped temperature and
isothermal plates, primary skin friction decreases whereas secondary skin friction
increases on increasing . This implies that, for both ramped temperature
and isothermal plates, Hall current, thermal buoyancy force, permeability of medium and
radiation tend to reduce primary skin friction whereas these physical quantities have
reverse effect on secondary skin friction. As time progresses, for both ramped temperature
and isothermal plates, there is reduction in primary skin friction whereas there is an
enhancement in secondary skin friction. It is observed from tables 7.7 and 7.8 that, for both
ramped temperature and isothermal plates, Nusselt number decreases on increasing
and it decreases on decreasing Prandtl number . Nusselt number increases for
ramped temperature plate and it decreases for isothermal plate on increasing t. This implies
that radiation and thermal diffusion tend to reduce rate of heat transfer at both ramped
temperature and isothermal plates. As time progresses, rate of heat transfer at ramped
rP
x
z
1, , , andrm G K N t 2 15M 0.71rP
Nu
, andrN P t
x z
1, , , andrm G K N t
Nu
N rP Nu
164
temperature plate is enhanced whereas it is reduced at isothermal plate. These numerical
results are in agreement with the results concluded from analytical expressions (7.26) and
(7.28).
7.5 Conclusions
A theoretical study of MHD natural convection flow past an impulsively moving vertical
plate with ramped temperature in the presence of Hall current and thermal radiation is
presented. Significant results are summarized below:
1. For both ramped temperature and isothermal plates:
Hall current, thermal buoyancy force, permeability of medium and radiation tend to
accelerate fluid flow in the primary and secondary flow directions in boundary layer
region. As time progresses, primary and secondary fluid velocities are getting
accelerated in the boundary layer region.
2. Primary and secondary fluid velocities are faster in case of isothermal plate than that
of ramped temperature plate.
3. For both ramped temperature and isothermal plates:
Radiation and thermal diffusion tend to enhance fluid temperature in the boundary
layer region. As time progresses, there is an enhancement in fluid temperature in the
boundary layer region.
4. Fluid temperature is lower for ramped temperature plate than that of isothermal plate.
5. For both ramped temperature and isothermal plates:
Hall current, thermal buoyancy force, permeability of medium and radiation tend to
reduce primary skin friction whereas these physical quantities have reverse effect on
secondary skin friction. As time progresses, there is a reduction in primary skin
friction whereas there is an enhancement in secondary skin friction.
6. Radiation and thermal diffusion tend to reduce rate of heat transfer at both ramped
temperature and isothermal plates. As time progresses, rate of heat transfer at ramped
temperature plate is enhanced whereas it is reduced at isothermal plate.
165
Fig. 7.1 Primary velocity profiles when Gr=6, K1=0.5, N=1 and t=0.4
Fig. 7.2 Secondary velocity profiles when Gr=6, K1=0.5, N=1 and t=0.4
m 0.5, 1, 1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0
0.2
0.4
0.6
0.8
1.0
m 0.5, 1, 1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.00
0.05
0.10
0.15
0.20
0.25
0.30
y
u
w
y
Ramped Temperature - - - - - Isothermal
Ramped Temperature - - - - - Isothermal
166
Fig. 7.3 Primary velocity profiles when m=0.5, K1=0.5, N=1 and t=0.4
Fig. 7.4 Secondary velocity profiles when m=0.5, K1=0.5, N=1 and t=0.4
Gr 4, 6, 8
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
Gr 4, 6, 8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
u
y
y
w
Ramped Temperature - - - - - Isothermal
Ramped Temperature - - - - - Isothermal
167
Fig. 7.5 Primary velocity profiles when m=0.5, Gr=6, N=1 and t=0.4
Fig. 7.6 Secondary velocity profiles when m=0.5, Gr=6, N=1 and t=0.4
K1 0.2, 0.5, 0.8
0.0 0.5 1.0 1.5 2.0 2.50.0
0.2
0.4
0.6
0.8
1.0
K1 0.2, 0.5, 0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
y
u
y
w
Ramped Temperature - - - - - Isothermal
Ramped Temperature - - - - - Isothermal
168
Fig. 7.7 Primary velocity profiles when m=0.5, K1=0.5, Gr=6 and t=0.4
Fig. 7.8 Secondary velocity profiles when m=0.5, K1=0.5, Gr=6 and t=0.4
N 1, 3, 5
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
N 1, 3, 5
0 1 2 3 4 50.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
y
u
y
w
Ramped Temperature - - - - - Isothermal
Ramped Temperature - - - - - Isothermal
169
Fig. 7.9 Primary velocity profiles when m=0.5, K1=0.5, Gr=6 and N=1
Fig. 7.10 Secondary velocity profiles when m=0.5, K1=0.5, Gr=6 and N=1
t 0.2, 0.4, 0.6
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
t 0.2, 0.4, 0.6
0 1 2 3 40.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
y
y
u
w
Ramped Temperature - - - - - Isothermal
Ramped Temperature - - - - - Isothermal
170
Fig. 7.11 Temperature profiles when t=0.4 and Pr=0.71
Fig. 7.12 Temperature profiles when t=0.4 and N=1
N 1, 3, 5
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
Pr 0.71, 0.5, 0.3
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
y
T
Ramped Temperature - - - - - Isothermal
y
T
Ramped Temperature - - - - - Isothermal
171
Fig. 7.13 Temperature profiles when N=1 and Pr=0.71
t 0.2, 0.4, 0.6
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
y
T
Ramped Temperature - - - - - Isothermal
172
Table 7.1 Skin friction at ramped temperature plate when K1=0.5, N=1 and t=0.4
↓ →
4 6 8 4 6 8
0.5 3.52773 3.38016 3.2326 0.822145 0.840905 0.859665 1 2.96664 2.80741 2.64819 1.2036 1.23458 1.26555
1.5 2.49560 2.32391 2.15222 1.29095 1.3273 1.36365
Table 7.2 Skin friction at isothermal plate when K1=0.5, N=1 and t=0.4
↓ →
4 6 8 4 6 8
0.5 3.08103 2.71011 2.3392 0.873748 0.918309 0.962871 1 2.47507 2.07006 1.66506 1.28524 1.35704 1.42884
1.5 1.94873 1.50359 1.05846 1.37326 1.45076 1.52826
Table 7.3 Skin friction at ramped temperature plate when Gr=6, N=1 and t=0.4
↓ →
0.2 0.5 0.8 0.2 0.5 0.8
0.5 3.76606 3.38016 3.27836 0.763887 0.840905 0.863658 1 3.2297 2.80741 2.69626 1.0949 1.23458 1.27702
1.5 2.7941 2.32391 2.1994 1.14304 1.3273 1.38548
Table 7.4 Skin friction at isothermal plate when Gr=6, N=1 and t=0.4
↓ →
0.2 0.5 0.8 0.2 0.5 0.8
0.5 3.13249 2.71011 2.59802 0.831435 0.918309 0.943612 1 2.54491 2.07006 1.9422 1.19984 1.35704 1.40451
1.5 2.05595 1.50359 1.34866 1.25277 1.45076 1.51236
Table 7.5 Skin friction at ramped temperature plate when m=0.5, Gr=6 and K1=0.5
↓ →
0.2 0.4 0.6 0.2 0.4 0.6
1 3.63395 3.38016 3.11496 0.793177 0.840905 0.886539 3 3.61757 3.35039 3.07493 0.796584 0.849511 0.899521 5 3.60943 3.33604 3.05594 0.798411 0.853922 0.905993
m rGx z
m rGx z
m 1Kx z
m 1Kx z
N tx z
173
Table 7.6 Skin friction at isothermal plate when m=0.5, Gr=6 and K1=0.5
↓ →
0.2 0.4 0.6 0.2 0.4 0.6
1 2.8951 2.71011 2.63569 0.823002 0.918309 0.954028 3 2.73568 2.5966 2.54322 0.887562 0.966504 0.993723 5 2.66206 2.54562 2.50195 0.916847 0.988774 1.01172
Table 7.7 Nusselt number -Nu when Pr=0.71
↓ →
For ramped temperature plate For isothermal plate
0.2 0.4 0.6 0.2 0.4 0.6 1 0.300666 0.425206 0.520769 0.751665 0.531507 0.433974 3 0.212603 0.300666 0.368239 0.531507 0.375832 0.306866 5 0.17359 0.245493 0.300666 0.433974 0.306866 0.250555
Table 7.8 Nusselt number -Nu when N=1
↓ → For ramped temperature plate For isothermal plate
0.2 0.4 0.6 0.2 0.4 0.6 0.71 0.300666 0.425206 0.520769 0.751665 0.531507 0.433974 0.5 0.252313 0.356825 0.437019 0.630783 0.446031 0.364183 0.3 0.195441 0.276395 0.338514 0.488603 0.345494 0.282095
*****
N tx z
N t
rP t
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