ANALYTICAL SOLUTIONS OF NONLINEAR BRINKMAN-FORCHHEIMEREXTENDED 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
1
kaurangini@yahoo.com (corresponding author)
Page 1 of 9
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 AlNimr 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 Brinkmanextended 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.
Page 2 of 9
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 x  -axis is taken along one of the plate while y  -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

d 2u u
C


un  P  0 ,
2
n
4
dy
Da
Da
(1)
The slip boundary conditions in dimensionless form are:
u  B  Kn
u  Kn
du
dy
du
dy
at y  0
at y  1
(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

d 2u u
C


uP0
2
4
dy
Da
Da n
(3)
Page 3 of 9
The solution of equation (3) was used in the presentations of the three situations in sections
(3.1)-(3.3)
3.1 COUETTE FLOW  P  0.0 and B  1.0
u  y   c1cosh( λ y)  c2sinh( λ y)
(4)
3.2 PRESSURE DRIVEN FLOW  B  0 and P  0.0
u  y   c1cosh( λ y)  c2sinh( λ y)  X 3
(5)
3.3 GENERALIZED COUETTE FLOW  B  1.0 and P  0.0
u  y   c3cosh( λ y)  c2sinh( λ y)  X 3
Where  
𝑋3 =
𝑃 √𝛾
𝜆2
1

(6)
1
C
X1 = Knλcosh(λ) − sinh(λ),

Da 4 Da n
,
, 𝑋4 = 1 −
𝐾𝑛𝜆𝑋2
𝐾𝑛𝜆𝑋3
𝑋1
𝑋1
, 𝑋5 = 𝐵 +
X2 = cosh(λ) − Knλsinh(λ),
𝑋5
𝑋2
𝑋3
− 𝑋3, 𝑋6 = 𝑋5 − 𝐵, 𝐶1 = 𝑋4, 𝐶2 = 𝐶1 (𝑋1) + 𝑋1,
𝑋6
𝐶3 = 𝑋4
Page 4 of 9
TABLE 1
B  1.0,   1.0, Da  0.01, G  0.0 & C  0.52
Jha and Kaurangini (2011)
Numerical solution
Y
0.0
1.00000
1.00000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.36443
0.13439
0.04959
0.01828
0.00673
0.00248
0.00091
0.00033
0.00011
0.00000
0.36457
0.13381
0.04923
0.01813
0.00668
0.00246
0.00090
0.00032
0.00010
0.00000
G=10.0
y
TABLE 2
B  0.0,   1.0, Da  0.01 & C  0.52
Jha and Kaurangini
Numerical solution
(2011)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00000
0.06306
0.08611
0.09450
0.09745
0.09817
0.09745
0.09450
0.08611
0.06306
0.00000
0.00000
0.06296
0.08606
0.09448
0.09744
0.09816
0.09744
0.09448
0.08606
0.06296
0.00000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00000
-0.06335
-0.08676
-0.09536
-0.09840
-0.09915
-0.09840
-0.09536
-0.08676
-0.06335
0.00000
0.00000
-0.06331
-0.08671
-0.09532
-0.09837
-0.09912
-0.09837
-0.09532
-0.08671
-0.06331
0.00000
G=10.0
Page 5 of 9
G=10.0 y
TABLE 3
B  1.0,   1.0, Da  0.01 & C  0.52
Jha and Kaurangini (2011)
Numerical solution
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.00000
0.42608
0.21942
0.14532
0.11546
0.10478
0.09988
0.09539
0.08643
0.06316
0.00000
1.00000
0.42646
0.21891
0.14313
0.11527
0.10470
0.09983
0.09536
0.08637
0.06306
0.00000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.00000
0.30250
0.04874
-0.04517
-0.07984
-0.09229
-0.09587
-0.09443
-0.08643
-0.06324
0.00000
1.00000
0.30234
0.04808
-004548
-0.07992
-0.09229
-0.09584
-0.09439
-0.08637
-0.06320
0.00000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.00000
0.36443
0.13439
0.04959
0.01828
0.00673
0.00248
0.00091
0.00033
0.00011
0.00000
1.00000
0.36457
0.13381
0.04923
0.01813
0.00668
0.00246
0.00090
0.00032
0.00010
0.00000
G=10.0
Gr=0.0
Page 6 of 9
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
Page 7 of 9
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
dp
 axial pressure gradient
dx
 H 3 dp 
P  dimensionless pressure gradient,  

2
  dx 
H  total width of the channel
Page 8 of 9
y   dimensional co-ordinate
 y 
y  dimensionless co-ordinate,  
H
n  index
C   inertia coefficient
  n  2 3 32n  
C  dimensionless inertia coefficient,  C  H   




U 0  motion of the channel wall at y   0
B  dimensionless motion of the channel wall at y   0
K  permeability of the porous medium
 K 
Da  Darcy number,  2 
H 
u   velocity of the fluid
 u H 

  
u  dimensionless velocity of the fluid, 
Greek symbols
 eff  effective kinematics viscosity of porous medium
  kinematics viscosity of fluid
  ratio of kinematics viscosity
Page 9 of 9