J. Comp. & Math. Sci. Vol. 1 (6), 732-739 (2010)
“The Effects of Variable Viscosity & Thermal Conductivity on Steady Free Convection Flow Along A Semi-Infinite Vertical Plate (in Presence of Uniform Transverse Magnetic Field).� G. BORAH1 and G. C. HAZARIKA2 Sr. Lecturer, Department of Mathematics1, Joya Gogoi College, Khumtai Dist. Golaghat (Assam) Prof. Department of Mathematics2, Dibrugarh University, Dibrugarh, (Assam) ABSTRACT Variable viscosity & thermal conductivity effects on steady free convection flow along a semi-infinite vertical plate in presence of uniform transverse magnetic field have been investigated. The governing boundary layer equations have been solved by taking series expansion of the stream function and temperature function. The resulting ordinary nonlinear coupled differential equations with the boundary conditions i.e. the boundary value problem (BVP) have been solved numerically on computer by using Runge Kutta shooting method. The numerical values of the functions are tabulated. The results are discussed numerically and presented graphically for various values of the nondimensional viscosity & thermal conductivity parameters. The effects of variable viscosity and thermal conductivity on the problem are found significant. Key Words: Steady free convection, Boundary layer flow, transverse magnetic field. AMS subject classification No.76XX
INTRODUCTION Viscosity is a measure of internal fluid friction due to which there is a resistance of fluid flow. Generally it is a function of temperature. Thermal Conductivity determines the quantity of heat passing per unit time per unit area at a temperature drop of 10C per unit length. It is also a function of temperature. Earlier, on the problem the viscosity & thermal conductivity of the fluid were assumed to be constant. However, it is known that these physical properties can charge significantly with temperature and subsequently the flow characteristics are changed compared to constant cases. Free convection viscous incompressible fluid flow past a different types of vertical bodies are investigated because of their wide application in thermal insulation of buildings, Geophysics, the extrusion of
polymer sheet from a dye, the cooling of an infinite metallic plate in a cooling path, glass blowing continuous casting, spinning of fibers, packed bed reactors and sensible heat storage beds. Sakiadis10 studied the boundary layer flow over a continuous solid surface, moving with constant velocity in an ambient fluid. Chakrabarta and Gupta2 investigated hydro magnetic flow, heat and mass transfer over a stretching sheet. Kumr et al4 studied hydro magnetic flow and heat transfer on a continuously moving vertical plate. Chamkha3 investigated thermal radiation and buoyancy effects on hydro magnetic flow over a heat source or sink. Reptis et al8 discussed the effect of thermal radiation on MHD asymmetric flow of any electrically conducting fluid past a semiinfinite plate.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
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G. Borah et al., J. Comp. & Math. Sci. Vol. 1(6), 732-739 (2010)
Beithon, N. et al1 investigated free convection flow of Newtonian fluid along a vertical plate Embedded in a double layer porous medium. The Lorentz force is also important. For example the boundary layers along the walls of MHD power generators are strongly influenced by such forces. Takhar considered the effect of radiation on free convection flow along semi-infinite vertical plate in presence of transverse magnetic field. Kumar4 studied hydro magnetic flow and heat transfer on a continuously moving vertical plate. Vajrawely and Hadjinicolaou12 studied the flow and heat transfer characteristic in an electrically conducting fluid near and isothermal stretching sheet. Sharma and Mathur11 investigated steady laminar free convection flow of an electrically conducting fluid along a porous hot vertical infinite plate in the presence of heat source or sink. The purpose of the present work is to study the effects of variable viscosity and thermal conductivity on steady free convection flow along a semi-infinite vertical plate in presence of uniform transverse magnetic field. Mathematical Formulation
(for low fluid velocity). A magnetic field of constant intensity is assumed to be applied normal to the vertical plate and the electrical conductivity of the fluid is assumed to be small so that the induced magnetic field can be neglected in comparison to applied magnetic field. Now we have the governing equations as (1) The
equation
of
continuity
∂u / ∂v′ + =0 ∂x ′ ∂y ′
(1)
(2) The equation of motion
∂u ′ ∂u ′ 1 ∂ ∂u ′ + v/ = µ ∂x ′ ∂y ′ ρ α ∂y ′ ∂ y ′ a β 2u ′ / (2) ′ + g β (T − Tα ) −
u′
ρα
(3) The equation of Energy
ρ α C p u ′
∂T ′ ∂T ′ ∂ ∂T ′ = K + v/ ∂x ′ ∂y ′ ∂y ′ ∂y ′ (3)
Consider a steady two-dimensional free convection flow of an incompressible viscous electrically conducting fluid past a semi-infinite vertical plate at constant temperature Tw/ in presence of uniform transverse magnetic field. The x/ axis is taken along the plate in upward direction and y/ axis is taken normal to it. The temperature of the fluid far from the plate is Tα/ . All the fluid properties are assumed to be constant except density variation with temperature. The viscous dissipative heat is assumed to be negligible
Where
ρα - density of fluid The magnetic intensity βViscosity of fluid µK– T/Cp –
Thermal conductivity Temperature of the fluid Electrical conductivity Specific heat at constant pressure
Other symbols have their usual meaning. The boundary conditions are at y/=0 , u ′ = v / = 0 , T/ = T/W
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
G. Borah et al., J. Comp. & Math. Sci. Vol. 1(6), 732-739 (2010)
y ′ → α , u → 0 , T ′ → Tα′
at
Method of Solution Introducing the following non-dimensional quantities
x′ U u′ L x = , y = y′ , u = , v = v/ L Lυα U Uυα θ =
T ′ − T α′ T w/ − T α /
Where u ′, v are velocity components along x ′, y ′ axes respectively, L the length of
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Lai and Kulacki6 have assumed that the viscosity and thermal conductivity of fluid to be an inverse linear function of temperature as
1
µ
=
[1 + γ (T −T α )], 1
1
µα
K
=
1 [1 + γ | (T −T α )] Kα
under this assumption, To obtain the desired solution, the partial differential equation (5), (6) is to be first reduced to ordinary differential equations we introduce the following similarity variables
/
the plate U =
gβ L(Tw/ − Tα/ ) is the quantity
with dimension of speed, θ the dimensionless fluid temperature and all the other symbols have their usual meanings. The equation (1), (2), (3) becomes in non-dimensional form
ψ = x3/ 4 f (η,ξ), η = x−14 y , ξ = x12 ,θ = θ(η,ξ),θ T −Tα = T −T w α Where ψ
u=
is stream function such that
∂ψ ∂ψ , v=− ∂y ∂x
(7)
It can be shown that =0
(4)
∂u ∂u 1 ∂µ ∂u ∂ u +v = +µ 2 ∂x ∂y µα ∂y ∂y ∂y (5) +θ − Mu 2
u
∂θ ∂θ 1 ∂ K ∂θ ∂ θ +v = +K . ∂x ∂ y K α Pr ∂ y ∂ y ∂ y 2 2
u
(6) The corresponding boundary conditions are at at
y = 0, u = v = 0, θ = 1 y → ∞ , u → 0, θ → 0
−1
x4 u =ξ f /, v = − 4
∂f 3 f − nf / + 2ξ dξ
equation (5) reduces to
f ′′′ −
θ − θc 1 2 θ − θ c 3 ff ′′ + f′ 4 θc 2 θc
∂f ∂ 2 f θ − θc − f ′′ − f′ + Mξ f ′ ∂ξ ∂η θ c 2 ∂ξ
ξ
θ − θc θc
θ − θ c θ f ′′ =0 −θ − θ θ θ − c c
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
(8)
G. Borah et al., J. Comp. & Math. Sci. Vol. 1(6), 732-739 (2010)
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equation (6) reduces to
2
θ − θ r ξ ∂θ 3 ∂f θ ′′ − Pr f θ ′ + Pr f ′ − θ ′ 4 ∂ξ θ r 2 ∂ξ 2
θ − θ r θ ′ θ r − θ − θ r = 0 (9) Where µ =
− µαθ c − K αθ r ,K= θ −θc θ −θ r
5
θr − θ0 3 θ1 2θ1/ / Pr. f 0θ0 + + 2 (θr − θ0 ) θr 2 θr
(13)
Where a prime denotes differentiation w.r.t. The corresponding boundary conditions at η = 0; f 0= f 0/ = f1 = f1/ = 0, θ 0 = 1, θ1 = 0
f = f o (η ) + ξ f | (η ) + ξ 2 f 2 (η )..............
θ = θ o (η ) + ξ θ | (η ) + ξ 2 θ 2 (η ) + ............. and substituting these in equation (8) and (9) we obtain the following set of differential equation, by equating co-efficient of powers of ξ
θ (θ − θ c ) θ 0/ f 0// f 0′2 + 0 0 + θc θ0 − θc
Pr
θθ + θ0/ θ r − θ0
f , θ in powers of ξ as follows
3 θ 0 − θ c 1 θ0 − θc f 0 f 0′′ − 4 θc 2 θc
3
θ1// = − Pr θ1/ f 0 − θ1 f 0/ + Pr θ0/ f1 2 4 4
/ 1 1
for series solution, we expand
f o′′′=
θ − θ r θ 0/ 3 (12) θ = Pr f 0θ 0/ 0 + 2 4 θ r (θ 0 − θ r ) // 0
(10)
3 5 3 f1/// = − f1// f0 − f| / f0/ + f1 f0// − Mf0/ 2 4 4 θc − θ0 θc − 2θ0 3 // / 2 θ1 − θ1 + f0 f0 − f0 4 θc θc θc θ /θ f 0// θ 1 f // − 0 1 − θ 1/ + 0 1 (11) θ 0 − θ c θ0 −θc θ0 −θc
at η → α ; f 0/ → 0, f1/ → 0, θ 0 → 0, θ1 → 0 (14) where M- Magnetic fluid parameter Pr-Prandtl Number Method of Solutions : To solve the boundary value problem governed by equations (10-13) subject to the boundary condition (14) for all values of parameters, we apply the Range-Kutta Method with shooting technique [9]; the missing initial values of f 0// (o), f1// (o),θ 01 (o), θ 1/ (o) etc. were estimated by an interactive scheme to some desired degree of accuracy. Hazarika5 showed that though there is no guarantee of convergence of the iterative scheme, if the initial guesses for the missing initial values are on opposite sides of the true value. The convergence is rapid and agrees well with other methods. Using shooting methods. The missing initial values f 0// (o), f1// (o),θ 01 (o), θ 1/ (o) etc. are estimated and tabulated for various values of the viscosity and thermal conductivity parameter; consequently the problem is solved.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
G. Borah et al., J. Comp. & Math. Sci. Vol. 1(6), 732-739 (2010)
RESULT AND DISCUSSION In this paper the system of differential equation (10-13) governed by the respective boundary conditions (14) is solved numerically by applying an efficient numerical technique based on the common Range-Kutta shooting method9 and an iterative procedure. The whole numerical scheme can be programmed and applied easily. It is found that the convergence of the iteration process is quite rapid. The estimated values of the missing initial values are arranged in different tables for various values of the viscosity and thermal conductivity parameter. The viscosity parameters as well as conductivity parameter are negative for liquid. These concepts of the parameter were first introduced by Ling and Dybbs7 in their study of convective flow in porous medium. We have obtained the velocity and temperature distribution for various values of viscosity & thermal conductivity parameter
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θ e ( = Tc ), θ r ( = Tr ) . The results of the problem are presented graphically. Our observations are as in the following In Fig.1 we have seen that the velocity profile decreases as viscosity parameter ( θ e ) increases In fig 2: The velocity profile increases
as thermal conductivity parameter ( θ r ) increases If fig. 3: It exhibits the variation of temperature profile for various values of
viscosity parameter ( θ e ). It is observed that temperature decreases for increase of viscosity parameter. If fig. 4: It shows the variation of temperature profile for various values of thermal conductivity parameter( θ r ). it is found that temperature increases for increase of the thermal conductivity parameter.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
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G. Borah et al., J. Comp. & Math. Sci. Vol. 1(6), 732-739 (2010)
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
G. Borah et al., J. Comp. & Math. Sci. Vol. 1(6), 732-739 (2010)
CONCLUSIONS The presented analysis has shown that fluid flow field is influenced by the viscosity temperature variation, thermal conductivity – temperature variation within the boundary layer. The results show that –The velocity of the fluid flow in our study decreases for increase of viscosity parameter whereas it increases for increase of the thermal conductivity parameter. The temperature of the fluid flow decreases when viscosity parameter increases and it increases when thermal conductivity increases. Hence we conclude that the variable viscosity and thermal conductivity effects on the problem is significant. The results can be used for better performance and accuracy in various fields.
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5. 6.
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Beithon, N. et. al investigated free convection flow of Newtonian fluid along a vertical plate Embedded in a double layer porous medium.
7.
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Chakrabarti, A. and Gupta A.S., Hydro magnetic flow, heat and mass transfer over a stretching sheet Qart. Appl. Math.33, 73-78 (1979). Chamkha, A. J., Thermal radiation and buoyancy effects on hydro magnetic flow over a heat source or sink. Int. J.Eng. Sci. 38, 1699-1712 (2000). Kumar B.R., Hydro magnetic flow and heat transfer on a continuously moving vertical plate. Acta Mechanics 153, 249253 (2002). Hazarika, G.C., Computer oriented Methods, (1985). Lai and Kulacki have assumed that the viscosity and thermal conductivity of fluid to be an inverse linear function of temperature, Int. J. Heat Mass Transfer 33, 1028-31 (1990). Ling, J. and Dybbs, A. Study of convective flow in porous medium. ASME paper 87 WA/HT-23 ASME, New York. (1987).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
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Reptis, A. et al. The effect of thermal radiation on MHD asymmetric flow of any electrically conducting fluid past a semi-infinite plate. Applied Mathematics and computation 153, 645-649 (2004). Runge-Kutta Method with Shooting technique. Computer oriented Methods. (1985). Sakiadis studied the boundary layer flow over a continuous solid surface, moving
11.
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with constant velocity in an ambient fluid. Sharma and Mathur investigated steady laminar free convection flow of an electrically conducting fluid along a porous hot vertical infinite plate in the presence of heat source or sink. Vajrawely and Hadjinicolaou studied the flow and heat transfer characteristic in an electrically conducting fluid near an d isothermal stretching sheet.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)