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 Total Downloads : 570
 Authors : Akhilesh K. Mishra
 Paper ID : IJERTV2IS3539
 Volume & Issue : Volume 02, Issue 03 (March 2013)
 Published (First Online): 26032013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mixed Convection Flow Through A Porous Medium Bounded By Two Vertical Walls With Slip Boundary Conditions
Akhilesh K. Mishra,
Department of Mathematics, Gombe State University, Gombe, Nigeria
Abstract
This paper investigates the steady fully developed mixed convection flow between two vertical walls filled with porous materials having slip boundary for the velocity and temperature. The analytic solutions for the velocity and temperature profiles has been obtained for different cases depending upon the values of the Darcy number, Rayleigh number and the ratio of the effective viscosity of the porous domain to the viscosity of the fluid. The effects of the various parameters entering into the problem, on the velocity and the temperature are depicted graphically and on the skin friction is depicted in tabular form, and discussed in detail.

Introduction
The study of flow and heat transfer through a porous medium has become of main interest in science and technology because of several engineering applications, particularly when the fluid flow is caused by shearing motion of a plate. The mechanism of mixed convection in the porous media has important applications in the utilization of geothermal energy. The recent books by Nield and Bejan [1] and Ingham and pop [2] have extensively documented the works devoted in this area.
Radiation Effects on Unsteady Flow through a Porous Medium Channel with Velocity and Temperature Slip Boundary Conditions has discussed by Chauhan and Kumar [3]. Mazumdar [4] studied the dispersion of solute in natural convection flow through a vertical channel with a linear axial temperature variation. Chen [5] has presented the effects of non Darcian flow phenomena on mixed convection in porous medium adjacent isothermal horizontal plates. Chen et. al. [6] have investigated the fully developed mixed convection in a vertical porous channel with a uniform heat flux imposed at the plates by using the BrinkmanForchheimer extended Darcy model. Sekhar, Reddy and Prasad [7] investigated chemically reacting on MHD oscillatory slip flow in a planer channel with varying temperature and concentration. Effects of slip conditions on forced convection and entropy generation in a circular channel occupied by a
highly porous medium: Darcy extended Brinkman Forchheimer model analysed by Chauhan and Kumar [8].
Jain and Sharma [9] and Jain and Gupta [10] have studied three dimensional coutte flows with slip boundary conditions and suction velocity varies sinusoidaly. Sharma [11] investigate the effect of periodic heat and mass transfer on the unsteady free convection flow past a vertical flat plate in slip flow regime when suction velocity oscillates in time. Chaudhary and Jha [12] studied the effects of chemical reactions on MHD micropolar fluid flow past a vertical plate in slipflow regime. Khaled [13] investigated the effect of slip condition on Stokes and Couette flows due to an oscillating wall. A more insight into the subject of the slip flow regime is given by Mahmoud, [14]. A series of investigations have been made on slip flow regime viz. Derek et.al [15].
In this paper, steady fully developed free and forced convection flow through porous bounded by two walls is considered. The boundary conditions for velocity and temperature are slip boundary conditions at both the walls. In which there is a uniform axial temperature variation along the walls and analytic solutions have been obtained for velocity and temperature profile for different cases and the effects of the pertinent parameters on the flow and temperature fields are examined and discussed.

Mathematical Analysis
Consider the fully developed laminar tree convection flow between two vertical walls filled with a fluid saturated porous medium under a constant pressure gradient. The walls are separated by a distance 2L apart and having an axial temperature variation. The xaxis is taken along the vertical direction while y axis is perpendicular to it. For fully developed laminar flow, the velocity has only the vertical component and is a function of y only.
As a result of these assumptions, the equation of motion in x direction and energy equation are obtained as follows
L3 1 p
Px
1 g x g Nx
(6)
t 0 x
The equation (1) and (2) in dimensionless form obtained as follows
d 2u u
Rv dy 2
Ra 1
Da
(7)
d 2
dy 2
u 0
(8)
In the above equations, Rv, Da and Ra are defined as
Figure 1. Physical configuration of the model
0 p g
2u
f
f
Rv
ff ,
f
Da k ,
L2
Ra
gNBL4
f
u
u
T
x
2T
eff
y 2
k
(1)
h L1 ,
1 L
h L2
2 L
(9)
u x
y 2
(2)
Where Rv= ratios of viscosities
The boundary conditions for the velocity and temperature fields are
Ra= Rayleigh Number Da= Darcy Nimber
u L1
du
,
T T0 Nx L2
dT
at y L
p= Velocity slip parameter
p= Temperature slip parameter.
dy dy
u L du ,
1 dy
T T Nx L dT
0 2 dy2
at y L
The boundary conditions for the model are as follows:
du d
u p dy , p dy , at y 1

u h du ,
h
d , at y 1
Under the usual Bossinesqs approximation the equation of state is assumed to be
1 dy
2 dy
(10)
1 T T
Now by using (7) and (8) resulted into a fourth order
0 0
differential equation in u as

d 4u
1 d 2u du
After assuming the uniform the uniform axial
Rv dy 4
Ra 0
Da dy 2 dy
(11)
temperature variation along the channel walls, the temperature of the fluid can be written as
The auxiliary roots m1, m2 , m3 and m4 of the above equations are as follows:
T T0 Nx (y )
Using (1) and (5) and introducing dimensionless quantities
m1, m2
1 A
2DaRv
(12)
y y ,
L
u uL ,
Px
NLPx
m , m – 1 A
3 4 2DaRv
Where,
A 1 4RaRvDa2
u(y) = B 1 cosh(Ey) B 2
1 1
1 1
( y) 1 B E cosh(Ey) B 2
(16)
(17)
Ra RaDa.
(13)
The Auxiliary equation roots given by the equations
(12) and shows that the solution for u and depends on
where E 1
DaRv,
, E 1
1 Rv E2
RaDa Ra
the values of Da2, Ra and Rv. Thus three different cases arise and the solutions have been obtained as under.
Case I: When 0 A 1.
The solution for u and by solving (7) and using (8) and (10) is obtained as
u( y) C1 cosh(m1 y) C2 cosh(m 2 y) (14)
E 2 (RaDaE 1 1) cosh(E), E 3 =(RaDaE1 h 2 Ep) sinh(E)
E 4 Eh 1 sinh(E) cosh(E),
B DaE 4 , B Da
2 3 2 3
2 3 2 3
2 (E E ) 1 (E E )
Case III: When A>1.
The velocity and temperature profiles for this case by using the boundary conditions are as follows:
( y) 1 A C cosh(m y) A C cosh(m y) (15)
Ra 1 1 1 2 2 2
u(y) = C3cosh(m1 y) C4 cos(m 2 y) (18)
where A
1 Rv m2
( y) 1 D C cosh(m y) D C cos(m y) (19)
1 RaDa Ra 1
Ra 1 3 1 2 4 2
A 1

Rv m2
Where D
1 m2 Rv
2 RaDa Ra 2
A 3 m1 h 1 cosh(m 2 ) sinh(m1) m 2 h 2 sinh(m 2 ) cosh(m1)
A 4 h 2 m 2 sinh(m 2 ) cosh(m 2 ) h 2 m 1 sinh(m1) cosh(m 2 )
1 RaDa 1 Ra
11 Rv
5 1 2 1 2 1 2 1 2 1 2 1 2
5 1 2 1 2 1 2 1 2 1 2 1 2
A (m 2 m 2 cosh(m ) cosh(m ) m m h h (m 2 m 2 ) sinh(m ) sinh(m )
D m2
2 RaDa 2 Ra
A 6 (m1 h 2 sinh(m1 ) cosh(m 2 ) m 2 h 1 sinh(m 2 ) cosh(m1 )
A 7 m1 h 2 sinh(m1 ) cosh(m 2 ) m 2 h 2 sinh(m 2 ) cosh(m1 )
A 1 ( A A ),
D 3 D 2 cosh(m 2 ) h 2 D 2 m 2 sinh(m 2 )
D 4 cosh(m1 ) m1 h 1 sinh(m1)
D 5 cosh(m 2 ) p m 2 sinh(m 2 )
8 Ra 3 4
9 5 1 6 2
9 5 1 6 2
y
y
A RvA m2 A m2 A
A 10 cosh(m1 ) m1 h 1 sinh(m1 )
D 6 D 1 cosh(m1 ) h 2 D 1 m 1 sinh(m1 )
D 7 Ra(D 3 D 4 D 5 D 6 )
D D
A 11 A8 A9 ,
C 5 , C
3 D
4
4 D
7 7
A12 cosh(m 2 ) p m 2 sinh (m 2 )
C A10 , C A12
2 A 1 A
The expressions for the skin friction in non
dimensional form for the different cases are obtained
11 11
Case II: When
A 1.
by using the relation
du
dy y 1
The velocity and temperature profiles for this case by using the boundary conditions are as follows:
They are as follows:
Case I
C1m1 sinh(m1) C2m2 sinh(m2 )
Case II
Case III
B1E1 sinh(E1 )
parabolic type up to certain values of Rayleigh number. As usual, the velocity increases with increase of Darcy number (Da) due to fact that Darcy number is directly proportional to permeability (K) of the medium, but the reverse situation occurs in the case of ratio of viscosities (Rv). It is also evident that velocity decreases with the increase of Rayleigh number.
C3m1 sinh(m1) C4m2 sin(m2 )
In figures 5 and 6 the impact of
(p, p ) on


Result and Discussion
In order to point out the effects of different parameters on velocity (u), following discussions are set out. Numerical calculations are carried out for different values of the Darcy number (Da), ratio of viscosities, (Rv), Rayleigh number (Ra), velocity and temperature slip parameters (p, p ) . The results have been shown graphically for the three cases A 1.
The Figure 1 illustrates the physical configuration of the model. Both the walls are separated by distance 2L and other parameters are explained there.
The velocity profiles for different values of Ra(10, 15 ) and Rv (1.0, 1.5, 2.0) are shown in figure 2, when Da=101 , for case I and figures 3 and 4 for case II and case III respectively, when Da=101 ,102and 103 and p and p is fixed. As expected the flow is symmetrical about y=0. These figures demonstrate that the flow is
the velocity profiles have been shown. Figures 5 and 6 illustrates the velocity profiles for the case I and case III respectively when Da=101, Ra=, p =0.4, 0.8, 1.2 and p = 0.2, 0.4, 0.6. It noticed that the velocity increases with the increase of slip parameter p, which, represents that the increase in the slip parameter has the tendency to reduce the friction forces which increases the fluid velocity. While reverse phenomena occurs in the case of temperature slip parameter p.

Conclusion
The present study investigates the fully developed mixed convection flow of an incompressible viscous fluid between two vertical walls filled with porous medium saturated by the same fluid. The Brinkman Darcy model is used to analyse the porous domain. The velocity profiles increase with the Darcy number while decreases with ratios of viscosities. The velocity slip parameter promotes the velocity profiles while temperature slip parameter has reverse impact on velocity profiles.
Figure 2: Velocity Profiles for Case I for different values of Ra and Rv, when Da=101.
Figure 4: Velocity Profiles for Case III for different values of Rv, when Da=101, 102 and 103
Figure 3: Velocity Profiles for Case II for different values of Rv, when Da=101, 102 and 103
Figure5: Velocity Profiles for Case I for different values of p and p, when Da=101
Figure 6: Velocity Profiles for Case III for different values of p and p, when Da=101

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