Free convection problem have attracted a considerable amount of interest because of its importance in atmospheric and oceanic circulations, nuclear reactors, power transformers etc. several authors viz. Sturat (1954), Greenspan (1969), Jana & Dutta (1977), Sinha & Gupta (1980), Gupta et al. (1983), Purohit & Sharma (1986), Palec & Daguenet (1987), Singh (1994) have discussed rotating flows, Seth & Banerjee (1996) have studied combined free and forced convection flow of a viscous fluid in rotating channel in the presence of a uniform transverse magnetic field applied parallel to the axis of rotation. Gebhart (1973), Debnath (1973), Acheson and Hide (1973), Reynolds (1975 a, 1975b), Khare (1977), Srinivasan & Kandaswami (1984), Kumar & Mala (1992) Varshney and Johri (1993), Sharma (1995), Varshney and Varshney (1996) etc. have discussed flow in rotating system in presence magnetic field. Singh et al. (2001) have studied free convection in MHD flow of a rotating viscous liquid in porous medium past a vertical porous plate. Dhiman (2000) have studied a uniform rotation and uniform magnetic field in thermohaline convection. Recently, Kumar et al. (2001) have presented a study of the hydrodynamic lubrication of a micropolar fluid between two rotating rollers. More recently, Singh et al. (2002) have studied hydromagnetic oscillatory flow of a viscous fluid past a vertical plate in a rotating system. Johri (2003) was investigated approximate solution of the miscible fluid flow through porous media using collocation method. In the present paper we shall discuss unsteady flow, with heat and mass transfer, in an incompressible, electrically conducting, viscous fluid through a time dependent porous medium past an infinite porous vertical plate with constant suction/injection in the presence of an uniform magnetic field applied perpendicular to the flow region. It is considered that the plate is subjected to a constant suction/injection velocity normal to the plate and the flow is through a non-homogeneous porous medium. The effects of various parameters on primary velocity, secondary velocity, temperature field and concentration field have been discussed with the help of figures while the effects of important parameters on in skin-friction due to primary and secondary velocity, temperature field and concentration field have been discussed with the help of figures while the effects of important parameters on in skin-friction due to primary and secondary velocities, rate of heat and mass transfer have been discussed with the help of tables. There are two figures showing effects of the important parameters on primary and secondary velocities and six tables showing the effects of various parameters on skin-friction due to primary velocity, secondary velocity, rate of heat transfer and rate of mass transfer.

u, V, w – The velocitise along x, y and z axis.  – Uniform angular velocity

Q – Constant heat source  – Density of a linear function g – Accleration due to gravity 0 – Volumetric coefficient of thermal expansion  – Electric permeability µe – Magnetic permeability H0 – Constant Magnetic Field k0 – Constant permeability µ – Coefficient of viscosity κ – Thermal conductivity Cp – Specificant at constant pressure T – Temperature Tp – Plate temperature T – Temperature far away the plate Cw – Concentration of species at plate C – Concentration of species far away from plate Nu – Nusslet No. 0 – Heat source parameter E – Rotation parameter n – Frequency parameter

We consider an unsteady heat and mass transfer flow of an incompressible, electrically conducting, viscous liquid flowing through porous medium, which depends on time such that int01ektk past an infinite, vertical, porous plate with constant heat source in the presence of transverse uniform magnetic field. Further we consider a Cartesian coordinate system choosing x-axis and y-axis in the plane of the porous plate and and the plate are considered in a state of rigid body rotation about z-axis with uniform angular velocity . We also assume that the uniform magnetic field

0, where 0,0,0HH

is applied in the z-direction and the magnetic Reynolds number is small. The constant heat source Q is assumed at z = 0. We take the heat source of absorption type TTQQ0. The suction velocity at the plate is 0ww where 0w is a positive real number and negative sign indicates that the suction is towards the plate. In this analysis buoyancy force, hall effect, effect due to perturbation of the field, induced magnetic field and polarization effect are ignored. Initially at t < 0 the plate and the fluid are at the same temperature T and species concentration is uniformly distributed in the flow region such that it is everywhere C. When t > 0 the temperature of the plate is raised to int1eTw and the concentration level is raised to int1eCw. For formulation of mathematical equations the following assumption have been made : (i) The physical properties of the fluid are constant excluding density in the buoyancy force term in the momentum equation. (ii) The density is a linear function of temperature and species concentration given by TT001 TT so that Boussinesq's approximation is taken into account. Following, Gebhart & Pera (1971), the species concentration is very low so that the Soret and Dofour effects are negligible. (i) The induced magnetic field and the heat due to viscous dissipation are negligible. (ii) The plate is infinite in length so that the physical quantities involved in the governing equations depend on z and t only. (iii) The magnetic field is not strong enough to cause Joule heating so that the term due to electrical dissipation is neglected in energy equation. Under above stated restrictions the equations of motion and energy are :

  

u02 2

02

0 ……(1)

ve202int02

2012   



……(2)



  

02

2

0……(3)

2 2



 



……(4)

The boundary conditions relevant to the problem are : ,1,0,1intint0eTTveUuw



,,0,0)(

01int

 

zasCC, ……(5) We introduce the following non-dimensional quantities :

 

ww**020*0

0*20*0*20*0*

,

,,,,,





Using the above stated non-dimensional quantities, the equations (1), (2), (3) and (4) after ignoring the stars over them, reduce to :

uEvzutumr







int0 22 2

1

12……(6)

v





  

int0 22 2

1 12

……(7)

T02

21  



……(8)

2

21

c

 



……(9)

where KCPpr (Prandtl number), DSC (Schmidt number)

200

0

TTgGwr

(Grashof number) 00wU (modified Grashof number), 20wE (Rotation parameter),

20

2022

HMe



(Magnetic parameter)

20

Q (Heat source parameter) Using q = u +iv in (6) and (7), we obtain

CGTGzuqiEekMzqtqmr



22int02211

….. (10)

The equation (8) can be written in the following form :

002

2  

Trr ….. (11)

02

2  



CCC ….. (12)

The boundary conditions (5) are transformed to : 0,1,1,1int2int1intzateLCeLTeq zasCTq,0,0,0 ….. (13)

w21

In order to solve the equations (10), (11) and (12), we assume the velocity, temperature and concentration in the neighbourhood of the plate as follows: int10,ezqqtzq ….. (14) int10,ezTTtzT ….. (15) and int10,ezCCtzC ….. (16) Using equation (14), (15) and (16) in equations (10), (11) and (12), we obtain following equations : zCGzTGzqiEMzqzqmr0001002….. (17)

zqkzCGm0011

….. (18)

00000zTzTPzTr ….. (19) 01011zTinPzTPzTrr ….. (20) 000zCSzCC ….. (21) 0111zCinSzCSzCCC ….. (22) Using (14), (15) and (16) in (13) the boundary conditions are reduced to : 0,1,,1,1,121011010zatLCCLTTqq zatCTTqq,0,0,0,0,001010….. (23) The solution of equations (17) to (22), under the boundary conditions (23) are : zHezT20 ….. (24) zHeLzT411 ….. (25) zSCezC0 ….. (26) zReLzC421 ….. (27) zHzSzHzHzHeeReeDezqC6626510 ….. (28) and zHzHzHeDeDeDzq6424321 zHzSzReReReRC64876 zHeRRRDDD88764321 ….. (29) Substituting the values of q0 (z) and q1 (z) in (14), the values of T0 (z) and T1 (z) in the equation (15) and the values of C0 (z) and C1 (z) in the equation (16) we obtain. zHzHzSzHzHzHeeeReeDetzqC6662651, zHzSzReReReRC64876

….. (30)

int142,eeLetzTzHzH ….. (31) int24,eeLetzCzRzSC ….. (32) From (30), the steady part of the primary velocity (u0) and the steady part of the secondary velocity (v0) are : zBQzBPeePeFzuzAzSzHC2222250sincos22 zBFzBFzBezA26252sincoscos2….. (33) and zBPzBQeeQeFzvzAzSzHC2222260sincos22 zBFzBFzBezA25262sincossin2….. (34) From (30), the unsteady part i.e. time dependent part of the primary velocity (u1) and time dependent part of the secondary velocity (v1) are :





  

  

41313 7314313 212210110191

sincos sincos sincossincos

1 21 11

zAezBQzBP22525sincos ….. (35)





  

  

41313 8313314 211212191101

sincos sincos sincossincos

1 21 21

zAezBPzBQ22525sincos ….. (36) Therefore substituting these values of u0 (z), v0 (z), u1 (z) and v1 (z) the primary velocity u (z, t) and secondary velocity v (z, t) can be written as ntvntuzutzusincos,110 ….. (37) ntuntvzvtzvsincos,110 ….. (38) Hence, from (31) and (32), the primary and secondary velocities at 2nt are :

zvzunzu102, ….. (39)

zuzvnzv102,





….. (40)

 



,421 ,424121 ,84121 ,4421 ,1,24

2114 12338 12226 02114 021022

    

 











,2

,2 ,2 ,2 ,2 ,2 ,2

3

11093

02

1872 1651

40

5558 120 5447 1424 2336 12225















      





 

 

,,412164122

1 2 1

,41644122

1

,41644122

1 2 1

,416422

1

,416422

1 2 ,2 1

2/112213 2/112212 2/112212 2/102222021 2/102222021

40

112114

    





 



,1622

1 2

,412164122

1

2/12221 2/112213

 

221ccc



22220 7522220

65

2210 542210

44

2322 3232322 223 2212221

12

2,2 ,2,2

,, ,4

2,4











 









,42,4

,, ,,

221622115 16262241424113 12220212221

rr

 

,22,2222022506282202260527nEkFnEFkFFFnEkFnEFkFFF ,222212111102121119bnEabnELGFbnEaaLGFrr



,2

2121220

bnEFFaF





,2

2121220

bnEFFaF



, ,1

5431210814 543119713

 



,2,2 ,2,22

,,

,2,2

,,

221262125 21241113 1121212121 22221111 12222221121211

    

and ,222227bnEPaQd

The non-dimensional skin-friction at the plate is :

q

  



 



 





int 0 1 0 0int

0 …..(41)

Hence, primary skin-friction (p) due to primary velocity is :

Also, secondary skin-friction s due to secondary velocity is : ntFntFFssincos171816 …..(43) where

 

, , , ,

122112101918214313318 122112111917214213317 2222262526216 2222252625215

   

The rate of heat transfer at the plate in terms of Nusselt number (Nu) is :

int 0 1 0 0int

TNzzzu

  

 

 

 

 

 

…..(44)

Hence, considering that real part only is of significance, the rate of heat transfer is : int11112sincosentLBntLAHNu …..(45) The rate of mass transfer at the plate in terms of Sherwood number (Sh) is

int 0 1 0 0int

CSzzzh

  

 

 

 

 

 

…..(46)

Hence, considering that real part only is of significance, the rate of mass transfer is : int2121sincosentLQntLPSSch …..(47)

at n = 5.0, t = 1.0 and  = 0.002)

(n = 5.0, t = 1.0 and  = 0.002)

Rate of heat transfer in terms of Nusselt Number

( = 0.005)

Rate of mass transfer in terms of Sherwood Number

( - 0.005)

We have observed the effects of Prandtl number parameter (Pr), Schmidt number (SC), constant permeability parameter (k0), heat source parameter (0) magnetic parameter (M), grashof number (Gr), modified Grash of number (Gm) and rotation parameter (E) on primary and secondary velocities. These effects are shown in figures. The effects of important parameter on rate of heat transfer, rate of mass transfer and skin-friction due to primary and secondary velocities have also been observed. These effects are computed in table. Figure -1 shows effects of magnetic parameter (M), Grahsof number (Gr), modified Grashof number (Om) and rotation parameter (E), on primary velocity (u) at Pr=0.71, Sc=0.66, k0=20.0, o=1.0, n=5.0, t=1.0, and  = 0.005. It is observed that primary velocity (u) increases as z increases and after attaining a maximum value near the plate, it decreases rapidly as z increases. It is also noted that an increase in M, Gr, or

Figure-2 shows effects of Prandtl number (Pr), Schmidt number (Se), permeability parameter (ko) and heat source parameter o on secondary velocity (v) at M = 1.0, Gr=5.0, Gm=8.0,  =1.0, n=5.0, t=1.0, and  = 0.005. It is observed that secondary velocity (v) decreases as z increases and after attaining a maximum value near the plate, it increases rapidly as z increases. It is also noted that an decrease in Pa or ko results in an decrease and an increase in secondary velocity respectively while an increase in o, k0 or Sc result in an increase in secondary velocity.

and rotation parameter (E), at n=5.0, t=1.0, and  = 0.005, on skin friction (p) due to primary velocity and skin-friction (s) due to secondary velocity. In the computation of numerical values for skin-friction due to primary velocity and secondary velocity, in the computation of numerical values for skin-friction due to primary velocity and secondary velocity. We have taken two important cases namely cooling case and heating case. These cases are of immense importance in astrophysical problems and industrial technology, where heating and cooling of the plates have economic applications. Therefore the cases of externally cooled plate (Gr>0) and externally heated plate (Gr<0) are studied taking numerical values of various parameter encountered in the equations of the skin-friction. The value of Prandtl number (Pr) is chosen as Pr=0.71 which corresponds to water, which correspond to air. The numerical values of the remaining parameters are choosen arbitrary. These effects are shown in tables (l to 4). The effects of Prandtl number (Pr), frequency parameter (n) and time parameter (t) on rate of heat transfer [expressed in terms of Nusselt number (Nu)] and the effects of Schmidt number (Sc), frequency parameter (n) and time parameter (t) on rate of mass transfer [expressed in terms of Sherwood number (Sh)] are numerically expressed in table-5 and table-6 respectively.

heat source parameter (o) Grahsof number (Gr), modified Grashof number (Gm) and rotation parameter (E), at n = 5.0, t =1.0, and  = 0.005 for cooling case. It is observed an increase in Pr, Sc, M, o or E decreases skin-friction due to primary velocity while an increase in ko, Gr or Gm increases skin-friction due to primary velocity in cooling case. number (Pr), Schmidt number (Sc), magnetic parameter (M), permeability parameter (k0), heat

= 5.0, t=1.0, and  = 0.005 for cooling case. It is observed increase in Pn So M or o increases skin-friction due to secondary velocity while an increase in k0, Gr, Gm or E decreases skin-friction due to secondary velocity in cooling case.

k0, o or Gm increases skin-friction due to primary velocity while an increase in Sc, M, Gr, or E decreases skin-friction due to primary velocity in heating case. For Gr, we are considering magnitude only due to heating case. Table-4 represents the skin-friction (s) due to secondary velocity to show the effects of Prandtl number (Pr), Schmidt number (Sc), magnetic parameter (M), permeability parameter (k0), heat source parameter (o) Grahsof number (Gr), modified Grashof number (Gm) and rotation parameter (E), at n=5.0, t=1.0, and  = 0.005 for heating case. It is observed an increase in Sc, M or Gr increases skin-friction due to secondary velocity while an increase in Pr, ko, o Gm or E decreases skin-friction due to secondary velocity in heating case. For Gr, we are considering magnitude only due to heating case. The effects of Pr, n and t on the rate of heat transfer, expressed in terms of Nusselt number at  = 0.005 are numerically represented in table-5. It is observed that a decrease in Pr increases the rate of heat transfer and vice-versa. It is also observed that the effects of increase in n or t are opposite to each others. Table-6 shows the effects of SC, n and t on the rate of mass transfer, expressed in terms of Sherwood number at  = 0.005. It is observed that a decrease in Sc increases the rate of mass transfer. It is also observed that the effects of increase in n or t mass transfer are reciprocal to each other.