ANALYTICAL SOLUTIONS OF NONLINEAR BRINKMAN-FORCHHEIMER-EXTENDED DARCY FLOW MODEL WITH SLIP BOUNDARY CONDITIONS
M.L. Kaurangini*[1] and Basant K. Jha**
*Department of Mathematics
Kano University of Science and Technology, Wudil- Nigeria
** Department of Mathematics
Ahmadu Bello University, Zaria- Nigeria
ABSTRACT
Steady flows in parallel-plates channels with boundary slip conditions filled with porous materials and governed by non-linear Brinkman-Forchheimer extended Darcy model for three different physical situations are presented. The approximate analytical solutions of the nonlinear model suggested in Jha and Kaurangini (2011) was utilized here. The results obtained are compared with those found in the literature. It is seen that when Knudsen number is zero in the solutions yield almost same values as obtained in the literature.
Key words: Non-linear, Darcy number, Knudsen number, steady flow, Forchheimer
1. INTRODUCTION
The study of fluid flow through channel with boundary slip conditions has applications in developing microelctrochemical system (MEMs) and biomedical applications such as drug delivery, DNA sequencing and bioMEMs. Recently studies on the subject were carried out in Haddad et al. (2005) and Gad el Hak (2001) theoretically.
Analytical studies of the fully developed natural convection in an open-ended vertical parallel plate microchannel with asymmetric wall temperature distributions are presented in Aydin and Avei (2006). Analytical investigation of transient hydrodynamics and thermal behaviors of fluid flow in an open-ended vertical parallel-plate microchannel under the effect of the hyperbolic heat conduction model was presented in Chen and Weng (2005). Also numerical investigation of developing hydro dynamical behaviors of free convection gas flow in a vertical open-ended parallel-plate microchannel filled with porous media was presented in Khadrawi et al (2005).
Porous media flow are characterized by high velocities in many modern applications, i.e., the Reynolds number based on mean pore size and is greater than unity. In such cases, it is necessary to account for deviation from linearity in the momentum equation for porous media. This is accounted for by the Forchheimer term representing the quadratic drag which is essential for large particle Reynolds numbers. Physically, quadratic drag appears in the momentum equation for porous media because of large filtration velocities. The form drag is due to the solid obstacles becomes comparable with the surface drag due to friction Nield and Bejan (1992). Recently Al-Nimr and Aldoss (2004) studied the mixed convection flow problems in porous media based on the model of Darcy law. Khadrawi and Al-Nimr (2003) proposed the classical boundary term in addition to Darcy’s law for flow through the porous media with solid boundary. This Brinkman-extended Darcy model analyzed the no slip boundary condition at the wall and showed that although the wall shear resistance has little influence on the pressure drop, it has drastic effects on stream-wise velocity component and heat transfer rate at the interface between porous media and solid boundary. Other results on the Brinkman-Forchheimer extended Darcy model are recently presented in and Abu-Hijleh and Al-Nimr (2001) and Kaurangini and Jha (2011). In all the results presented above, no attempt was made to solve the nonlinear equations analytically with boundary slip conditions.
The present work is an extension of Jha and Kaurangini (2011) to boundary slip conditions.
2. MATHEMATICAL MODEL
A steady laminar fully developed fluid flow between two infinitely long horizontal micro parallel plates with slip boundary conditions and filled with porous material is considered. The fluid flow is driven either by pressure gradient or (and) by the movement of one of the bounding plates. The fluid is assumed to be Newtonian with uniform properties and the porous medium is isotropic and homogeneous. The -axis is taken along one of the plate while -axis is normal to it. Under these assumptions and using the dimensionless parameters given in the nomenclature, the equation of fluid motion which accounts for the non-linear inertia term as in Jha and Kaurangini (2011) is
, (1)
The slip boundary conditions in dimensionless form are:
(2)
Hence, the momentum transfer in the channel is governed by steady Brinkman-Forchheimer extended Darcy model.
3. ANALYTICAL SOLUTIONS
The assumption used in solving the flow equation introduced in Jha and Kaurangini (2011) was utilized here in equation (1) and it becomes
(3)
The solution of equation (3) was used in the presentations of the three situations in sections (3.1)-(3.3)
3.1 COUETTE FLOW
(4)
3.2 PRESSURE DRIVEN FLOW
(5)
3.3 GENERALIZED COUETTE FLOW
(6)
Where , X1=Knλcoshλ-sinh(λ), X2=coshλ-Knλsinh(λ), X3=Pγλ2, X4=1-KnλX2X1, X5=B+KnλX3X1-X3, X6=X5-B, C1=X5X4, C2=C1X2X1+X3X1, C3=X6X4
TABLE 1
Y / Jha and Kaurangini (2011) / Numerical solution0.0 / 1.00000 / 1.00000
0.1 / 0.36443 / 0.36457
0.2 / 0.13439 / 0.13381
0.3 / 0.04959 / 0.04923
0.4 / 0.01828 / 0.01813
0.5 / 0.00673 / 0.00668
0.6 / 0.00248 / 0.00246
0.7 / 0.00091 / 0.00090
0.8 / 0.00033 / 0.00032
0.9 / 0.00011 / 0.00010
1.0 / 0.00000 / 0.00000
TABLE 2
G=10.0 / y / Jha and Kaurangini (2011) / Numerical solution0.0 / 0.00000 / 0.00000
0.1 / 0.06306 / 0.06296
0.2 / 0.08611 / 0.08606
0.3 / 0.09450 / 0.09448
0.4 / 0.09745 / 0.09744
0.5 / 0.09817 / 0.09816
0.6 / 0.09745 / 0.09744
0.7 / 0.09450 / 0.09448
0.8 / 0.08611 / 0.08606
0.9 / 0.06306 / 0.06296
1.0 / 0.00000 / 0.00000
G=-10.0
0.0 / 0.00000 / 0.00000
0.1 / -0.06335 / -0.06331
0.2 / -0.08676 / -0.08671
0.3 / -0.09536 / -0.09532
0.4 / -0.09840 / -0.09837
0.5 / -0.09915 / -0.09912
0.6 / -0.09840 / -0.09837
0.7 / -0.09536 / -0.09532
0.8 / -0.08676 / -0.08671
0.9 / -0.06335 / -0.06331
1.0 / 0.00000 / 0.00000
TABLE 3
G=10.0 / y / Jha and Kaurangini (2011) / Numerical solution0.0 / 1.00000 / 1.00000
0.1 / 0.42608 / 0.42646
0.2 / 0.21942 / 0.21891
0.3 / 0.14532 / 0.14313
0.4 / 0.11546 / 0.11527
0.5 / 0.10478 / 0.10470
0.6 / 0.09988 / 0.09983
0.7 / 0.09539 / 0.09536
0.8 / 0.08643 / 0.08637
0.9 / 0.06316 / 0.06306
1.0 / 0.00000 / 0.00000
G=-10.0
0.0 / 1.00000 / 1.00000
0.1 / 0.30250 / 0.30234
0.2 / 0.04874 / 0.04808
0.3 / -0.04517 / -004548
0.4 / -0.07984 / -0.07992
0.5 / -0.09229 / -0.09229
0.6 / -0.09587 / -0.09584
0.7 / -0.09443 / -0.09439
0.8 / -0.08643 / -0.08637
0.9 / -0.06324 / -0.06320
1.0 / 0.00000 / 0.00000
Gr=0.0
0.0 / 1.00000 / 1.00000
0.1 / 0.36443 / 0.36457
0.2 / 0.13439 / 0.13381
0.3 / 0.04959 / 0.04923
0.4 / 0.01828 / 0.01813
0.5 / 0.00673 / 0.00668
0.6 / 0.00248 / 0.00246
0.7 / 0.00091 / 0.00090
0.8 / 0.00033 / 0.00032
0.9 / 0.00011 / 0.00010
1.0 / 0.00000 / 0.00000
CONCLUSION
An attempt was made to solve analytically the Brinkman-Forchheimer extended Darcy flow formation in a parallel plate channels with boundary slip conditions and filled with fluid saturated porous media is presented utilizing the transformation suggested in Jha and Kaurangini (2011). In particular it was noticed:
· There was excellent agreement with the result in Jha and Kaurangini (2011) presented in tables 1, 2 and 3
· There was also excellent agreement with the result found using the traditional numerical method (finite difference method).
· The result will help in getting analytical solution of nonlinear differential equations
REFERENCES
1. M.A. Al-Nimr and T.K Aldoss, “The effect of the macroscopic local inertial term on the non-Newtonian fluid flow in channels filled with porous medium”, Int. J. of Heat and Mass Transfer, Vol. 47, 2004, pp. 125-133.
2. B.A. Abu-Hijleh, M.A Al-Nimr, “The effect of the local inertial term on the fluid flow in channels partially filled with porous material” Int. J. Heat Mass Transfer Vol. 44, 2001, pp. 1565-1572.
3. A.F. Khadrawi, M.A. Al-Nimr “The effect of the local inertial term on the free convection fluid flow in vertical channels partially filled with porous media” J. Porous Media Vol. 6, No. 1, 2003, pp.1-12.
4. M.L.Kaurangini and Basant K. Jha (2011) Effect of inertial on the Generalized Couette Flow in Composite parallel-plates with Uniform Porous Medium in the presence of Suction/Injection Modelling, Simulation and Control, series B, Mechanics and Thermics Vol. 80 No. 2 (AMSE) Spain pp. 53-70
5. D.A. Nield and A. Bejan., Convection in Porous Media, Springer- Verlag, New-York 1992.
6. Basant K. Jha and M.L.Kaurangini “Approximate analytical solutions for the nonlinear Brinkman-Forchheimer-extended Darcy flow model” Applied Mathematics, , Vol. 2, No 2, 2011, 1432-1436, DOI:10.4236/am.2011.212202
7. Haddad, O. M., Abuzaid, M.M. and Al-Nimr, M.A. Developing Free Convection Gas Flow in a vertical Open-Ended Microchannel Filled with Porous Media. Numerical Heat Transfer, Part A, 2005, 48: 693-710.
8. Gad-el-Hak, M. (ed.) The MEMS Handbook CRC Press New York, 2001
9. Aydin, O. and Avei, M. Heat and Fluid Flow Characteristics of Gases in Micro pipes. International Journal of Heat Mass Transfer, 2006, 49, 1723-1730.
10. Chen, C. K. and Weng, H. C. Natural Convection in a vertical microchannel. ASME Journal of Heat Transfer, 2005, 127:1053-1056.
11. Khadrawi, A. F., Othman, A and Al-Nimr, M.A. Transient Free Convection Fluid Flow in a vertical Microchannel as Described by the Hyperbolic Heat Conduction Model. International Journal of Thermophysics, 2005, 26(3): 905-918.
NOMENCLATURE
axial pressure gradient
dimensionless pressure gradient,
total width of the channel
dimensional co-ordinate
dimensionless co-ordinate,
index
inertia coefficient
dimensionless inertia coefficient,
motion of the channel wall at
dimensionless motion of the channel wall at
permeability of the porous medium
Darcy number,
velocity of the fluid
dimensionless velocity of the fluid,
Greek symbols
effective kinematics viscosity of porous medium
kinematics viscosity of fluid
ratio of kinematics viscosity
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[1] (corresponding author)