mhd natural convection flow past an impulsively moving...

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

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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

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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.

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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

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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.


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.


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


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



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



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



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



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


y

T


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


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

mhd natural convection flow past an impulsively moving...

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