Cmjv03i04p0426

J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

MHD Natural Convection between Vertical Parallel Plates with Oscillatory Wall Temperature S. DAS1, B. C. SARKAR2 and R. N. JANA3 1

Department of Mathematics, University of Gour Banga, Malda 732103, INDIA. 2,3 Department of Applied Mathematics, Vidyasagar University, Midnapore 721102, INDIA. (Received on: February 9, 2012) ABSTRACT The unsteady MHD natural convection flow of a viscous dissipative incompressible fluid with Joule dissipation between two infinite vertical parallel plates is studied. The temperature of one of the plates is oscillating about a constant non-zero mean temperature as the other plate maintains uniform temperature. The governing non-linear partial differential equations are solved analytically. The velocity and temperature distributions are presented graphically whereas the shear stress, the rate of heat transfer and their amplitudes and tangent of phase are given in tabular form. It is observed that both the velocity and temperature of the fluid decrease with an increase in either magnetic parameter or frequency parameter or phase angle. Further, the absolute value of the shear stress decreases while the rate of heat transfer increases with an increase in magnetic parameter for the fixed values of frequency parameter.

Keywords: MHD, Natural Convection, Prandtl Number, Eckert Number, Frequency Parameter and Rate of Heat Transfer.

1. INTRODUCTION MHD natural convection flows between two long vertical plates have attracted substantial interest in the context of metallurgical fluid dynamics, nuclear reactors, heat exchangers, re-entry aerother-

modynamics, astronautics, geophysics, cooling appliances in electronic instruments and applied mathematics. Investigation of MHD natural convection transport processes due to the coupling of the fluid flow and heat transfer is a challenging as well as interesting phenomenon. It has been

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

extensively studied between vertical plates because of its importance in many engineering applications. Ostrach1 first presented the steady laminar free-convective flow of a viscous incompressible fluid between two vertical walls. Ostrach2 studied the combined effects of a steady free and forced convective laminar flow and heat transfer between two vertical walls. Singer3 assessed the unsteady free convection heat transfer with magnetohydrodynamic effects in a channel regime. Soundalgekar and Bhat4 investigated the oscillatory MHD channel flow and heat transfer. Sacheti et al.5 discussed an exact solution for unsteady magnetohydrodynamic free convection flow with constant heat flux. The transient free convection flow between two vertical parallel plates was investigated by Singh et al.6 An exact solutions for transient free convection MHD Couette channel flow with impulsive motion of one of the plates was obtained by Jha7. Chamkha8 studied the unsteady laminar hydromagnetic flow and heat transfer in porous channels with temperature-dependent properties and inertial effects on the convection regime. Jha et al.9 analyzed the transient free convection flow in a vertical channel due to symmetric heating. Singh and Paul10 described the transient natural convection between two vertical walls heated/cooled asymmetrically.

Compo et al.11 considered natural convection for heated iso-flux boundaries of the channel containing a low Prandtl number fluid. The unsteady free convection oscillatory Couette flow through a porous medium with periodic wall temperature was presented by Sharma et al.12. Narahari13 studied the oscillatory plate temperature effects of free convection

flow of dissipative fluid between long vertical parallel plates. The effect of radiation on unsteady convective heat transfer of a viscous fluid through a porous medium in a vertical channel with oscillatory wall temperatures was investigated by Suseelavathy et al.14. More recently, Ahmed et al.15 studied the effect of magnetic field on free convective oscillatory flow between two vertical parallel plates with periodic plate temperature and dissipative heat.

In this paper, we study the unsteady MHD natural convection flow of a viscous dissipative incompressible fluid with Joule dissipation confined between two infinite vertical parallel plates when the temperature of one of the plates oscillates and the other one maintains uniform temperature. It is observed that the fluid velocity u decreases with an increase in either magnetic parameter M 2 or frequency parameter ω or phase angle ωτ . A decrease in fluid temperature θ occurs due to an increase in either frequency parameter or phase angle.

Further, the magnitude of the shear stress at the plate η = 1 increases with an increase in Eckert number Ec while decreases with an increase in either magnetic parameter M 2 or phase angle ωτ for a fixed value of frequency parameter ω . It is also seen that the rate of heat transfer −θ ' (1) increases with an increase in magnetic parameter M 2 whereas it decreases with an increase in either phase angle ωτ or Eckert number Ec for the fixed values of frequency parameter ω . Further, it is seen that for fixed values of phase angle ωτ , Eckert number Ec and magnetic parameter M 2 , the rate of heat

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428

transfer −θ ' (1) increases with an increase in frequency parameter ω .

Then under the usual Boussinesq's approximation, the fluid flow be governed by the following system of equations:

2. FORMULATION OF THE PROBLEM AND ITS SOLUTIONS

∂u∗ ∂ 2u∗ σ B02 ∗ = g β (T − T1 ) +ν − u , ∂t ∂y 2 ρ

Consider the unsteady MHD natural convective flow of a viscous incompressible fluid between two infinite vertical parallel plates separated by a distance h in the presence of a transverse magnetic field. The x - axis is taken along one of the plates in the vertically upward direction and the y axis is taken normal to the plates [See Fig.1].

The temperature of the plate at y = 0 is T = T1 and the temperature of the plate at y = h is T = T2 + ε (T2 − T1 ) cos ω∗t , ω ∗ being the frequency of the plate temperature oscillations. A uniform magnetic field of strength B0 is imposed perpendicular to the plates. The transverse applied magnetic field and magnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible. Also, it is assumed that there is no applied voltage, so that the electric field is absent.

(1)

2

 ∂u ∗  ∂T ∂ 2T 2 ∗2 ρ cp = k 2 + µ  + σ B0 u , ∂t ∂y  ∂y 

(2)

where u ∗ is the velocity of the fluid in the x -direction, g the acceleration due to gravity, β the coefficient of volume expansion, µ the coefficient of viscosity, ν the kinematic coefficient of viscosity, ρ the fluid density, k the thermal conductivity and cp the specific heat at constant pressure.

The velocity and temperature boundary conditions are u∗ = 0, T = T1 at y = 0, u∗ = 0, T = T2 + ε (T2 − T1 ) cos ω∗t at y = h, (3)

where ε is a small real constant quantity << 1 .

We introduce the non-dimensional variables y h

η= , τ=

νt h

2

, u=

T − T1 u ∗ν , θ= . h g β (T2 − T1 ) T2 − T1 2

(4) On the use of equation (4), equations (1) and (2) become

Fig.1. Geometry of the problem.

∂u ∂ 2u = θ + 2 − M 2u, ∂τ ∂η

(5)

  ∂u  2  ∂θ ∂ 2θ 2 2 Pr = + PrEc   + M u , 2 ∂τ ∂η  ∂η  

(6)

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

M2 =

where

σ B02 h2 ρν ρν c p

parameter, Pr =

k

is

the

magnetic

The corresponding boundary conditions for u and θ are u = 0 , θ = 0 at η = 0, u = 0, θ = 1 + ε cos ωτ at η = 1,

ω ∗h2 ν

(7)

is the non-dimensional

frequency of plate temperature oscillation.

Equations (5) and (6) are coupled nonlinear equations. we assume u (η ) = u0 (η ) + ε eiωτ u1 (η )

θ (η ) = θ 0 (η ) + ε e θ1 (η ) iωτ

u0 = 0, u1 = 0, θ 0 = 0, θ1 = 0 at η = 0,

(13)

u0 = 0, u1 = 0, θ 0 = 1, θ1 = 1 at η = 1,

(14)

the Prandtl number

h 4 g 2 β 2 (T2 − T1 ) and Ec = the Eckert number. c pν 2

where ω =

with the following boundary conditions

 , 

(8)

where u0 the mean velocity, u1 the fluctuating part of the fluid velocity, θ0 the mean temperature and θ1 the fluctuating part of the fluid temperature.

Substituting equation (8) in the equations (5) to (7), equating the harmonic and non-harmonic terms and neglecting terms containing ε 2 and higher powers of ε , we have

where primes denote differentiation with respect to η .

For incompressible fluids the Eckert number Ec is always << 1 . Using multiparameter perturbation technique, we make the following substitutions using Ec as the perturbation parameter u0 = u01 + Ecu02 , θ 0 = θ 01 + Ecθ 02 , u1 = u11 + Ecu12 , θ1 = θ11 + Ecθ12 .

     

(15)

On the use of (15) in equations (9) to (14), equating the coefficients of different powers of Eckert number Ec , neglecting coefficients of Ec 2 and higher power of Ec , we get ′ − M 2 u 01 = −θ 01 , u 01

′ − ( M 2 + iω ) u11 = − θ11 , u11

′ − M u 02 = −θ 02 , u 02

′ − ( M + iω ) u12 = − θ12 , u12

2

′ = 0, θ 01

2

θ11′ = i ω Pr θ11 ,

′′ = − Pr ( u 0′ 2 + M 2 u 012 ), θ 02 ′ u11 ′ + M 2 u 01 u11 ) θ12′′ − iω Pr θ12 = −2 Pr ( u 01

(16)

u`′′− (M + i ω)u1 = −θ1 ,

(10)

with the following boundary conditions: η = 0, u01 = 0, u02 = 0; θ01 = 0,θ02 = 0; u11 = 0, u12 = 0; θ11 = 0,θ12 = 0;

θ0′′ = −Pr Ec (u0′2 + M 2 u02 ),

(11)

η = 1,

θ1′′− i ω Pr θ1 = −2PrEc(u0′ u1′ + M 2 u0u1 )

(12)

u0′′ − M 2u0 = −θ0 , 2

(9)

       

u01 = 0, u02 = 0; u11 = 0, u12 = 0;

θ01 = 1,θ02 = 0; θ11 = 1,θ12 = 0;

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

      

(17)

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

The solution of the equation (16) subject to the boundary conditions (17) are

 1 η2 1 + A6  −M2 + 2  2a 2 4 a1  1 

θ 01 = η ,

  M 2η  −  a1η −  sinh a1η  2a1   

u01 =

(18)

1 M2

sinh M η   η − , sinh M  

(19)  2 M 2 a2 + A7  a2 − A5 

θ 02 = A1 (η − 1) + A2η

1 4

4M sinh 2M

cosh 2Mη +

  Ma − M 2  + 2 1  cosh(a1 − M )η   M − 2Ma1  

2  cosh Mη  , M sinh M  5

(20) A1 Aη (η − 1) + 2 2 M2 M A sinh( M (η − 1)) − A4 sinh Mη + 3 sinh M η+2  + Pr  sinh M η 6  2 M sinh M 1 3   2 − η +  cosh M η 2 M 5 sinh M  2M 2  1 + cosh 2 M η 12 M 6 sinh 2M

 Ma + M 2  + A9  2 2  cosh(a2 + M )η  M + 2Ma2 + A5 

u02 =

−

1 M4

η4 3 η2 3  + 4  ,  + 2 M   12 2 M

sinh a1η θ11 = , sinh a1 u11 =

1  sinh a1η sinh a2η  −  , A5  sinh a1 sinh a2 

θ12 =

A10 sinh(a1 (η − 1)) − A11 sinh a1η sinh a1

  2  cosh a2η + M η sinh a2η   

 Ma + M 2  + A8  2 1  cosh(a1 + M )η  M + 2Ma1 

 η2 η4 2η sinh M η − Pr  + − 4 4 2 2 12 sinh M M M M 

+

   cosh a1η 

  Ma − M 2  + 2 2  cosh(a2 − M )η  (24)  M − 2 Ma2 + A5   u12 =

A10 sinh(a1 (η − 1)) − A11 sinh a1η A5 sinh a1 +

(21)

+

A6 A5

A12 sinh(a2 (η − 1)) − A13 sinh a2η sinh a2

 1 2a12  1 4a2 η 2   − − M 2  2 + 21 +   cosh a1η  2    4a1 A5  2a1 A5

(22)

    1 2a1  +η  M 2  −  − a1  sinh a1η    2a1 A5    

(23)

 1  1 η2 1  + A7  −M2 + + 3   cosh a2η  4a   2  2a2 A5 4a2 8a2      1 1  1 + η  M 2  2 +  −  sinh a2η      4a2 A5  2  

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

 Ma1 + M 2 − A8  2 cosh(a1 + M )η 2  (M + 2Ma1 )(M + 2Ma1 − A5 )

+

 Ma1 − M 2 cosh( a1 − M )η  2 2 ( M − 2 Ma1 )( M − 2 Ma1 − A5 ) 

 Ma2 + M 2 − A9  2 cosh(a2 + M )η 2  ( M + 2Ma2 )(M + 2Ma2 + A5 )

+

 Ma2 − M 2 cosh(a2 − M )η  , ( M − 2Ma2 )( M 2 − 2Ma2 + A5 )  2

A9 =

− Pr , A5 M sinh M sinh a2 2

  1 M2   2a M 2  A10 =  A6  − 2  + A7  a2 − 2  A5    2a1 4a1    Ma + M 2 Ma − M 2  + A8  2 1 + 21   M + 2 Ma1 M − 2 Ma1 

 Ma + M 2 Ma − M 2   + A9  2 2 + 2 2   , (26)  M + 2Ma2 + A5 M − 2Ma2 + A5  

(25)  1 1 1  A11 = A6  − M 2  + 2   cosh a1  2a  1  2 4a1  

where a1 = i ω Pr , a2 = M 2 + iω ,

1 1   A1 = −2 Pr  5 + 4 , 2  M sinh M 8M sinh M 

  M2  −  a1 −  sinh a1  2a1   

 1 1 3  A2 = Pr  − 2 2   2M  6 M  1 2  + cosh 2M + 5 cosh M  , 4 2 4M sinh M M sinh M 

 2M 2 a2 + A7  a2 − A5 

 Ma + M 2  + A8  2 1  cosh(a1 + M )  M + 2Ma1 

1 3 3   A1 A3 = Pr  − 8− 6 7 − M2 , 2 12 M M M 4 M sinh M sinh  

A4 =

A2  1 1 3  − Pr  4  + 4  2 M  M  12 M 

 1 1  1  + + 1 + 5 2  7  2M sinh M  2M  2M sinh M 1  − cosh 2M  , 6 12M sinh 2M 

A5 = [M 2 + iω (1 − Pr )], A6 =

Pr , A5 a1 M 2 sinh a1

  cosh M 

  Ma − M 2  + 2 1  cosh( a1 − M )    M − 2Ma1   Ma + M 2  + A9  2 2  cosh(a2 + M )  M + 2Ma2 + A5    Ma − M 2  + 2 2  cosh(a2 − M )  ,  M − 2Ma2 + A5   A12 =

Pr , A5 M 2 sinh M sinh a1

A6 A5

 1 2a12 4a12   2 1 − − M +   2  A52    4a1  2a1 A5

 1  1 1  + A7  −M2 + 3   4a2  2a2 A5 8a2    Ma1 + M 2 − A8  2 2  ( M + 2 Ma1 )( M + 2 Ma1 − A5 )

2 Pr A7 = 2 2 , A5 M sinh a2 A8 =

  2  cosh a2 + M sinh a2   

+

 Ma1 − M 2  ( M − 2 Ma1 )( M 2 − 2 Ma1 − A5 )  2

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

432

S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)  Ma2 + M 2 − A9  2 2  ( M + 2 Ma2 )( M + 2 Ma2 + A5 )

+

 A10 Ma2 − M 2 , − 2 2 ( M − 2 Ma2 )( M − 2 Ma2 + A5 )  A5

A6 A5

A13 =

 1 2a12  1 4a 2 1   − − M 2  2 + 21 +   cosh a1  A5 2    4a1  2a1 A5

    1 2a1  + M 2  −  − a1  sinh a1    2a1 A5      1  1 1 1  + A7  −M2 + + 3   cosh a2  4a   2  2a2 A5 4a2 8a2      1 1  1 +  M 2  2 +  −  sinh a2      4a2 A5  2   2  Ma1 + M − A8  2 cosh(a1 + M ) 2  ( M + 2Ma1 )( M + 2Ma1 − A5 ) +

 Ma1 − M 2 cosh( a1 − M )  2 2 ( M − 2 Ma1 )( M − 2 Ma1 − A5 ) 

 Ma2 + M 2 − A9  2 cosh(a2 + M ) 2  ( M + 2Ma2 )( M + 2Ma2 + A5 ) +

field normal to the flow in an electrically conducting fluid introduces a Lorentz force which acts against the flow. This resistive force tends to slow down the flow and hence the fluid velocity decreases with the increase of the magnetic field parameter as observed in Fig.2. This trend is consistent with many classical studies on magneto-convection flow. Fig.3 shows that an increase in frequency parameter ω leads to fall in the values of fluid velocity u which means that the momentum diffusivity of the fluid decreases as compared to its thermal diffusivity. In Figs.4 it is seen that the velocity u decreases with an increase in phase angle ωτ . Fig.5 and Fig.6 represent the temperature profiles θ against η for several values of frequency parameter ω and phase angle ωτ . It is observed from Figs. 5 and 6 that the fluid temperature θ decreases with an increase in either frequency parameter ω or phase angle ωτ .

 A Ma2 − M 2 cosh( a2 − M )  − 11 .

2 ( M − 2Ma2 )( M − 2Ma2 + A5 )  A5 2

3. RESULTS AND DISCUSSION We have plotted the nondimensional velocity and temperature profiles for several values of magnetic parameter M 2 , frequency parameter ω and phase angle ωτ when Eckert number Ec = 0.01 and ε = 0.2 in Figs.2-6. Figs.2-4 represent the fluid velocity u against η for several values of magnetic parameter M 2 , frequency parameter ω and phase angle ωτ .

It is seen from Fig.2 that the fluid velocity u decreases with an increase in magnetic parameter M 2 . The presence of a magnetic

2

Fig.2. Velocity profiles for M when

Pr = 0.3 , ω = 5 and ωτ =

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π

4

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

Fig.3. Velocity profiles for ω when

M 2 = 5 , Pr = 0.3 and ωτ =

π

4

Fig.4. Velocity profiles for ωτ when M 2 = 5 , Pr = 0.3 and ω = 5

Fig.5. Temperature profiles for ω when

M 2 = 5 , Pr = 0.3 and ωτ =

π

4

Fig.6. Temperature profiles for ωτ when M 2 = 5 , Pr = 0.3 and ω = 5

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

The non-dimensional shear stress τ u at the plate (η = 1) is given by

where a1 , a2 and A1 to A13 are given by (26).

Equation (27) can be expressed in terms of amplitude and phase of shear stress as

 1 A + A =  2 (1 − M coth M ) + Ec  1 2 2  M M + M ( A3 cosech M − A4 coth M )

 5 13  + 2  3 2M

        

 a coth a1 − a2 coth a2 +ε eiωτ  1 A5  + Ec {a2 ( A12 cosecha2 − A13 coth a2 ) +

a1 ( A10 cosecha1 − A11 coth a1 ) A5 +

A6 A5

 1 2a13    − a1 −  A5   2

a 1 2a1 4a13   −M 2  1 − + + 2   sinh a1 A5    2 4a1 A5     1 2a 2  −  M 2  + 1  + a12  cosh a1      2 A5     1  1 1 1 1 + A7   M 2  + 2 −  −  sinh a2  2  A5 4a2 2  2      a 1  a2  − M 2  2 −  −  cosh a2    A5 4a2  2     (a1 + M )( Ma1 + M 2 ) − A8  sinh(a1 + M ) 2 2  ( M + 2Ma1 )( M + 2Ma1 − A5 )

+

 (a2 − M )( Ma2 − M 2 )  sinh( a2 − M )   , 2 2 ( M − 2Ma2 )( M − 2Ma2 + A5 )   

(27)

∂u  ∂u  ∂u τu = =  0 + ε eiωτ 1  ∂η η =1  ∂η ∂η η =1

 5 1 + Pr  coth M − 5 2M 4  6M

+

 (a1 − M )( Ma1 − M 2 ) sinh(a1 − M )  2 2 ( M − 2Ma1 )( M − 2Ma1 − A5 ) 

 (a2 + M )( Ma2 + M 2 ) − A9  sinh(a2 + M ) 2 2  ( M + 2Ma2 )( M + 2 Ma2 + A5 )

 A + A2

 1

τ u =  2 (1 − M coth M ) + Ec  1 2  M M + M ( A3 cosechM − A4 coth M )

 5 1  5 13 + Pr  coth M − + 5 4  2 6 M 2 M  3 2M  +ε | R | cos(ωτ + α ),

        

(28) 1

where | R |= ( Rr2 + Ri2 ) 2 is the amplitude of the unsteady part of the shear stress and tangent of phase tan α = Ri /Rr , Rr and Ri are respectively the real and imaginary parts of  ∂u1    in (27).  ∂η η =1

Numerical results of the shear stress at the right plate (η = 1) against the frequency parameter ω are presented in Table 1 for various values of phase angle ωτ , magnetic parameter M 2 and Eckert number Ec . Table 1 shows that the absolute value of the shear stress increases with an increase in Eckert number Ec while it decreases with an increase in either magnetic parameter M 2 or phase angle ωτ for fixed value of frequency parameter ω . It is seen that for fixed value of phase angle ωτ , magnetic parameter M 2

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and Eckert number Ec magnitude of τ u increases with an increase in frequency parameter ω . It is observed from Table 2 that the absolute value of the tangent of phase tan α remains same with an increase in phase angle ωτ but it decreases with an increase in either magnetic parameter M 2 or

Eckert number Ec for fixed value of frequency parameter ω . Further, it is seen that for fixed values of phase angle ωτ , magnetic parameter M 2 and Eckert number Ec , the absolute value of the tangent of phase tan α increases with an increase in frequency parameter ω .

Table 1. Shear stress −τ u at the plate (η = 1) when Pr = 0.3

ωτ

0.01

Ec 0.03

0.05

0.29757 0.29881 0.29966 0.30014

0.29762 0.29886 0.29972 0.30019

0.29767 0.29892 0.29977 0.30025

M2

ω

π /4

π /3

π /2

5

10

15

2 3 4 5

0.29743 0.29868 0.29954 0.30003

0.28789 0.28978 0.29130 0.29247

0.26348 0.26622 0.26873 0.27097

0.29757 0.29881 0.29966 0.30014

0.25052 0.25146 0.25220 0.25274

0.22067 0.22141 0.22202 0.22252

Table 2. Tangent of phase − tan α of shear stress at the plate (η = 1) when Pr = 0.3

ωτ ω

π /4

π /3

2 3 4 5

0.11580 0.17317 0.22970 0.28514

0.11580 0.17317 0.22970 0.28514

π /2

5

10

15

0.01

Ec 0.03

0.05

0.11580 0.17317 0.22970 0.28514

0.11580 0.17317 0.22970 0.28514

0.08908 0.13332 0.17715 0.22042

0.07348 0.11002 0.14628 0.18219

0.11580 0.17317 0.22970 0.28514

0.11550 0.17302 0.22963 0.28511

0.11520 0.17287 0.22955 0.28507

M2

Now in the field of temperature, we can obtain the expression for the rate of heat transfer θ ′(1) at the plate (η = 1) as −θ ′(1) = −

∂θ  ∂θ  ∂θ =  0 + ε ei ωτ 1  ∂η η =1  ∂η ∂η η =1

  1  1 = 1 + Ec  A1 + A2 − Pr  4 + 2 2M sinh 2M M   1  1 + − 2 2 2 3 M 2 M M sinh 

  1   − M 3 coth M     

+ε eiωτ  a1 coth a1 + Ec {a1 ( A10 cosecha1 − A11 coth a1 )

   M 2  − A6  −  M 2 + a1  a1 −   cosh a1   2a1       1 1 1  M2 − a1  − M 2  + 2  − a1 +  2a1  2 4a1   2a1

   sinh a1   

  2M 2 a2 − A7   M 2 + a2  a2 −  A5  

+ M 2 a2 cosh a2 )

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   sinh a2 

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 (a + M )( Ma1 + M 2 ) sinh(a1 + M ) − A8  1 ( M 2 + 2Ma1 )  −

respectively the real and imaginary parts of  ∂θ1    in (29).  ∂η η =1

 (a1 − M )( Ma1 − M 2 ) sinh(a1 − M )  2 ( M − 2Ma1 ) 

 (a + M )( Ma2 + M 2 ) − A9  2 2 sinh(a2 + M )  ( M + 2Ma2 + A5 )   (a − M )( Ma2 − M 2 ) − 2 2 sinh(a2 − M )   , ( M − 2 Ma2 + A5 )  

(29)

where a1 , a2 and A1 to A11 are given by (26).

The rate of heat transfer θ ′(1) at the plate (η = 1) can be expressed in terms of amplitude and phase of heat transfer as   1  1 −θ ′(1) = 1 + Ec  A1 + A2 − Pr  4 + 2 2 M 2 M sinh M   

1  1 + − 2 2 2M sinh 2M  3M

  1   − M 3 coth M     

−ε | φ | cos(ωτ + β1 ),

where | φ |= (φr2 + φi2 )

1/2

(30)

is the amplitude of the

unsteady part of the heat transfer and tangent of phase tan β1 = φi /φr . φr and φi are

Table 3 shows that the rate of heat transfer −θ ′(1) at the plate (η = 1) decreases with an increase in either phase angle ωτ or Eckert number Ec for fixed values of frequency parameter ω . It is seen that for fixed values of phase angle ωτ and Eckert number Ec the absolute value of the rate of heat transfer increases with an increase in frequency parameter ω . It is observed from Table 4 that the tangent of phase tan β1 remains same with an increase in phase angle ωτ where as it increases with an increase in Eckert number Ec for fixed values of frequency parameter ω . It is also seen that for fixed values of phase angle and Eckert number, tan β1 increases with an increase in frequency parameter. Table 5 shows that the rate of heat transfer −θ ′(1) increases while the tangent of phase tan β1 decreases with an increase in magnetic parameter M 2 for fixed values of frequency parameter ω . Further, it is seen that the rate of heat transfer −θ ′(1) and the tangent of phase tan β1 increases both with an increase in frequency parameter for fixed values of magnetic parameter M 2 .

Table 3. Rate of heat transfer −θ ′(1) at the plate (η = 1) when Pr = 0.3 and M 2 = 5

ωτ

ω

π /6

π /4

2 3 4 5

1.17491 1.17663 1.17901 1.18202

1.14185 1.14325 1.14518 1.14762

Ec

π /3

π /2

1.10027 1.10126 1.10262 1.10435

1.00013 1.00013 1.00014 1.00014

0.01 1.14251 1.14391 1.14585 1.14831

0.03 1.14244 1.14384 1.14578 1.14824

0.05 1.14237 1.14377 1.14571 1.14817

0.07 1.14229 1.14369 1.14563 1.14810

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S. Das, et al., J. Comp. & Math. Sci. Vol.3 (4), 426-438 (2012)

Table 4. Tangent of phase tan β1 of heat transfer at the plate (η = 1) when Pr = 0.3 and M 2 = 5

ωτ

ω

π /6

2 3 4 5

π /4

0.09899 0.14663 0.19215 0.23503

0.09899 0.14663 0.19215 0.23503

Ec

π /3 0.09899 0.14663 0.19215 0.23503

π /2 0.09899 0.14663 0.19215 0.23503

0.01 0.09899 0.14663 0.19215 0.23503

Table 5. −θ ′(1) and tan β1 at the plate (η = 1) when ωτ =

ω\M 2 3 4 5

2

5

1.14251 1.14391 1.14585 1.14831

−θ ′(1) 10 15 1.14252 1.14253 1.14392 1.14392 1.14586 1.14586 1.14832 1.14832

20

5

1.14253 1.14393 1.14587 1.14832

0.09899 0.14663 0.19215 0.23503

0.03 0.09902 0.14666 0.19219 0.23507

π 4

0.05 0.09904 0.14669 0.19223 0.23512

0.07 0.09906 0.14673 0.19227 0.23516

, Pr = 0.3 and Ec = 0.01

tan β1 15 0.09899 0.09899 0.14663 0.14663 0.19215 0.19214 0.23502 0.23502 10

20

0.09899 0.14662 0.19214 0.23502

4. CONCLUSION

5. REFERENCES

The unsteady MHD natural convection flow of a viscous dissipative incompressible fluid with Joule dissipation between two infinite vertical parallel plates in which the temperature of one of the plates is oscillatory where as that of the other plate is uniform has been considered. It is observed that the velocity profiles exhibit a parabolic nature within the channel. From the above observations we see that the application of the magnetic field helps in controlling the flow. An increase in magnetic field force retards the main flow field. An increase in frequency parameter reduces the temperature field. A rise in Eckert number leads to a rise in shear stress and leads to a fall in heat flux at the upper plate surface.

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