J. Comp. & Math. Sci. Vol.2 (2), 304-314 (2011)
Unsteady Hydromagnetic Mixed Convection Flow in a Vertical Channel with Travelling Thermal Wave and Quadratic Temperature Variation S. JAFARUNNISA, V. RAGHAVENDRA PRASAD, U. RAJESWARA RAO and D.R.V. PRASADA RAO Department of Mathematics, S.K. University, Anantapur – 515 055 A.P., India. ABSTRACT In this paper we analyze Magneto Hydro Dynamic (MHD) unsteady convective heat transfer in a vertical channel on whose walls the Traveling thermal wave is imposed. A uniform magnetic field of strength H0 is applied transverse to the boundaries. A quadratic density distribution is taken in the equation of state. The equations governing the flow and heat transfer have been solved by employing a regular perturbation technique with δ, the aspect ratio as a perturbation parameter. The velocity and temperature are analysed for different values of the governing parameters G, M, γ, x + γt. Also the shear stress and the rate of heat transfer on the walls are numerically evaluated. Key words: Convective Heat transfer, Vertical channel, Travelling thermal wave, Magnetic field.
1. INTRODUCTION The Heat transfer in fluid flows has gained significance in recent times because of its applications in recent advancement of space technology. For example the problem of controlling the skin-friction and aerodynamic heat transfer around the high speed vehicles has vital important with the advent of rocketary and supersonic flights. The heat transfer Problems are generally solved for two types of configurations. 1. problems where the semi-infinite fluid is bounded by a non-porous (or porous) rigid
boundary with a free stream which moves either with a uniform or time dependent fluctuation velocity. 2. problems where in the fluid is confined between non-porous (or porous) rigid boundaries. It is intuitively evident that the temperature distribution around a hot boundary in a fluid stream gives rise to a thermal boundary layer across which the temperature gradient is large. Hence apart from the velocity distribution in the flow field the study of the thermal boundary layer and the influence of different forces (or kinematical factors) on this boundary layer is a major aspect of the heat
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transfer problems. When the buoyancy force is disregarded and the heat is transferred due to conduction alone the velocity field no longer depends on the temperature field, although the dependence of the temperature field on the velocity field still persists. This happens at large Reynolds number and small temperature differences. Such flows being termed an forced flows while the temperature dependent buoyancy caused flow is known as Free (or Natural) convection flows. Convection fluid flows generated by traveling thermal waves have also received attention due to applications on physical problems. The linearised analysis of these flows has shown that a Traveling Thermal Wave (TTW) can generate a mean shear flow within a layer of fluid, and the induced mean flow is proportional to the square of the amplitude of the wave. From a physical pt of view, the motion induced by TTW is quite interesting as a purely fluid dynamical problem and can be used as a possible explanation for the observed four-day retrograde zonal motion of the upper atmosphere of venus. Also, the heat transfer results will have a definite bearing on the design of oil or gas fixed boilers. Problems where the fluid is semiinfinite in extent bounded by a rigid well and the free-stream oscillations with timedependent velocity are often encountered in engineering applications viz., aerodynamics of a helicopter or in a fluttering aerofoil or in a variety of Bio-engineering problems. The analysis of the forced oscillatory (with free stream oscillations) flow has been dealt in detail by several authors. Taking the buoyancy force into consideration, this study has been extended to include the effects of
the free convection on the oscillatory flows by Soundalgekar10, and Murthy6. Most of these investigations mentioned above are based on Lighthill method5, which reduces the governing coupled non-linear partial differential equations into two sets of ordinary differential equations which can be solved using the standard technique (analytical or numerical method). Vajravelu and Debnath12have made an interesting study of non-linear convection heat transfer and fluid flows induced by TTW. The TTW problem was investigated both analytically and experimentally by white head 14 by populating series expansions in the square of the aspect ratio (assumed small) for both the temp and flow fields. White head 14obtained an analytical solution for the mean flow produced by a moving source theoretically prediction regarding the ratio of the mean flow velocity to the source speed were found to be good agreement with experimental observations in Mercury which therefore justified the validity of the asymptotic expansion a posteriori Ravindra7 has analysed the mixed convection flow of a viscous fluid through a porous medium in a vertical channel. The thermal buoyancy in the flow field is created by a TTW imposed on the boundaries. In all the above investigations, the variation of density is taken in the linear form
∆ρ = - ρβ (∆T)
(1.1)
where β is the coefficient of thermal expansion and is 2,07 x 104 (oC)-1+. This is valid for temperature variation near 20oC. But this analysis is not applicable to the study of the flow of water at 4oC past a vertical plate. This is because, at 4oC, the density of water is a maximum at
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atmospheric pressure and the above relations (1.1) dies not hold good. The modified form of (1.1) applicable to water at 4oC is given by ∆ρ = - ργ (∆T)2 (1.2) Where γ = 8x10-6(0c)-2. Taking this fact into account, Goren showed in this case, similarity solutions for he free convection flow of water at 4oC past a semi-infinite vertical plate exist. Govindarajulu 4 showed that a similarity solution exist for the free convection flow of water at 4oC from vertical and horizontal plates in the presence of suction and injection. Soundalgekar10 obtained an approximate solution of the same problem using Kraman-Pohlhausen integral method. Sastri and Vajravelu have solved the problem of free convection between vertical walls by taking the nonlinear density temperature variation, viz.,
∆ρ = - ρβ g (T - Te) - ρβ1 (T – Te)2
(1.3)
where β0 and β1 are the constants. This relation includes both the relationships (1.1) and (1.2). Gilpin 5has used a density temperature relation which is similar to relation (1.3) and has shown the existence of Quasi-stead modes of convection for some temperature below 4oC. Using the relation (1.2) Sinha9 Agarwal and Upamanyu1 have analysed the problem of fully developed free convection flow between vertical plates and in a circular pipe respectively. Sree Ramachandra Murthy11 investigated the effects of free convection on the flow of a viscous electrically conduction fluid through a porous medium in the cylindrical annulus under the influence of radial magnetic field. Devika rani et al.2 have analysed the Mixed convection flow through a porous
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medium in a annual region with quadratic temperature variation. Recently Rukhsana begum8 has analysed the effect of radiation on convective heat transfer through a porous medium in a cylindrical annulus with quadratic density temperature variation. In this paper we make an attempt to investigate Magneto Hydro Dynamic (MHD) unsteady convective heat transfer in a vertical channel on whose walls Traveling thermal wave is imposed. A uniform magnetic field of strength H0 is applied transverse to the boundaries. A quadratic density distribution is taken in the equation of state. The equations governing the flow and heat transfer have been solved by employing a regular perturbation technique with δ, the aspect ratio as a perturbation parameter. The velocity and temperature are analysed for different values of the governing parameters G, M, γ, x + γt. Also the shear stress and the rate of heat transfer on the walls are numerically evaluated. 2. FORMULATION OF THE PROBLEM We analyse the motion of a viscous,elerctrically conducting fluid in a vertical channel bounded by flat walls . The thermal buoyancy in the flow field is created by a traveling thermal wave imposed on the boundary wall at y=L while the boundary at y = -L is maintained at constant temperature T1. The walls are maintained at constant concentrations.A uniform magnetic field of strength H0 is applied normal to the boundaries. Assuming the magnetic Reynolds to be small we neglect the induced magnetic field. Boussinesq approximation is used so that the density variation will be considered only in the buoyancy force. Also
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the kinematic viscosity n,the thermal conducting k are treated as constants. We choose a rectangular Cartesian system 0(x, y) with x-axis in the vertical direction and y-axis normal to the walls. The walls of the channel are at y = ± L. The equations governing the unsteady flow and heat transfer are Equation of linear momentum ∂u ∂u ∂u ∂p ρe ( + u + v ) = − + ∂t ∂x ∂y ∂x (2.1) ∂ 2u ∂ 2u σ µ 2H 2 µ ( 2 + 2 ) − ρ g − ( e o )u ∂x ∂y ρ0 ρe ( µ(
∂v ∂v ∂v ∂p +u +v ) = − + ∂t ∂x ∂y ∂y
∂ 2v ∂ 2v + ) ∂x 2 ∂y 2
(2.2)
(2.3)
Equation of energy ρ eC p ( = λ(
∂T ∂T ∂T +u +v ) ∂t ∂x ∂y
∂ 2T ∂ 2T )+Q + ∂x 2 ∂y 2
u = 0 , v = 0 , T = T2 + (T1 − T2 ) Sin(mx + nt ) on
Equation of continuity
∂u ∂v + =0 ∂x ∂y
permeability of the β is the coefficient of thermal expansion, and Q is the strength of the constant internal heat source. The flow is maintained by a constant volume flux for which a characteristic velocity is defined as 1 L Q= (2.6) ∫ u d y. 2L − L The boundary conditions for the velocity and temperature fields are u= 0, v = 0 , t=t1 on y = – L
(2.4)
Equation of state
ρ − ρ e = − βρ e (T − Te ) 2 where ρ e is the density of the fluid in the equilibrium state, Te is the temperature and in the equilibrium state,(u,v)are the velocity components along O(x,y) directions, p is the pressure, T is the temperature in the flow region,ρis the density of the fluid,µ is the constant coefficient of viscosity, Cp is the specific heat at constant pressure, λ is the coefficient of thermal conductivity , σ is the electrical conductivity, µ e is the magnetic
y = L
Sin ( mx + nt )
(2.7) is the imposed traveling
thermal wave In view of the continuity equation we define the stream function ψ as u = -ψy, v=ψx (2.8) Eliminating pressure p from equations (2.2) & (2.3) and using the equations governing the flow in terms of ψ are2 [(∇ 2ψ ) t + ψ x (∇ 2ψ ) y − ψ y (∇ 2ψ ) x ] = ν ∇ 4ψ − β g ( T − T0 ) 2 y −(
(2.9)
σµ H ∂ ψ ) 2 ∂y ρ0 2 e
ρ eC p (
2 o
2
∂θ ∂ψ ∂θ ∂ψ ∂θ + − ) = λ ∇ 2θ ∂t ∂y ∂x ∂x ∂y
(2.10)
Introducing the non-dimensional variables in (2.9 ) & (2.10) as x′ = mx , y ′ = y / L , t ′ = tν m2 , T − T2 e Ψ′ = Ψ / ν , θ = T1 − T2
the governing equations dimensional form (after dashes) are
(2.11) in the nondropping the
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-383)
S. Jafarunnisa, et al., J. Comp. & Math. Sci. Vol.2 (2), 304-314 (2011) δ R(δ (∇12ψ )t + ∇14ψ +
∂(ψ , ∇12ψ ) )= ∂ ( x, y )
2G ∂ 2ψ (θ θ y ) − M 12 2 R ∂y
(2.12)
∂y ∂x
∂x ∂y
where
R=
UL
ν βg (∆Te ) 2 L3 G= ν2
(Reynolds number)
Ρ=
(Prandtl number),
µ cp k1
σµ e2 H o2 L2 M = ν2 δ =mL 2 1
γ =
(Grashof number)
(Hartmann number), (Aspect ratio)
n (non-dimensional thermal wave νm 2 velocity)
∇12 = δ 2
in accordance with the prescribed arbitrary function t . 3. ANALYSIS OF THE FLOW
The energy equation in the non-dimensional form is ∂ψ ∂ψ ∂θ ∂ψ ∂θ (2.13) δ P(δ + − ) = ∇12θ ∂t
308
∂2 ∂2 + ∂x 2 ∂y 2
The corresponding boundary conditions are
ψ ( +1) − ψ (−1) = −1 ∂ψ ∂ψ = 0, = 0 at y = ±1 (2.14) ∂x ∂y on y = – 1 θ ( x, y ) = 1 on y = 1 θ ( x, y ) = Sin ( x + γt ) ∂θ = 0 at y = 0 (2.15) ∂y The value of ψ on the boundary assumes the constant volumetric flow in consistent with the hypothesis (2.7). Also the wall temperature varies in the axial direction
The main aim of the analysis is to discuss the perturbations created over a combined free and forced convection flow due to traveling thermal wave imposed on the boundaries. The perturbation analysis is carried out by assuming that the aspect ratio δ to be small. We adopt the perturbation scheme and write ψ ( x, y , t ) =ψ 0 ( x, y, t ) + δψ 1 ( x, y, t ) + δ 2ψ 2 ( x, y, t ) + − − − − θ ( x, y, t ) = θ 0 ( x, y, t ) + δ θ1 ( x, y, t ) +δ 2 θ 2 ( x, y , t ) + − − − − −
(3.1)
On substituting (3.1) in (2.12)& (2.13) and separating the like powers of δ the equations and respective conditions to the zeroth order are
ψ 0, y y y y y − M12ψ 0, y y = −G(θ o θ0, y ) / R (3.2) (3.3) θ o , yy = 0 with ψ 0(+1) – Ψ (– 1) = 1 , ψ 0 , Y = 0, ψ 0,X = 0 on y = ±1
θo = 1 on y = −1 θ o = Sin( x + γt ) on y = 1
(3.4) (3.5)
and to the first order are ψ1, y y y y y − M12ψ1, y y = −2G(θoθ1, yy + θ1 θo, y )/ R + (ψ0, y ψ0, x y y −ψ0, x ψ0, y y y )
(3.6)
θ 1, yy = P (ψ 0, xθ o , y − ψ 0, yθ ox )
(3.7)
with ψ 1(+1) – ψ1(–1) = 0 ψ 1,Y = 0, ψ 1, X = 0 on y = ±1 θ1(±1) = 0, on y = ± 1
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(3.8) (3.9)
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Equations 3.2-3.9 are solved subject to the
boundary conditions.
4. SOLUTION OF THE PROBLEM Solving the equations (3.3)- (3.7) subject to the relevant boundary conditions we obtain α Sin( D1 ) − 1 θ o ( y , t ) = ( )(1 − y 2 ) + ( )y + 2 Sin( D1 ) + 1 ( ) 2
2
ψ o ( y , t ) = a1 0 + a1 1 y + a1 2 C h ( M 1 y ) + a1 3 S h ( M 1 y ) + a 5 y 3 + φ ( y )φ ( y ) = a 6 y 2 + a 7 y 3 + a 8 y 4 + a 9 y 5
θ 1 ( y , t ) = a16 ( y 2 − 1) / 2 + a17 ( y 3 − y ) / 6 + a18 ( y 4 − 1) / 12 + a19 ( y 5 − y ) / 20 + a 20 ( y 6 − 1) / 30 + + a 21 ( y 7 − y ) / 42 + a 22 ( y 8 − 1) / 56
ψ 1 = B1 + B2 y + B3 Ch ( M 1 y ) + B4 Sh ( M 1 y ) + φ1 ( y ) φ 1 ( y ) = − a55 y 2 a56 y 3 − a57 y 4 − a58 y 5 − a59 y 6 − a60 y 7 − a61 y 8 − − a62 y10 + ( ya63 + y 2 a65 + y 3 a67 + y 4 a69 + y 5 a71 + y 6 a73 )Ch ( M 1 y ) + + ( ya64 + y 2 a66 + y 3 a68 + y 4 a70 + y 5 a72 + y 6 a74 ) Sh ( M 1 y )
w h ere a 1 , a 2 , ...............a 74 , B1 , B 2 , ............. B4
a re co n s tan ts .
5. RATE OF HEAT TRANSFER:
where a93,……….a97 constants.
Knowing the temperature derivations the local rate of heat transfer coefficient (nusselt number nu) on the walls has been calculated using the formula
In the absence of magnetic field (m=0) the results are in good agreement with reddy et al.13
∂θ 1 Nu = ( ) y = ±1 θ m − θ w ∂y
6. DISCUSSION OF THE RESULTS:
and the corresponding expressions are (a93 + δ a95 ) ( N u ) y = +1 =
(a97 − Sin( D1 )
( N u ) y = −1 =
(a94 + δ a96 ) (a97 − 1)
In this analysis we discuss the convective heat transfer of a viscous, electrically conducting fluid confined in a vertical channel bounded by flat walls on which a traveling thermal wave is imposed. The velocity and temperature distributions are analysed for different sets of governing parameters G, M, γ and x + γt and are represented in figs.1-12.
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S. Jafarunnisa, et al., J. Comp. & Math. Sci. Vol.2 (2), 304-314 (2011) 0. 0
0.0 3
- 0. 2
G =1 0
- 0. 4
G =2 x 1 0 G =3 x 1 0
- 0. 6
G =- 1 0
3 3
G =- 3 x 1 0
-0.6
3 3
-0.8
u
u
- 1. 0
-0.4
3
G =- 2 x 1 0
- 0. 8
x+γt=π/4 x+γt=π/2 x+γt=π x+γt=2π
-0.2
- 1. 2
-1.0
- 1. 4
-1.2
- 1. 6
-1.4
- 1. 8
-1.6 - 2. 0 -1 .0
-0 .5
0.0
0 .5
1 .0
y
-1.8 -1.0
-0.5
0.0
0.5
y
Fig(1) Variation of u with G
Fig(4) Variation of u with x+γt x+
Fig(2) Variation of u with M
Fig(5). Variation of v with G
0.0 -0.2
γ= 5 γ=8 γ = 10
-0.4 -0.6
u
-0.8 -1.0 -1.2 -1.4 -1.6 -1.0
-0.5
0.0
0.5
1.0
y
Fig(3) Variation of u with γ
Fig(6) Variation of v with M.
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1.0
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Fig(7) Variation of v with γ
Fig(10) Variation of θ with M
Fig (8) Variation of v with x + γt
Fig(11) Variation of θ with γ
Fig(9) Variation of θ with G
Fig(12) Variation of θ with x+γt x+
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In the following discussion we use the notation that the channel walls are heated or cooled according as G>0 or G<0.We take P=0.71 and δ=0.01. It is found that the temperature variation on the boundary contributes substantially on the flow field. This contribution may be represented as perturbation over the mixed convection flow generated in the state of uniform wall temperature. These perturbations not only depend on the wall temperature but also on the mixed convection flow.In general we find that the creation of the reversed flow in the flow region depends on whether the free convection effects dominates over the forced flow or vice versa.If the free convection effects are sufficiently large as to create reversal flow. The variation in the wall temperature affects the flow remarkably. Fig.1 represents the variation of the axial velocity u with Grashof number G. u <0 is the actual axial flow and u>0 is the reversal flow.It is found that no reversal flow exists anywhere in the fluid region for any variation. We find that the magnitude of u experiences an enhancement everywhere in the flow region with maximum in the midhalf with an increase in G>0 except in the vicinity of the boundaries y = ±1 at which u depreciates while an increase in G<0 decreases u anywhere in the region except in the neighbourhood of y = ±1 at which it experiences an enhancement. From fig.2 we notice that higher the Lorentz force smaller u in the region − 0.4 ≤ y ≤ 0.4 ,larger u in the vicinity of y = ±1 . An increase in thermal wave velocity γ enhances u in the right half and smaller u in the left half. For higher γ ≥ 10 a reversed effect is noticed(fig.3). The variation of u
312
with phase x+γ t shows that an increase in x+γ t ≤ π /2 enhances u in the left half and reduces in the − 0.2 ≤ y ≤ 0.4 and enhances u in the vanity of the boundaries.and this behaviour gets reversed at x+γ t =π and again for higher x+γ t ≥ 2π, u enhances in the vanity of y = ±1 (fig.5). The secondary velocity (v) which arises due to the non-uniform temperature on the boundaries is shown in figs.5-8.It is found that for G>0 the secondary velocity is directed towards the midregion and is towards the boundary for G<0. v enhances with increase in G>2x103 except in the vicinity of y = ±1 and for higher
G ≥ 3x10 3 ,v enhances throughout the region while an increase in G<0 enhances v anywhere in the fluid region..The variation of v with M shows that higher the Lorentz force larger v in the entire region and for still higher values of M smaller v in the flow region(fig.6).An increase in the thermal wave velocity γ ≤ 8 enhances v except in the vicinity of the boundaries and for higher γ ≥ 10 , v enhances everywhere in the flow region(fig.7).From fig.8 we find that the secondary velocity depreciates with increase in x+γ t <π /2 and enhances for higher x+γ t >π . The non-dimensional temperature (θ) is exhibited in figs.9-12 for different values of G,M,γ and x+ γt.It is found from fig.9 that the actual temperature depreciates in the flow region for G ≤ 2x 103 and for higher G ≥ 3x 103,the actual temperature experiences a depreciation in the entire flow region.An increase in G<0 enhances the actual temperature everywhere in the flow region.The behaviour of θ with M shows
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that higher the Lorentz force larger the actual temperature and for further enhancing force smaller the actual temperature in the flow field (fig.10).Also an increase in γ ≤ 8 reduces θ in the flow region and for higher γ≥10,it experiences an enhancement in the region(fig.11). The behaviour of θ with phase x+γt of the boundary temperature enhances with increase in x+γ t < π /2 and depreciates for further increase in x+γ t ≥ π except in a narrow region adjacent to y=1 in which the temperature experiences an enhancement(fig.12). The Nusselt Number(Nu) which measures the rate of heat transfer at the walls is shown in tables 1&2 for different values of G,M.,γ, x+γt.It is found that the rate of
heat transfer depreciates with increase in G>0 and enhances with G<0.Higher the Lorentz force smaller the magnitude of Nu at y =-1 and lager Nu at y=1 for all G.An increase in the thermal wave velocity γ reduces Nu at both the walls..From tables. 1&2 show that the rate of heat transfer enhances at y = -1 enhances with increase with increase in the phase x+ γt ≤π /2 and reduces with higher values of x+ γt = π and again enhances at x+ γt = 2π while at y=1,Nu enhances in the heating case and reduces in the cooling case with increase in x+ γt ≤π /2 and for higher x+γt ≥ π the rate of heat transfer experiences an enhancement for all G.
Table 1 Nusselt Number (Nu) at y = -1 [ G=103
I -1.1827
II -0.0764
III -0.0681
IV -2.5581
V 1.8832
3 X103
-0.0081
-103 -3X103 M γ
0.0716 -1.3729 2 5
x+γt
π/4
VI 0.6219
VII 0.72
0.0589
-0.0512
-0.0101
-0.0287
0.0165
0.0443
-0.157 -1.4414 4 5
-0.0879 -0.1061 6 5
0.0746 -1.3979 2 10
0.7816 1.6846 2 5
-0.2273 -1.0013 2 5
-0.5409 -1.0858 2 5
π/4
π/4
π/4
π/2
π
2π
Table 2 Nusselt Number (Nu) at y = 1 G/Nu G=103 3 X103 -103 -3X103 M γ x+γt
I 0.2388 0.0172 -0.0569 0.1678 2 5 π/4
II 0.2499 0.1651 0.3568 0.1871 4 5 π/4
III 0.3495 0.2288 0.2865 0.2449 6 5 π/4
IV 4.354 0.01745 -0.0669 0.2578 2 10 π/4
V -0.2641 -0.0066 -0.0002 0.0069 2 5 π/2
VI -0.4441 -0.1597 -0.1621 0.0044 2 5 π
VII 0.6751 0.2543 -0.6891 0.0796 2 5 2π
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6. REFERENCES 1. Agarwal R.S. and Upamanyu K.G.‘Laminar free convection with and without heat sources in a circular pipe’ Bull, Calcutta, Math. Society, 68, pp.285 (1976). 2. Devika Rani B – ‘Mixed convection flow through a porous medium in a annual region with quadratic temperature variation’ - J. Pure & Appl. Phys., Vol. 21, No.3, July-Sep,2009, pp. 335-346 (2009). 3. Gilpin–‘R. R Cooling of a horizontal cylinder of water through its max density point at 4oC’- Int. J. Heat and mass Transfer, 18, pp.1307 .(1975). 4. Govindarajulu - ‘J. Free Convection flow of water at 4oC on vertical and horizontal plate’- Chem.. Engg. Sci. 25. pp.1827 (1970). 5. Lighthill, M. J. ‘Proc. Roy. Soc. (London), A., pp. 224 (1954). 6. Murthy, K. N. J. Pure. Appl. Math., 10 (9), pp.1051 (1979). 7. Ravindra M –‘MHD Convection flow through porous medium with nonuniform wall temperature’-Ph.D. Thesis, S. K. U., Anantapur (India) (1994).
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8. Rukhsana Begum A – ‘Effect of radiation on connective Heat Transfer through a porous medium in a cylindrical annulus with quadratic density temperature variation’ - J.Pure & Appl. Phys., Vol. 21, No. 3, (2009). 9. Sinha, P.C - Chem. Engg. Sci. 24, p.33 (1969). 10. Soundalgekar, V.M -Acta Mechanica, V.16, p.77-91 (1973). 11. Sree Ramachandra Murthy A ‘Buoyancy included hydromagnetic fluid through porous medium’-Ph.D. Thesis, S. K. Uni versity, Anantapur, ( India), pp. 81–89 (1992). 12. Vajravelu K and Debnath L – ‘Non linear study of convective heat transfer and fluid flows induced by TTW’- Acta. Mech., U59, pp 233-249(1986). 13. Vijaya bhaskar reddy P, Devika Rani B and Prasada Rao DRV – ‘Unsteady convective Heat and transfer in a vertical channel with Travelling thermal waves and Quadratic temperature Variation.- J. Pure&Appl.phys. Vol.21, N0.4, pp. 639-651 (2009). 14. Whitehead J. A. Geo-phy, Fluid Dynamics V.3, pp.161-180 (1972).
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