J. Comp. & Math. Sci. Vol.2 (2), 286-295 (2011)
Effect of Radiation on Unsteady Convective Heat Transfer Flow of a Viscous Fluid Through a Porous Medium in a Vertical Channel with Oscillatory Wall Temperatures B. SUSEELAVATHY, S. JAFARUNNISA, U. RAJESWARA RAO and D.R.V. PRASADA RAO Department of Mathematics, S.K. University, Anantapur-515055 India email: drv_atp@yahoo.co.in, jafaruni.phd@gmail.com.
ABSTRACT We analyse the effect of radiation on unsteady convective Heat transfer flow of a viscous fluid through a porous medium in a vertical channel bounded by flat walls with oscillatory wall temperature. The equations governing the flow and Heat transfer are solved by regular perturbation techniques. The stress and the rate of heat transfer on the boundaries are evaluated numerically for different variation of parameters. Keywords: Radiation, Porous medium, Viscous fluid , Vertical channel and Oscillatory wall temperature.
1. INTRODUCTION The time dependent thermal convection flows have applications in chemical engineering, space technology etc. These flows can also be achieved by either time dependent movement of the boundary or unsteady temperature of the boundary. The energy crisis has been a topic of great importance in recent years all over the world. This has resulted in an unabated exploration for new ideas and avenues in harnessing various conventional energy sources like tidal waves, wind power and geothermal energy. It is well known power and geothermal energy. It is well known that in order to harness maximal geothermal energy one should have complete and
precise knowledge of quanta of perturbation needed to initiate convection currents in mineral fluids embedded in the earth’s crest enables one to use mineral energy to extract the minerals. The role of thermal radiation of major importance of in some industrial applications such as glass production and furnace design and in space technology applications, such as cosmical flight aerodynamics rocket, propulsion system, plasma physics and space craft reentry aero thermo dynamics which operate at high temperature. Keeping these applications in view we make an attempt to investigate the effect of the radiation on unsteady convective heat
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transfer through a porous medium in vertical channel. Raptis10 analyzed the thermal radiation and free convection flow through a porous medium by using perturbation technique. Bakier and Gorla1 investigated the effect of thermal radiation on mixed convection from horizontal surfaces in saturated porous media. With regard to thermal radiation heat transfer flows in porous media, Chamkha2 studied the solar radiation effects on porous media supported by a vertical plate. Forest fire spread also constitutes an important application of radiative convective heat transfer as described by Meroney6. More recently Chitraphiromsri and Kuznetsov3 have studied the influence of high-intensity radiation in unsteady thermofluid transport in porous wet fabrics as a model of fire fighter protective clothing under intensive flash fires. Impulsive flows with thermal radiation effects and in porous media are important in chemical engineering systems, aerodynamic blowing processes and geophysical energy modeling. Such flows are transient and therefore temporal velocity and temperature gradients have to be included in the analysis. Raptis and Singh9 studied numerically the natural convection boundary layer flow past an impulsively started vertical plate in a Darcian porous medium. The thermal radiation effects on heat transfer in magneto – aerodynamic boundary layers has also received some attention, owing to astronautical re-entry, plasma flows in astrophysics, the planetary magneto-boundary layer and MHD propulsion systems. Shateyi et al.11 have analyzed the Thermal Radiation and Buoyancy Effects on Heat and Mass
Transfer over a Semi-Infinite Strectching Surface with Suction and Blowing. Dulal Pal et al.4 have analyzed unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. Ahmed et al.2a have analyzed the thermal radiation effect on a transient MHD flow with mass transfer past an impulsively fixed infinite vertical plate. Rajesh et al.8 have considered the radiation effects on MHD flow through a porous medium with variable temperature or variable mass diffusion.Soret & duffer effect. Ganesam and Loganathan5 studied the effect of the radiation and mass transfer effects on flow past a moving vertical cylinder using Rossel and approximation by the crank–Nicolsan finite difference method. 2. FORMULATION OF THE PROBLEM We consider the flow of a viscous incompressible fluid through a porous medium in a vertical channel bounded by flat walls in the presence of constant heat sources. A uniform magnetic field strength H0 is applied normal to the boundaries. We choose a Cartesian coordinate system 0(x y) with walls at y = ± 1 by using Boussinesq approximation we consider the density variation only on the buoyancy term.
∂u µ ∂ 2u µ = − u − ρg ∂t ρ ∂y 2 k ρ0cp
(2.1)
∂T ∂2T µ ∂ = k f 2 + Q + 2µ(uy2 ) + (u2 ) − (qr ) (2.2) ∂t ∂y k ∂y
ρ - ρ0 = -β 0(T – T0)
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The boundary conditions are u = 0, T = T1 at y = -L u = 0, T = T1 + ∈(T2 – T1) cos ωt (2.4) By using Rosseland approximation for the radiative flux we get
qR =
4σ ∂ (T ′ ) 3β R ∂y ∗
4
(2.5)
and expanding T ′ 4 about Te by using Taylor’s series and neglecting higher order terms we get
T ′4 ≅ 4Te3T − eTe4
(2.6)
where σ* is the Stefan – Boltzmann constant and βR is the extinction coefficient. Using (2.5) in (2.6) the energy equation reduces to ρ 0C +
µ
p
∂T ∂t
(u
k
2
= k
) +
f
∂ 2T ∂y2
the
2 y
non
dimensional
u T − T1 , y′ = y/L, θ = , t′ = ωt, γ /L T2 − T1
Equations 2.1 & 2.7 reduce to (dropping the dashes)
γ2
∂u ∂ 2u = Gθ + 2 − ( D −1 )u ∂t ∂y
Pγ 2
∂θ ∂ 2θ = 2 + α + PEc (u 2y ) ∂t ∂y
4 ∂ 2θ + P Ec D −1 ( u 2 ) + 3N ∂y 2
where
G = β gL3
(T2 − T1 )
γ
2
α=
θ .L2 kf
(Darcy parameter) (Prandtl number) (Heat source parameter)
N=
3β R 4σ ∗Te3
(Radiation parameter)
γ2 =
ωL2 ν
(Wormsely Number)
P1 =
3 NP 3 Nα α1 = 3N + 4 3N + 4
The transformed boundary conditions are u = 0, θ = 0, at y = -1 u =0, θ = 1 + ∈ cos ωt at y = +1 (2.10)
)
(2.7)
1 6 σ ∗ T e3 ∂ ∂ 2 T 3β R ∂y2
on introducing variables
u′ =
+ Q + 2 µ (u
L2 D = k µC P P= Kf −1
(2.8)
(2.9)
(Grashoff number)
in view of the boundary conditions (2.4) we assume
u = u 0 + ∈ e it u1 + .......... θ = θ 0 + ∈ e itθ1 + ..........
(2.11)
Substituting these expansions in the above equations and separating the like powers we obtain the following equations
∂ 2 u0 − M 12u0 = G (θ ) 2 ∂y
(2.12)
∂ 2u1 − ( M 12 + iγ 2 )u1 = −G (θ1 ) 2 ∂y
(2.13)
∂ 2θ0 4 ∂ 2u 0 1+ + α + PEc 2 + 2 ∂y 3 N1 ∂y −1
PEc D ( u ) = 0 2 0
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∂ 2θ 1 4 2 1 + − iP γ θ 1 + ∂y 2 3N1 ∂ u 0 ∂u1 2PEc . + 2 P E c D −1 (u 0 u1 ) ∂y ∂y
11 u00 − M 12u 01 = −Gθ 01 , u01 (± 1) = 0 (2.19)
(2.15) =0
Since the equations (2.11 – 2.14) are non linear coupled equations. Therefore assuming Ec << 1 take u0 = u00 + Ec u01 u1 = u10 + Ec u11 θ0 = θ00 + Ec θ01 (2.16) θ1 = θ10 + Ec θ11
−1 2 θ 0011 + P1u12 00 = P1 D u 00 = 0 ,
θ01 (- 1) = 0, θ01
(2.20)
11 u10 − ( M 12 + iγ 2 )u10 = −Gθ10 ,
u10 (± 1) = 0
(2.21)
11 u10 − iP1γ 2θ10 = 0 , θ10(-1) = 0,
θ10(+1) = 1
(2.22)
11 u11 − ( M 12 + iγ 2 )u11 = −Gθ11 ,
Substituting( 2.1) in equations (2.11 – 2.14) and separating the like terms we get 11 u00 − M 12 u00 = −Gθ 00 , u00 (± 1) = 0 (2.17)
u11 (± 1) = 0
θ 0011 − α1 = 0 , θ0(-1) = 0, θ0 (+ 1) = 1 (2.18)
θ(± 1) = 0
(2.23)
1 1 −1 θ1111 − iP1γ 2θ11 = −2 Pu 1 00 u10 − 2 P1 D (u00 u10 ) ,
The solutions of (2.17), (2.18), (2.19), (2.20), (2.21) (2.22) (2.23) & (2.24) are
ch (M1 y ( 2 ch( M1 y ) sh (M1 y ) + a2 y − u00 = a3 + a1 y − ch M − 1 ch M sh M 1 1 1 ch( M1 y ) ch( M 1 y ) − a41 ch(2 M 1 y ) − ch 2 M 1 + u01 = − a39 y 2 − ch M1 ch M 1 ch( M 1 y ) + a45 ( y 2 − 1)ch( M 1 y ) + a43 y sh( M 1 y ) − sh 2 M 1 ch M1 sh( M1 y ) sh( M 1 y ) − a40 sh(2 M 1 y ) − sh 2 M 1 − a38 y − sh M1 sh M1 sh( M 1 y ) + a49 ( y 2 − 1) sh( M 1 y ) + a42 y ch( M 1 y ) − chM 1 shM 1 shµ ( M 1 y ) + a46 y 3ch( M 1 y ) − chM 1 sh M 1 Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
(2.24)
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θ01 = a23(y2 – 1) – a24(y3 – y) – a25(y4 – 1) – a26(y5 – y) – a27(y6 – y) – a28[y2 sh (M1y) – y sh M1] + a29(y2 ch M1y – ch M1) + a30(y ch M1y – y chM1) + a31(y sh M1y – sh M1) – a32(y ch M1y – ch M1) – a33(sh M1y – y sh M1) + a34(ch 2 M1y – ch 2 M1) + a35(sh 2 M1y – y sh 2M1) + ½ (y+1)
1 ch β1 y sh β1 y θ10 = + 2 ch β1 sh β1 u11 = b41 ch (β 2 y) + b42 sh (β2 y)+ φ3(y) θ11 = b21 ch (β 1 y) + b22 sh (β1 y) + φ2(y) φ3(y) = – b21 ch (β 3 y) – b22 ch (β 4 y) – b23 sh (β 3 y) – b24 sh (β 4 y) + b25 ch (β 5 y) – b26 ch(β 6 y) – b27 sh (β 5 y) – b28 sh (β 6 y) + b29 y3 ch (β1 y) + b30 y3 sh (β 1 y) + b31 y2 ch (β 1 y) + b32 y2 sh (β 1 y) – b33 y2 sh (β 2 y) – b34 y2 ch (β 2 y) + b35 y ch (β 1 y) + b36 y sh (β1 y) + b37 y sh (β 2 y) + b38 y ch (β 2 y) + b39 y sh (β 2 y) + b40 y ch (β2 y). φ2(y) = b1 ch (β 3 y) + b2 ch (β 4 y) + b3 sh (β 3 y) + b24 sh (β 4 y) – b5 ch (β 5 y) + b6 ch (β 6 y) + b7 sh (β 5 y) + b8 sh (β 6 y) + b9 y3 ch (β 1 y) – - b10 y3 sh (β 1 y)+ b11 y2 ch (β2 y) + b12 y2 sh (β 2 y) + b13 y2 sh (β 1 y) + b14 y2 ch (β 1 y)+ b15 y sh (β2 y) + b16 y ch (β 2 y) + b17 y ch (β 1 y) + b18 y sh (β 1 y) + b19 ch (β2 y)+ b20 sh (β 2 y). Where a1,a2,………………………………,b1,…………..b20 are constants.
3. SHEAR STRESS AND NUSSELT NUMBER The Shear stress and the rate of heat transfer are determined from the following formulas
0 -1 -0.8 -0.6 -0.4 -0.2 0
0.2 0.4 0.6 0.8
1
-0.05
-0.1
I
u
II III
-0.15
du dθ τ • = , qw = Nu±1 = dy y=±1 dy y=±1
-0.2
-0.25 y
For D-1=0 , α=0 and N-1=0 the results are in good agreement with Narahari7.
Fig. 1 : Variation of u with G G
I 103
II 2x103
III 3x103
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Fig. 2 : Variation of u with D-1 I 103
D-1
II 2x103
III 3x103
Fig. 4 : Variation of u with N1
IV 5x103
N1
I 0.5
II 4
III 10
IV 100
region except in the vicinity of the boundary y = ± 1 where |u| enhances. 3. An increase in |α|| (<>) leads to an increase in |u| in the entire fluid region. 4. An increase N1 ≤ 4 leads to a depreciation in |u| while it enhances with higher N1 ≥ = 10.
Fig. 3 : Variation of u with α α
I 2
II 4
III 6
IV -2
V -4
VI -6
1. Enhancement with increase in G, with maximum |u| attained at y =0.8. 2. Lesser the permeability of the porous medium, larger |u| in the flow region and for further lowering of the permeability smaller |u| and for still lowering of permeability smaller |u| in the flow
Fig. 5 : Variation of u with γ γ
I 0.5
II 1.5
III 2.5
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Fig. 8 : Variation of θ with G I II III G 103 2x103 3x103
Fig. 6 : Variation of u with Ec I 0.001
Ec
II 0.003
III 0.005
IV 0.007
7. Lesser the permeability the porous medium smaller the actual temperature in the flow region and for further lowering of the permeability smaller the actual temperature. 8. The region of transition from negative to positive θ enlarges with increase in α > 0. Also actual temperature reduces with α ≤ 4 and depreciate with a higher α ≥ 6.
Fig. 7 : Variation of u with t
t
I π/4
II π/2
III π
IV 2π
5. |u| depreciate in the left half and enhances in the right half with increase Ec. 6. The actual temperature experiences an enhancement with G ≤ 3 x 103 and depreciation with higher G ≥ 5 x 103.
Fig. 9 : Variation of θ with D-1 D-1
I 103
II 2x103
III 3x103
IV 5x103
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Fig. 10 : Variation of θ with α α
I 2
II 4
III 6
IV V VI -2 -44 -6
Fig. 12 : Variation of θ with γ
γ
Fig. 11 : Variation of θ with N1 N1
I 0.5
II 4
III 10
IV 100
9. Lesser the permeability the porous medium smaller the actual temperature in the flow region and for further lowering of the permeability smaller the actual temperature. 10. The region of transition from negative to positive θ enlarges with increase in α > 0.
I 0.5
II 1.5
III 2.5
Fig. 13 : Variation of θ with Ec
Ec
I 0.001
II 0.003
III 0.005
IV 0.007
Also actual temperature reduces with α ≤ 4 and depreciate with a higher α ≥ 6. 11. An increase N1 ≤ 4 the actual temperature depreciates everywhere in the region, for N1 = 10 the actual temperature enhances in the left half and reduces in the right half and for higher
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N1 ≥ 100 the actual temperature exhibits an increasing tendency in entire flow region. 12. the actual temperature depreciate in the left half and enhances in the right half with increase in Ec. Table – 1 Nusselt number (Nu) at y = 1
G 1x103 3x103 1x103 -3x103 D-1
I -0.55480 -0.55143 -0.55480 -0.55143 1
II -0.55314 0.55314 -0.53648 0.53648 -0.55314 0.55314 -0.53648 0.53648 3
III -0.54717 -0.48277 -0.54717 -0.48277 5
Fig. 14 : Variation of θ with t
I π/4
t
II π/2
III π
IV 2π 2
Table – 2 Nusselt number (Nu) at y = 1
G 1x103 3x103 -1x103 -3x103 α
I -0.55314 0.55314 -0.53648 0.53648 -0.55314 0.55314 -0.53648 0.53648 2
II -1.58190 -1.32474 -1.58190 -1.32474 4
III -2.60058 -2.02230 -2.60058 -2.02230 6
IV 1.57043 1.63443 1.57043 1.63443 -2
I 3.75221 4.32930 3.75221 4.32930 1.5
II -0.97371 -0.79079 -0.97371 -0.79079 4
III -1.22416 1.22416 -0.92642 0.92642 -1.22416 1.22416 -0.92642 0.92642 10
–
G 1X103 3x103 -1X103 -3x103 Ec
IV -1.41838 -1.00194 -1.41838 -1.00194 100
IMPORTANT CONCLUSION –
VI 3.75221 4.32930 3.75221 4.32930 -6
Table – 4 Nusselt number (Nu) at y = 1
Table – 3 Nusselt number (Nu) at y = 1 G 1x103 3x103 -1x103 -3x103 N1
V 2.65330 2.90965 2.65330 2.90965 -4
The rate of heat transfer experiences an enhancement with increase in G. While it depreciates with D-1. An increase in the strength of the heat source / sink exhibits a increasing tendency |Nu| at y = ± 1.
I -0.55027 -0.48588 -0.55027 -0.48588 0.001
II -0.53007 -0.33687 -0.53007 -0.33687 0.003
III -0.51297 0.51297 -0.19097 0.19097 -0.51297 0.51297 -0.19097 0.19097 0.005
IV -0.49587 -0.04507 -0.49587 -0.04507 0.007
–
An increase in the radiation parameter N1 ≤ 4, |Nu| depreciate and enhances with higher N1 ≥ 10.
–
Higher the dissipative forces smaller the magnitude of the he rate of the heat transfer at both the boundary.
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4. REFERENCES 1. Bakier A.Y and Gorla R.S.R. Thermal radiation effects on mixed convection from horizontal surfaces in porous media, Transport in Porous Media, Vol.23 pp 357-362 (1996). 2. Chamkha A. J., Takhar H. S. and Soundalgekar V. M.; Radiation effects on free convection flow past a semiinfinite vertical plate with mass transfer, Chem. Engg. J, Vol.84, pp 335-342. 3. Chitrphiromsri, P and Kuznetsov, A.V. Porous medium modmel for investigating transient heat and moisture transport in firefighter protective clothing under high intensity thermal exposure, J. Porous Media, Vol 8,5, pp 10-26 (2005). 4. Dulal Pal and Babulal Talukdar: Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction, Communications in Nonlinear Science and Numerical Simulation, Volume 15, Issue 7, July 2010, P.18131830 (2010). 5. Ganesan.P and Loganothan. P. Radiation and Mass transfer effects on flow of an incompressible viscous fluid past a moving cylinder. Int. J. H & M
transfer, Vol. 45, P.4281-4288 (1972). 6. Meroney, R.N. Fires in Porous Media, May 5-15, Kiev, Ukraine (2004). 7. Narahari M. Free convection flow between two long vertical parallel plates with variable temperature at on boundary. Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME 2008), Johar Bahru, Malaysia (2008). 8. Rajesh. V and S.V. K. Varma. Radiation effects on MHD flow through a porous medium with variable temperature or variable mass diffusion, Int. J. of Appl. Math and Mech. 6 (1). p.39-57 (2010). 9. Raptis, A. A and Singh, A. K. Free convection flow past an impulsively started vertical plate in a porous medium by finite difference method, Astrophysics Space Science J., Vol. 112, pp 259-265 (1985). Radiation and free 10. Raptis, A. A. convection flow through a porous medium, Int. commun. Heat Mass Transfer, Vol. 25, pp 289-295 (1998). 11. Shateyi. S. Thermal Radiation and Buoyancy Effects on Heat and Mass Transfer over a Semi-Infinite Strectching Surface with Suction and Blowing, Journal of Applied Mathematics, V. 2008, Article id. 414830, 12 pages (2008).
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