Mixed Convection Flow Through A Porous Medium Bounded By Two Vertical Walls With Slip Boundary Conditions

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.

1. 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 Brinkman-Forchheimer 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 slip-flow 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.

2. 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 x-axis 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

   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

   2. 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

3. 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=10-1 , for case I and figures 3 and 4 for case II and case III respectively, when Da=10-1 ,10-2and 10-3 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=10-1, 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.

4. 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=10-1.

   Figure 4: Velocity Profiles for Case III for different values of Rv, when Da=10-1, 10-2 and 10-3

   Figure 3: Velocity Profiles for Case II for different values of Rv, when Da=10-1, 10-2 and 10-3

   Figure5: Velocity Profiles for Case I for different values of p and p, when Da=10-1

   Figure 6: Velocity Profiles for Case III for different values of p and p, when Da=10-1

5. References

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2. Ingham DB and Pop I (1998), Transport phenomena in porous media, Pergamon press, Oxford.

3. Chauhan DS and Kumar V, (2012) Radiation Effects on Unsteady Flow through a Porous Medium Channel with Velocity and Temperature Slip Boundary Conditions has discussed, Applied Mathematical Sciences, 36(6), 1759 1769.

4. Mazumdar BS (1981), Taylors diffusion for a natural convection flow through a vertical channel. Int. J. Engineering Sciences 19, 771-779.

5. Chen CH (1997), Analysis of non-Darcian flow phenomena on mixed convection in porous medium adjacent isothermal horizontal plates, Int. J. Heat Mass transfer 40, 2993-2997.

6. Chen YC, Chung JN, Wu CS, Lue YF (2000) Non Darcy mixed convection in vertical channel filled with porous medium, Int. J. Heat Mass transfer 43, 2421- 2429.

7. Sekhar KR, Reddy GVR and Prasad BD, (2012) Chemically reacting on MHD oscillatory slip flow in a planer channel with varying temperature and concentration, Advances in Applied Science Research, , 3 (5):2652-2659.

   Jain NC and Gupta P., (2006), Three dimensional free convection coutte flow with transpiration cooling. J. Zhejiang Univ. SCIENCE A, 7(3), p.p 1-8.

8. Chauhan DS and Kumar V,(2009) Darcy extended Brinkman-Forchheimer model.Turkish J. Eng. Ev. Sci, (33),91-104.

9. Jain NC and Sharma B, (2009), On three dimensional free convection coutte flow with transpiration cooling and tempera-ture jump boundary condition. Int. J. App. Mech. Eng., 14(3), 715-732.

10. Jain NC and Gupta P., (2006), Three dimensional free convection coutte flow with transpiration cooling. J. Zhejiang Univ. SCIENCE A, 7(3), p.p 1-8.

11. Sharma, P K., (2005) 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 Latin American Applied Research, 35, 313-319,.

12. Chaudhary, RC and Jha, AK, (2008), Effects of chemical reactions on MHD micropolar fluid flow past a vertical plate in slip-flow regime, Applied Mathematics and Mechanics, 29 (9), 1179-1194.

13. Khaled, A.R.A. and Vafai, K. (2004) The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions, Int. J. Non-Linear Mech., 39(5), 795-809.

14. Mahmoud, M.A.A. (2010). Chemical reaction and variable viscosity effects on flow and mass transfer of a non-Newtonian visco-elastic fluid past a stretching surface embedded in a porous medium. Journal Meccanica, 10(1007):9292

15. Darek C, Tretheway DC and Mainhart CD, (2002), A series of investigations on slip flow regime, Physics of the fluids. 14, L9.