mhd natural convection flow with hall effects, radiation and ... - INDJSRT

Ind. J. Sci. Res. and Tech. 2015 3(5):10-22/Seth et al Online Available at: http://www.indjsrt.com Research Article

ISSN:-2321-9262 (Online)

MHD NATURAL CONVECTION FLOW WITH HALL EFFECTS, RADIATION AND HEAT ABSORPTION OVER AN EXPONENTIALLY ACCELERATED VERTICAL PLATE WITHRAMPED TEMPERATURE *

G. S. Seth, A. K. Singha and R. Sharma Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India *Author for Correspondence ABSTRACT An investigation of unsteady MHD natural convection flow of an electrically conducting, viscous, incompressible, heat absorbing and radiating fluid past an exponentially accelerated vertical plate with ramped temperature through a porous medium taking Hall effects into account is carried out. Exact solution for fluid velocity and fluid temperature is obtained in closed form by Laplace transform technique. The expressions for skin friction and Nusselt number are also obtained for both ramped temperature and isothermal plates. The numerical values of fluid velocity and fluid temperature are displayed graphically whereas numerical values of skin friction and Nusselt number are presented in tabular form for various values of pertinent flow parameters. The numerical results of natural convection flow near a ramped temperature plate are also compared with that of natural convection flow near an isothermal plate. Key Words: Magnetic Field, Hall Current, Natural Convection Flow, Ramped Temperature, Heat Absorption and Optically Thick Radiating Fluid INTRODUCTION Investigation of hydromagnetic natural convection flow in porous and non-porous media has drawn considerable attentions of several researchers owing to its applications in astrophysics, geophysics, aeronautics, electronics, metallurgy, chemical and petroleum industries. Magnetohydrodynamic (MHD) natural convection flow of an electrically conducting fluid in a fluid saturated porous medium has also been successfully exploited in crystal formation. Oreper and Szekely (1983) have found that the presence of a magnetic field can suppress natural convection currents and the strength of magnetic field is one of the important factors in reducing non-uniform composition thereby enhancing quality of the crystal. Keeping in view of such facts, Hossain and Mandal (1985), Jha (1991), Aldoss et al., (1995), Helmy (1998), Kim (2000) and Ibrahim et al., (2004) studied hydromagnetic natural convection flow past through a vertical plate considering different aspects of the problem. Effects of heat generation/absorption have significant contribution in the heat transfer characteristics of several physical problems of practical interest viz. convection in Earth’s mantle, post-accident heat removal, fire and combustion modelling, fluids undergoing exothermic and/or endothermic chemical reaction, development of metal waste from spent nuclear fuel, applications in the field of nuclear energy etc. Taking into consideration of this fact, Sparrow and Cess (1961), Moalem (1976), Chamkha (1995, 2004), Ramadan and Chamkha (2000), Kamel (2001), Rao et al., (2012) and Ravikumar et al. (2014) investigated fluid flow problem taking heat generation/absorption into account. It is widely known that thermal radiation effects on hydromagnetic natural convection flow with heat transfer plays a vital role in manufacturing industries for glass production, furnace design, casting and levitation, steel rolling etc. Moreover, several engineering processes occur at very high temperatures where the knowledge of radiative heat transfer becomes indispensable for the design of important equipments. Nuclear power plants, gas turbines, spacecraft re-entry aerothermodynamics and various propulsion devices foraircraft, missiles, satellites and space vehicles are examplesof such engineering areas. Keeping in view of its significance, Raptis and Massalas (1998), Chamkha(2000), Azam (2002), Cookey et al. ,(2003), Prasad et al., (2006), Chandrakala and Bhaskar (2009), Shankar et al., (2010), Suneetha and Reddy (2011), Prakash et al., (2013), Das and Jana (2015) and Daniel and Daniel (2015) analyzed hydromagnetic fluid flow problem with thermal radiation considering different aspects of the problem. It is noticed that when the density of an electrically conducting fluid is low and/or applied magnetic field is strong, Hall current plays a vital role in determining the flow-features of the fluid flow problems because it induces secondary flow in the flow-field (Sutton and Sherman, 1965). Taking into account of this fact, Takhar and Ram (1991) investigated effects of Hall current on hydromagnetic natural convection boundary layer flow of heat generating fluid past an infinite plate in a porous medium using harmonic analysis. Sharma et al., (2006) analyzed unsteady MHD free and forced convection flow past an infinite vertical porous flat plate in a porous medium in the presence of a heat source/sink with Hall effects.

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Convective flows with heat transfer are generally modelled by researchers under the consideration of uniform surface temperature or uniform heat flux. However, there exist several problems of practical interest which may require non-uniform or arbitrary thermal conditions at the surface of the body. Keeping in viewthis fact, several researchers investigated natural convection flow from a vertical plate with ramped temperature considering different aspects of the problem. Studies on this topic are due to Chandran et al., (2005), Rajesh (2010), Seth and Ansari (2010), Patra et al., (2012), Das (2012), Nandkeolyar and Das (2014), Nandkeolyar et al., (2013), Kundu et al., (2014), Das et al., (2014), Samiulhaq et al., (2014), Rajesh and Chamkha (2014) and Seth and Sarkar (2015). Aim of the present investigation is to study unsteady hydromagnetic natural convection flow with Hall effects of a viscous, incompressible, electrically conducting and temperature dependent heat absorbing and radiating fluid past an exponentially accelerated vertical plate through fluid saturated porous medium when temperature of the plate has a temporarily ramped profile. This problem has not yet received any attention from the researchers although natural convection flow with heat transferof a heat absorbing and radiating fluid resulting from ramped temperature profile of a plate moving with time dependent velocity may have strong bearings on numerous problems of practical interest where initial temperature profiles are of much significance in designing of so many hydromagnetic devices and in several industrial processes where heat absorption and radiation play a vital role on the fluid flow. MATHEMATICAL FORMULATION Consider unsteady hydromagnetic natural convection flow of an electrically conducting, viscous, incompressible, temperature dependent heat absorbing and optically thick heat radiating fluid past an infinite moving vertical plate embedded in a porous medium taking Hall effects into account. Coordinate system is chosen in such a way that x  axis is along length of the plate in the upward direction and y   axis is normal to the plane of the plate in the fluid. Fluid flow is permeated by an uniform transverse magnetic field B0 which is applied in the direction of y   axis. Initially i.e. at time t  0 , both the fluid and plate are at rest and at uniform temperature T . At time t   0 , plate is exponentially accelerated with velocity U 0 e

bt 

in x  direction. The temperature of the plate is raised or

lowered to T  (Tw  T )t  / t0 when 0  t   t0 and it is maintained at uniform temperature Tw when t   t0 . b and

U 0 are, respectively,an arbitrary constant and uniform velocity of the plate i.e. velocity for impulsive movement of the plate ( b  0 ).Since plate is of infinite extent along x  and z  directions and is electrically non-conducting, all physical quantities except pressure depend on y  and t  only. It is assumed that the induced magnetic field produced by fluid motion is negligible in comparison to applied one. This assumption is valid because magnetic Reynolds number is very small for metallic liquids and partially ionized fluids which are generally used in industrial processes.

Figure 1: Physical model of the problem

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Taking into consideration the assumption made above, the governing equations for unsteady hydromagnetic natural convection flow of an electrically conducting, viscous, incompressible, temperature dependent heat absorbing and optically thick heat radiating fluid in a uniform porous medium, under Boussinesq approximation, taking Hall effects into account are given by u   2u   B 2 (u   mw) u v 2  0  v  g  (T   T ), (1) 2   t k y  (1  m )

w  2 w   B02 (mu   w) w v 2   v , (2) t  k y   (1  m2 ) T   2T  Q 1 qr    2  0 (T   T )  , (3) t  cp  c p y  y 

where m  e e is Hall current parameter. u, w, T , K , c p , Q0 , qr , v,  ,  , g,  ,  , e and  e are, respectively, fluid velocity in x  direction, fluid

velocity along z   direction, fluid temperature, permeability of porous medium, specific heat at constant pressure, heat absorption coefficient ,radiating flux vector, kinematic coefficient of viscosity, electrical conductivity, fluid density, acceleration due to gravity, coefficient of thermal expansion, thermal diffusivity, cyclotron frequency and electron collision time. Initial and boundary conditions are u '  0, w '  0, T '  T for y  0 and t   0, (4a) u '  U0 ebt  , w '  0

(4c)

T '  Tw

u '  0, w '  0, T '  T

at y  0 for t   0, (4b) T '  T  (Tw  T )t  / t0

at y  0 for t   t0 , as y   for t   0. (4e)

at y  0 for 0  t   t0 ,

(4d)

For an optically thick heat radiating fluid, in addition to emission there is also self –absorption and usually the absorption coefficient is wavelength dependent and large so we can adopt the Rosseland approximation for radiative flux vector qr (Azam, 2002).Thus qr is given by

qr  

4 * T 4 , (5) 3k * y 

where k * is mean absorption coefficient and  * is Stefan-Boltzmann constant. It is assumed that the temperature difference within the fluid in the boundary layer region and free stream is sufficiently small so that T 4 may be expressed as a linear function of T  . This is accomplished by expanding T 4 in a Taylor series about free stream temperature T . Neglecting second and higher order terms, T 4 is expressed as T 4  4T 3T   3T 4. (6) Using equations (5) and (6), in equation (3), we obtain T   2T  Q 1 16 *T 3  2T     2  0 (T   T )  . (7) t  cp  c p 3k * y 2 y 

In order to present equations (1), (2), (7) and initial and boundary conditions (4a) to (4e) in non-dimensional form, the following non-dimensional quantities and pertinent flow parameters are introduced y  y/U 0 t0 , u  u  / U 0 , w  w / U 0 , t  t  / t0 ,   2 2     T  (T  T ) / (Tw  T ), M   B0 v / U 0 ,   (8) 2 2 3 K1  K U 0 / v , Gr  g  v(Tw  T ) / U 0 , Pr  v /  ,  *   vQ0 /  c pU 02 , N  16 *T 3 / 3kk and b  bt0 ,  where M , K1 , Gr , Pr ,  and N are, respectively, magnetic parameter, permeability parameter, Grashof number, Prandtl number, heat absorption parameter and radiation parameter. In view of (8), equations (1), (2), and (7), in non-dimensional form, reduce to

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u  2u M (u  mw) u     Gr T ,(9) t y 2 K1 (1  m2 ) w  2 w M (mu  w) w  2   , (10) t K1 y (1  m2 )

T  1  N   2T   T . (11)  t  Pr  y 2 It is to be noted that characteristic time t0 is defined according to the non-dimensional process mentioned above as t0  v / U 02 . Making use of (8), initial and boundary conditions (4a) to (4e) in non-dimensional form are given by at y  0 for t  0, (12b) u  0, w  0, T  0 for y  0 and t  0, (12a) u  ebt , w  0 T t at y  0 for 0  t  1, (12c) T  1 at y  0 for t  1, (12d) u  0, w  0, T  0 as y   for t  0. (12e) Combining equations (9) and (10), we obtain F  2 F MF F  2    Gr T , t (1  im) K1 y (13) where F  u  iw.

In compact form, initial and boundary conditions (12a) to (12e) are given by F  0, T  0 for y  0 and t  0, (14a) F  ebt T t at y  0 for 0  t  1, (14c) T  1 F  0, T  0 as y   for t  0. (14e)

at y  0 for t  0, (14b) at y  0 for t  1, (14d)

It is noticed from equations (11) and (13) that energy equation (11) is uncoupled from the momentum equation (13). Therefore, we have to obtain first the solution for fluid temperature T  y, t  by solving equation (11) and then using it in equation (13) the solution for fluid velocity F  y, t  is to be obtained. Using Laplace transform technique, exact solution for fluid temperature T  y, t  and fluid velocity F  y, t  is obtained and is presented in the following form T ( y, t )  P( y, t )  H (t  1) P( y, t 1), (15) F ( y, t ) 

ebt 2

 y e 

b 

 y erfc   2 t



 b    t   e y

b





 y erfc   2 t

b    t  

  2 G  y, t   H  t  1 G  y, t  1  ,  P( y, t ) 

1  y a y  t  e 2  2  

G ( y, t ) 

1  t  y e e 2  

y

a

(16)where

 y a   y a  y erfc   t    t  e 2 t   2   

 y  erfc   (    )t   e y 2 t 

y a  erfc   (   )t   e y 2 t 

(   ) a

y

 y a  erfc    t   , 2 t   

 y  erfc   (    )t   e y 2 t 

 y a  erfc   (   )t    2 t  

y   1 y  e y  erfc   t     t   2 t   2  

a

 1 y   t      2 

   

 y  y   1 y a  erfc    t   e  t   2 t    2    

13

(   ) a





y a   erfc    t   , 2 t   P G   a M 1  ,  where, a  r ,   r ,   . 1  im K1 1 N a 1 1 a a

ey

y a   1 y a  y erfc    t    t    e 2 t    2 

a

H (t  1) and erfc( x) are, respectively, unit step function and complementary error function. Solution when fluid is in contact to isothermal plate Analytical solution, presented by (15) and (16), is solution for fluid temperature and fluid velocity for hydromagnetic natural convection flow of an electrically conducting, viscous, incompressible, temperature dependent heat absorbing and optically thick heat radiating fluid past an exponentially accelerated moving vertical plate with ramped temperature taking Hall effects into account. In order to emphasize the effects of ramped temperature distribution within the plate on fluid flow, it is be justified to compare such a flow with the one past exponentially accelerated moving vertical plate with uniform temperature. Keeping in view the assumptions made earlier, the solution for fluid temperature and fluid velocity for the flow past exponentially accelerated moving isothermal vertical plate is obtained and is presented in the following form   1 y a   y a y a  T  y, t   e y a erfc   t  e erfc    t     , (17)  2 2 t  2 t    

F  y, t   

ebt  y e 2 

 t  y e e 2  

e y

a (   )



 e y



 

b 

 y  erfc   (b   )t   e y 2 t 

 y  erfc   (    )t   e y 2 t 

y a  erfc   (   )t   e y 2 t  

a (   )

 

b

 y  erfc   (b   )t   2 t 

 y  erfc   (    )t  2 t 

y a   erfc   (   )t   2 t   

 y   y  erfc    t   e y  erfc   t  2 t  2 t 

y a  y a   erfc    t  e y a erfc    t   . (18) 2 t  2 t   Skin friction and Nusselt number Expressions for the 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 ramped temperature and isothermal plates. (i) For ramped temperature plate: 1  (b   )t      ebt  (b   ) erfc (b   )t  1  e    2 G2 (0, t )  H (t  1)G2 (0, t  1)  , (19) t   Nu  P2 (0, t )  H (t  1) P2 (0, t  1), (20) e y

a

 

 

where

   x  i z ,

 

 G2 (0, t )  e t     erfc 

erfc 

 

(   )t  1 

 

(    )t  1 

1

t

e (   )t  a(   ) 

a  (  )t    1   1   e    t   .2    t 2        

14



2  1    t  1  a    t   e   t   .2 a    t        1  a t  erfc  t  1 2  t   e ,    t    1 a at t P2  0, t    t a  e .  erfc  t  1  2     (ii) For isothermal plate: ebt 1 t    (b   ) erfc (b   )t  1   e  2  t

erfc   t  1 

   

   

  (    ) erfc 



 t  e  



 a erfc  

 

 

 

(    )t  1  a(   ) erfc

  t  1   erfc   t  1    t  1 

Nu  a erfc

 

(   )t  1  

a t  e , t 

(21)

a t e . (22) t

Validation of result In the absence ofradiation ( N  0 ) and plate acceleration parameter ( b  0 ), we have compared our numerical results with the results of Nandkeolyar et al (2013). Our results are in excellent agreement with the results of Nandkeolyar et al (2013) which are shown in the Fig. 2. 1.0 Nandkeolyar et al.(2013) P resent Results

0.8

u

0.6

0.4 Ramped Temperature Isothermal

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

y

Figure 2 Temperature profiles when b  0, N  0, t  0.5 and Pr  0.71.

RESULTS AND DISCUSSION In order to highlight the influenceof plate acceleration parameter, Hall current, thermal buoyancy force, radiation, heat absorption and time on the flow-field, the numerical values of primary and secondary fluid velocities in the boundary layer region, computed from the analytical solutions (16) and (18), are displayed graphically versus boundary layer coordinate y for various values of plate acceleration parameter b , Hall current parameter m , Grashofnumber Gr , radiation parameter N , heat absorption parameter  and time t taking magnetic parameter M  15 , Prandtl number Pr  0.71 (ionized air) and permeability parameter K1  0.2 in Figs. 3 to 14. We have taken M  15 , i.e. strong magnetic field to take into account the effects of Hall current. It is revealed from Figs.3 to14 that, for ramped temperature and isothermal plates, secondary fluid velocity w attains a distinctive maximum value near the surface of the plate and then decreases properly on increasing boundary layer coordinate y to approach free stream value. Also the primary and secondary fluid velocities are faster in case of isothermal plate than that of ramped temperature plate.

15



Figure 3 Primary velocity profiles when m  0.5, Gr  10, N  2,   3and t  0.5.

Figure 4 Secondary velocity profiles when m  0.5, Gr  10, N  2,   3and t  0.5.

Figure 5 Primary velocity profiles when

Figure 6 Secondary velocity profiles when

b  0.4, Gr  10, N  2,   3and t  0.5.

b  0.4, Gr  10, N  2,   3and t  0.5.



m  0.5, b  0.4, N  2,   3 and t  0.5.

m  0.5, b  0.4, N  2,   3 and t  0.5.

Figs. 3 and 4 depict the effect of plate acceleration parameteron the primary and secondary fluid velocities u and w respectively for both ramped temperature and isothermal plates. It is observed from Figs. 3 and 4 that, for both ramped temperature and isothermal plates, u and w increase on increasing b. This means that plate acceleration parameter tends to accelerate primary and secondary fluid velocities for both ramped temperature and isothermal plates. Figs. 5 and 6 reveal the influence of Hall current on the primary and secondary fluid velocities for both ramped temperature and isothermal plates. It is evident from Figs. 5 and 6 that u and w increase on increasing m for both ramped temperature and isothermal plates. This implies that Hall current tends to accelerate primary and secondary fluid velocities for both ramped temperature and isothermal plates. Figs. 7 and 8 depict the effect of thermal buoyancy force on the primary and secondary fluid velocities for both ramped temperature and isothermal plates. It is observed from Figs. 7 and 8 that, for both ramped temperature and isothermal plates, u and w increase on increasing Gr . Physically Gr presents relative strength of thermal buoyancy force to viscous force, Gr increases

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on increasing thermal buoyancy force. This implies that thermal buoyancy force tends to accelerate primary and secondary fluid velocities for both ramped temperature and isothermal plates.


Figure10 Secondary velocity profiles when

m  0.5, Gr  10, b  0.4,   3 and t  0.5.

m  0.5, Gr  10, b  0.4,   3 and t  0.5.

Figs. 9 and 10 show the influence of radiation on the primary and secondary fluid velocities for both ramped temperature and isothermal plates. It is noticed from Figs. 8 and 9 that u and w increase on increasing N for both ramped temperature and isothermal plates. This means that, for both ramped temperature and isothermal plates, radiation has a tendency to accelerate primary and secondary fluid velocities.



m  0.5, Gr  10, b  0.4, N  2and t  0.5.

m  0.5, Gr  10, b  0.4, N  2 and t  0.5.

`



m  0.5, Gr  10, b  0.4, N  2and   3.

m  0.5, Gr  10, b  0.4, N  2 and   3.

Figs. 11 and 12 illustrate the effect of heat absorption on the primary and secondary fluid velocities for both ramped temperature and isothermal plates. It is noticed from Figs. 11and 12 that, for both ramped temperature and isothermal plates, u and w decrease on increasing  . This implies that heat absorption tends to retard primary and secondary fluid velocities for both ramped temperature and isothermal plates.Figs.13 and 14exhibit the effect of time on the primary and secondary fluid velocities for both ramped temperature and isothermal plates. It is evident from

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Figs. 13 and 14 that u and w increase on increasing t for both ramped temperature and isothermal plates. This means that primary and secondary fluid velocities are getting accelerated with the progress of time for both ramped temperature and isothermal plates. The numerical values for fluid temperature T , computed from analytical solutions (15) and (17), are displayed graphically versus boundary layer coordinate y in Figs. 15 to 17 for various values of heat absorption parameter  , time t and radiation parameter N , taking Prandtl number Pr  0.71 . It is evident from Figs. 15 to 17 that fluid temperature T decreases on increasing  whereas it increases on increasing either t or N for both ramped temperature and isothermal plates. This implies that, for both ramped temperature and isothermal plates, heat absorption has a tendency to reduce fluid temperature whereas thermal radiation has a reverse effect on it. As time progress, there is an enhancement in fluid temperature for both ramped temperature and isothermal plates. It is noticed from Figs. 15 to 17 that,for both ramped temperature and isothermal plates, fluid temperature is maximum at the surface of the plate and it decreases properly on increasing boundary layer coordinate y to approach free stream value. Also fluid temperature is higher in case of isothermal plate than that of ramped temperature plate. .

Figure 15 Temperature profiles when t  0.5 and N  2.

Figure 17 Temperature profiles when

Figure 16 Temperature profiles when

t  0.5 and   3.   3 and N  2. The numerical values of skin frictions  x and  z due to primary and secondary flows respectively for both ramped temperature and isothermal plates, computed from the analytical expressions (19) and (21), are presented in tabular form in Tables 1 to 6 for various values of b, m, N , t , Gr ,  and t , taking M  15 and Pr  0.71 whereas those of

Nusselt number N u for both ramped temperature and isothermal plates, computed from analytical expressions (20) and (22), are provided in Table 7 for different values of N ,  and t , taking Pr  0.71. Table 1: Skin friction at ramped temperature plate when Gr  10, N  2,   3 and t  0.5.

bm 0.4 0.8 1.2

 x 0.5 4.28435 5.49680 6.99203

z 1 3.60609 4.68891 6.02702

1.5 3.05015 4.03212 5.24842

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0.5 0.97087 1.15129 1.36957

1 1.39948 1.65163 1.95608

1.5 1.46921 1.72530 2.03384



Table 2: Skin friction at isothermal plate when Gr  10, N  2,   3 and t  0.5.

bm 0.4 0.8 1.2

 x 0.5 3.22937 4.44183 5.93706

z 1 2.46139 3.54421 4.88232

1.5 1.79907 2.78104 3.99734

0.5 1.13846 1.31888 1.53716

1 1.69143 1.94358 2.24803

1.5 1.83697 2.09305 2.40159

Table 3: Skin friction at ramped temperature plate when m  0.5, Gr  10,   3 and b  0.4. N t 

2 4 6

 x 0.3 4.27254 4.24241 4.21850

z 0.5 4.28435 4.24616 4.23101

0.7 4.32624 4.26935 4.23703

0.3 0.85054 0.85646 0.86003

0.5 0.97087 0.98151 0.98783

0.7 1.09792 1.11305 1.12197

Table 4: Skin friction at isothermal plate when m  0.5, Gr  10,   3 and b  0.4. N t 

2 4 6

 x 0.3 2.85797 2.77681 2.73113

z 0.5 3.22937 3.15369 3.11115

0.7 3.65291 3.57874 3.53703

0.3 1.05578 1.08039 1.09482

0.5 1.13846 1.16129 1.17454

0.7 1.21420 1.23636 1.24922

Table 5: Skin friction at ramped temperature plate when m  0.5, b  0.4, t  0.5 and N  2. Gr   

2 6 10

 x 1 4.98476 4.61838 4.25201

z 3 4.99122 4.63778 4.28435

5 4.99657 4.65383 4.31110

1 0.88844 0.93332 0.97819

3 0.88698 0.92892 0.97087

5 0.88578 0.92533 0.96488

Table 6: Skin friction at isothermal plate when m  0.5, b  0.4, t  0.5 and N  2. Gr   

2 6 10

 x 1 4.75921 3.94175 3.12428

Table 7: Nusselt number  Nu N  2 3 2 3 2 3 2 1 2 3 2 5 2 3 4 3 6 3

z 3 4.78023 4.00480 3.22937

t 0.3 0.5 0.7 0.5 0.5 0.5 0.5 0.5 0.5

5 4.79614 4.05254 3.30894

1 0.92617 1.04649 1.16682

Ramped Temperature 0.38368 0.55828 0.72887 0.44983 0.55828 0.65207 0.55828 0.43244 0.36548

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3 0.92050 1.02948 1.13846

Isothermal plate 0.89492 0.85907 0.84872 0.56755 0.85907 1.09210 0.85907 0.66543 0.56239

5 0.91625 1.01673 1.11721



It is evident from Tables 1 and 2 that, for both ramped temperature and isothermal plates, primary skin friction

x

and secondary skin friction  z increase on increasing b. Primary skin friction  x decreases whereas secondary skin friction  z increases on increasing m. It is noticed from Tables 3 to 6 that  x decreases on increasing either N or

Gr whereas it increases on increasing either t or  for both ramped temperature and isothermal plates,  z increases on increasing either N or t or Gr whereas it decreases on increasing  for both ramped temperature and isothermal plates. This implies that, for both ramped temperature and isothermal plates, plate acceleration parameter tends toenhance both the primary and secondary skin frictions. Hall current, radiation and thermal buoyancy force tend to reduce primary skin friction whereas thesehave reverse effect on the secondary skin friction. Heat absorption tends to enhance primary skin friction whereas it has a reverse effect on the secondary skin friction. Primary and secondary skin frictions are getting enhancedwith the progress of time. It is evident from Table 7 that Nusselt number N u increases on increasing  whereas it decreases on increasing N for both ramped temperature and isothermal plates. On increasing time t , Nusselt number N u increases for ramped temperature plate whereas it decreases for isothermal plate. This implies that, for both ramped temperature and isothermal plates, heat absorption tends to enhance rate of heat transfer at the plate whereas radiation has a reverse effect on it. As time progress, rate of heat transfer at the plate is getting enhanced for ramped temperature plate whereas it is getting reduced for isothermal plate. CONCLUSION Present investigation deals with unsteady MHD natural convection flow taking Hall current into account of a heat absorbing and radiating fluid past an exponentially accelerated vertical plate with ramped temperature. The significant results are summarized below: For both ramped temperature and isothermal plates:  Hall current, thermal buoyancy force and radiation tend to accelerate primary and secondary fluid velocities whereas heat absorption has a reverse effect on it. Plate acceleration tends to accelerate primary and secondary fluid velocities. Primary and secondary fluid velocities are getting accelerated with the progress of time. Heat absorption has a tendency to reduce fluid temperature whereas thermal radiation has a reverse effect on it. There is an enhancement in fluid temperature with the progress of time. Hall current, thermal buoyancy force and radiation tend to reduce primary skin friction whereas these agencies have reverse effect on the secondary skin friction. Plate acceleration tends to enhance primary and secondary skin frictions. Primary and secondary skin frictionsare getting enhanced with the progress of time.Heat absorption tends to enhance rate of heat transfer at the plate whereas thermal radiation has a reverse effect on it.  As time progress, rate of heat transfer at the plate is getting enhanced for ramped temperature plate whereas it is getting reduced for isothermal plate. REFERENCES Aldoss TK, Al-Nimr MA, Jarah MA & Al-Shaer BJ (1995). Magnetohydrodynamic mixed convection from a vertical plate embedded in a porous medium. Numerical Heat Transfer, 28 635-645. Azam GEA (2002). Radiation effects on the MHD mixed free forced convective flows past a semi-infinite moving vertical plate for high temperature differences. Physica Scripta, 66 71-76. Chamkha AJ (1995). Hydromagnetic flow and heat transfer of a heat-generating fluid over a surface embedded in a porous medium. International Communications in Heat and Mass Transfer, 24 815-825. Chamkha AJ (2000). Thermal radiation and buoyancy effects on hydromagnetic flow over an accelerating permeable surface with heat source or sink. International Journal of Engineering Science, 38 1699-1712. Chamkha AJ (2004). Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption. International Journal of Engineering Science, 42 217-230. Chandrakala P & Bhaskar PN (2009). Thermal radiation effects on MHD flow past a vertical oscillating plate. International Journal of Applied Mechanics and Engineering, 14 379-358. Chandran P, Sacheti NC & Singh AK (2005). Natural convection near a vertical plate with ramped wall temperature. Heat Mass Transfer, 41 459-464. Cookey CI, Ogulu A & Omubo-Pepple VB (2003). Influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction. International Journal of Heat and Mass Transfer, 46 2305-2311.

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