J. Comp. & Math. Sci. Vol. 1 (6), 748-753 (2010)
Effects of Radiation and MHD on Mixed Convection along a vertical moving Surface NAVNEET JOSHI, MANOJ KUMAR and LALIT MOHAN PATHAK Department of Mathematics, Statistics &Computer Science G.B. Pant University of Agriculture and Technology, Pantnagar-263 145 Uttarakhand, India E-mail: navneet.nimt@gmail.com ; mnj_kumar2004@yahoo.com ABSTRACT Radiation and MHD on laminar mixed convection boundary layer flow and heat transfer on continuously moving vertical surface have been studied. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The system of non-linear partial differential equations has been obtained and finally transformed into a set of ordinary differential equations with the help of similarity transformations involved in the problem. This set of equations has been solved and results have carried out for different values of the various physical parameters. The results showing the effect of involved parameters on velocity and temperature have been computed and presented graphically to discuss them in details. Keywords: MHD mixed convection, vertical moving surface, radiation, Runge-Kutta method shooting technique.
INTRODUCTION Continuously moving surface through an otherwise quiescent medium has many applications in manufacturing processes. Such processes are wire drawing, metal extrusion, and paper production,11 and13.The pioneering work in this area was carried out by Sakiadis1 who developed a numerical solution for the boundary layer flow field of a stretched surface. Many authors have reviewed this problem to study the hydrodynamic and thermal boundary layer due to moving surface.2,4 and5. Suction or injuction of a uniform surface was introduced by Fox et al.12 and Erickson et al.7 for stretched surface velocity and temperature and by Gupta and Gupta10 for linearly moving surface. Magyari et al.3 have reported analytical and computational solution when the surface moves with rapidly decreasing velocities using the self- similar method. Many researchers
considered the effect of constant viscosity on boundary layers developed by continuously moving surface. It is known that the fluid viscosity changes with temperature as in6. Recently, in8 the effect of variable viscosity on a mixed convection heat transfer along a vertical moving surface was studied. Present investigation deals the effects of MHD and radiation and to get precious information about the flow and temperature. MATHEMATICAL FORMULATION Let us consider a steady two dimensional laminar flow due to vertically moving isothermal surface. Using Boussinesq approximation for incompressible viscous fluid, the fluid viscosity is assumed to vary as an inverse linear function of temperature.
1
µ
=
1
µ∞
[1 + γ (Τ − Τ∞ )] , or
1
µ
= a (Τ − Τ∞ )
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Navneet Joshi et al., J. Comp. & Math. Sci. Vol. 1 (6), 748-753 (2010)
749 Where a =
1 γ and Τr = Τ∞ − µ∞ γ
The equations governing variable viscosity fluid flow are Equation of continuity:
for
convective
∂u ∂u + =0 ∂x ∂y
(1)
Equation of momentum: ∂u ∂u 1 ∂ ∂u σB02 u µ − +v = Sgβ (T − T∞ ) + + ρ ∞ ∂y ∂y ρ ∞ ∂x ∂y
u
Fig 1.The surface moving upwards in the X-direction
(2) Equation of energy:
∂q r ∂T ∂T ∂ T 1 +v =α − 2 ∂x ∂y ρ ∞ C p ∂y ∂y 2
u
(3) Subject to the following boundary conditions: u = U w v = 0 T = Tw at y → 0
T → T∞ as y → ∞ (4)
u→0
The x-coordinate is measured along the moving surface from the point where the surface originates and y coordinate is measured normal to it (fig1).Where u and v are the velocity components in x-and y-directions respectively. S is a dummy parameter stands for 0, +1, -1. The stream function and following transformation have been used. 1 2 x
ψ = 2ν ∞ Re f (η )
θ (η ) =
(T − T∞ ) (Tw − T∞ )
η=
y x 2
Re
1 2 x
The radiative heat flux qr under Rosseland approximation by Brewster (1992) has the form:
qr = −
4σ∂T 4 3χ∂y
Where σ is Stefan-boltzmann constant and χ is the mean absorption coefficient. The temperature differences within the flow are so small that T 4 can be expressed as a linear function of T∞ . This is obtained by expending
T 4 in Taylor series about T∞ and neglecting the higher order terms. Thus we get:
T 4 = 4T∞3T ′ − 3T∞4 and
u=
ν∞ x
Re x =
Re x f ′(η ) Uwx
ν∞ 1
v=
ν ∞ Re x 2 x 2
( f η′ − f )
(5)
Where f′ and θ are the dimensionless velocity and temperature respectively, η is similarity variable. Putting all the values in equations (2)- (4), we get
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Navneet Joshi et al., J. Comp. & Math. Sci. Vol. 1 (6), 748-753 (2010)
f ′′′ −
(θ − θ r ) θr
θ − 2δ
ff ′′ −
(θ − θr ) θr
(θ − θr ) f ′′θ ′ − 2λ θr (θ − θ r )
C f Re x = Nu x
− 2Mf ′ = 0
Re x
=−
750
2θ r f ′′(0, θ r ) (θ r − 1) 1 2
θ ′(0, θ r )
(6)
θ ′′(1 + R ) + Pr fθ ′ = 0
(7)
The transformed boundary conditions are given by:
f ′(0) = 1 f ′(∞ ) → 0
f (0) = 0 θ (∞) → 0
(8)
Tr − T∞ 1 =− Tw − T∞ γ (Tw − T∞ )
SGr Re x
M =
2
Gr =
16σT∞3 σΒ 02 x R= ρ ∞U w 3kχ
− θ ' ( 0)
θr
0.44651189 0.44998875 0.45196875
Pr
0.71 7
-0.7636335 -0.7928991
0.44651189 1.27831222
M
1 2 5
-0.7636335 -1.437735635 -0.7614701
0.44651189 0.42275189 0.37084991
10
-0.7611201
0.35384961
2
-0.23834932
0.47371222
4
0.75665932
0.5189322
RESULTS AND DISCUSSION
ν2
Pr =
f ' ' (0) -0.7636335 -0.6694365 -0.5945276
λ
gβ (Tw − T∞ )x 3
Values
2 4 10
R
Introducing the following non-dimensional parameters:
λ=
Physical parameters
θ (0) = 1
Where θ r is constant viscosity/temperature parameter defined by
θr =
Table: Skin friction coefficient f ' ' (0) and local Nusselt number − θ ' (0)
ν∞ α (9)
Where λ is buoyancy parameter, M is the magnetic parameter, Pr is the prandtle number, R is the radiation parameter, Gr is the Grashof number for heat transfer. Keeping in view of engineering aspects, the most important characteristics of the flow are rate of skin-friction and heat transfer coefficient, which can be written as
The system of ordinary differential equations (6)-(7) subject to the boundary conditions (8) has been solved by the fourthorder Runge-Kutta scheme with the shooting method. In order to get a clear insight of the physical problem, numerical results have been depicted graphically. Numerical results for the velocity profile f ' (η ) , temperature profile θ (η ) , local skin friction coefficient C f and the local Nusselt number Nu x have been obtained for the various values of parameters, viz.,
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
751
Navneet Joshi et al., J. Comp. & Math. Sci. Vol. 1 (6), 748-753 (2010)
Viscosity/temperature parameter (=2,4,10), Prandtl number (=0.71,7,70), Magnetic parameter (=1,2,3), Radiation parameter (=5,10,15) respectively. It is worth mentioning that small values of Pr (<<1) physically correspond to liquid metals, which have high thermal conductivity but low viscosity, while Pr ~ 1 corresponds to di-atomic gases including air. On the other hand, large values of Pr (>> 1) correspond to high-viscosity oils. The numerical values of the local skin-friction coefficient C f and the local Nusselt number
Nu x are calculated from the equation (9).
The velocity and temperature profiles for various values of Magnetic parameter are presented in figures 2 and 3. These figures show that the velocity decreases with an incease in the values of magnetic parameter, while temperature increases with increase in the value of magnetic parameter. Figure 4 and 5 show that the velocity and temperature profile decreases with increase the value of prandtl number. Figure 6 and 7 show that velocity increases with an increase in the values of buoyancy parameter while temperature decreases with increase in the buoyancy parameter. It is observed that in the presence buoyancy parameter the skin-friction coefficient and the Nusselt number increases. There is no significant variation observed in constant viscosity/temperature parameter and radiation parameter.
Fig 2. Velocity profile with M for θr=2, Pr=0.71, R=1, λ=1 Fig 4. Velocity profile with Pr for θr=2, M=1, R=1, λ=1
Fig 3. Temperature profile with M for θr=2, Pr=0.71, R=1, λ=1
Fig 5. Temperature profile with Pr for θr=2, M=1, R=1, λ=1
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Navneet Joshi et al., J. Comp. & Math. Sci. Vol. 1 (6), 748-753 (2010)
Fig 6. Velocity profile with 位 for 胃r=2, M=1, R=1, Pr=0.71
752
Fig 7. Temperature profile with Pr for 胃r=2, M=1, R=1, 位=1
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Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
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Navneet Joshi et al., J. Comp. & Math. Sci. Vol. 1 (6), 748-753 (2010)
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