Applied Mathematics, 2011, 2, 1432-1436
doi:10.4236/am.2011.212202 Published Online December 2011 (http://www.SciRP.org/journal/am)
Approximate Analytical Solutions for the Nonlinear
Brinkman-Forchheimer-Extended Darcy Flow Model
Basant K. Jha1, Muhammad L. Kaurangini2*
1Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria
2Department of Mathematical Sciences, Kano University of Science and Technology, Wudil, Nigeria
Received September 6, 2011; revised November 6, 2011; accepted November 13, 2011
Abstract
New approximate analytical solutions for steady flow in parallel-plates channels filled with porous materials
governed by non-linear Brinkman-Forchheimer extended Darcy model for three different physical situations
are presented. These results are compared with those obtained from an implicit finite-difference solution of
the corresponding time dependent flow problem. It is seen that the time dependent flow solutions yield the
almost same steady state values as obtained by using the new approximate analytical solutions
Keywords: Non-Linear, Darcy, Time Dependent Flow, Steady State
1. Introduction
The behaviour of fluid flow in porous media has achi-
eved considerab le attention due to its important practical
applications. Applications include packed-bed catalytic
reactor, geothermal reservoir, drying of porous solids,
shell-side flow model in shell-and tube heat exchanger,
petroleum resources and many others. Reference [1] and
[2] studied the mixed convection flow problems in
porous media based on the model of Darcy law. For flow
through the porous media with solid boundary, [3] pro-
posed the classical boundary term in addition to Darcy’s
law. 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.
In many modern applications, porous media are cha-
racterized by high velocities, i.e., the Reynolds number
based on mean pore size is grater than unity. In such
cases, it is necessary to account for deviation from li-
nearity in the momentum equation for porous media.
This deviation is accounted for by the Forchheimer term
representing the quadratic drag which is essential for lar-
ge particle Reynolds numbers. From the physical point
quadratic drag appears in the momentum equation for
porous media because of large filtration velocities, the
form drag due to the solid obstacles becomes comparable
with the surface drag due to friction [4].
Reference [5] also presented a closed form solution of
the Brinkman-Forchheimer-extended Darcy momentum
equation and the associated heat transfer equation for the
case of fully developed flow with uniform heat flux at
the boundary. They assumed a boundary-layer-type de-
veloped flow and as a consequence their solution is inac-
curate when the inertia parameter is small and Darcy
number approaches and exceeds the value of unity. Refe-
rence [ 6] reco nsidered th e analysis pr esented in [5] with-
out invoking their boundary-layer assumption and de-
rived a more general theoretical solution.
Recent results on the model (Brinkman-Forchheimer
extended Darcy) presented are in [7-10]. In all the resu lts
presented above, no attempt was made to solve the non-
linear equations analytically.
In the present work, flow formation in a parallel plate
channels filled with a fluid saturated porous media is
analyzed analytically and numerically. The flow is des-
cribed by the Brinkman-Forchheimer extended Darcy
equation.
2. Mathematical Model
The physical problem under consideration consists of a
steady laminar fully developed flow between two infi-
nitely long horizontal parallel plates filled with porous
material. The flow formation is caused either by pressure
gradient or (and) by the movement of one of the bound-
ing plates. The fluid is assumed to be Newtonian with
uniform properties and the porous medium is isotropic

B. K. JHA ET AL.1433
and homogeneous. The
x
-axis is taken along one of the
plate while -axis is normal to it. Under the above
mentioned assumption and using the dimensionless pa-
rameters given in the nomenclature, the equation of mo-
tion in porous media which accounts for the boundary
and non-linear inertia term is
y
2
24
d0
d
n
n
uu C
uG
Da
yDa
  , (1)
The first term in the left-hand side of Equation (1) is
the Brinkman term, second is the Darcy and third is the
Forchheimer term


2n, hence the momentum trans-
fer in the porous media is governed by steady Brinkman-
Forchheimer extended Darcy model.
The boundary conditions in dimensionless form are:
at 0
0 at y1
uB y
u


(2)
The above equations have been rendered in dimen-
sionless form by using the non-dimensional parameters
defined in nomenclature.
3. Analytical Solutions
By introducing the assumption into Equation
(1) it becomes
1n
u 


2
24
d0
dn
uu CuG
Da yDa


 
(3)
Equation (3) has the so lutions in Subsections 3.1-3.3.
3.1. Couette Flow [G = 0.0 and B = 1.0]
 


Sinh 1.
Sinh
y
uy



 (4)
where 4
11
n
C
Da Da

 
3.2. Pressure Driven Flow [B = 0 and G ≠ 0.0]
  

2
SinhSinh 1.1.
Sinh
yy
G
uy

 
 









(5)
3.3. Generalized Couette Flow [B = 1.0 and G ≠
0.0]
 


 



2
Sinh 1.
Sinh
Sinh 1.Sinh 1.
Sinh Sinh
y
uy
yy
G













The Equations (4) to (6) can be used to find the values
of the dimensionless velocity as a function of dimen-
sionless distance in the interval [0,1] at the iteration
u
y


1i

in terms of value of

at the iteration . It
should be noted here that i

is function of

which
really stands for


y
i
u. Thus Equation s (4 ) to (6) can be
written in the following algorithmic form

 
1, i
uyFyuy

 
i
(7)
4. Numerical Solution
The analytical solutions of the previous section are valid
for steady state momentum transfer in porous medium
containing Darcy, Brinkman and Forchheimer terms. To
explore the limits of validity of these analytical solutions
and to extend our investigation to time dependent mo-
mentum transfer in porous medium, numerical solution
of the time dependent problem is obtained using implicit
finite difference approach.
Consider the dimensionless form of time dependent
momentum equation
2
240
n
n
uuuC
uG
tDa yDa



 
 (8)
The first term in the right-hand side of Equation (8) is
the Brinkman term, second is the Darcy and third is the
Forchheimer term


2n

, hence the momentum trans-
fer in the porous media is governed by time dependent
Brinkman-Forchheimer extended Darcy model.
The initial and boundary conditions in dimensionless
form for the present problem are:
0, for all when 0,
at 0: ,
at1: 0, for 0
uy
yuB
yu t
t



 
(9)
The equations above also have been rendered in di-
mensionless form by using the non-dimensional parame-
ters defined in nomenclature.
The numerical solution of Equation (8) using the ini-
tial and boundary conditions (9) is obtained by discreti-
zation of the momentum Equation (8) into the finite defe-
rence equation at the grid points . They are in or-
der as follows:

,ij




(6)









 

2
4
,,1 1,2,,
,
,n
n
uij uijuijuij uij
ty
Cui j
uij G
Da Da

1




 

(10)
Here the index i refers to y and j to t. The partial time
derivative is approximated by the backward difference

B. K. JHA ET AL. 1434
formula, while the second-order partial space derivative
is approximated by the central difference formula. The
above equation is solved by Thomas algorithm by ma-
nipulating into a system of linear algebraic equations in
the tri-diagonal form.
In each time step, the process of numerical integratio n
for every dependent variable starts from the first neigh-
boring grid point of the plate at and proceeds
towards the another plate using the tri-diagonal form of
the finite difference Equation (10) until it reaches at im-
0y
mediate grid point of the plate at . 1y
In each time step the velocity field is obtained. The
process of computation is advanced until a steady state is
approached by satisfying the following convergence cri-
terion:
,1 ,6
max
10
ij ij
AA
MA


 (12)
with respect to velocity field.
Here ,ij
A
represents the velocity field,
M
is the
number of interior grid points and max A is the maxi-
mum absolute value of ,ij
A
.
In the numerical computation special attention is need-
ed to specify t

to get a steady state solution as rapidly
as possible, yet small enough to avoid instabilities.
It is set, which is suitable for present computation, as

2
tStabr y

 (13)
The parameter is determined by numerical
experimentation in order to achieve convergence and sta-
bility of the solution procedure. Numerical experiments
show that the value 2 is suitable for numerical computa-
tions.
Stabr
5. Results and Discussion
For the Brinkman-Forchheimer extension of Darcy equa-
tion to model the flow in a porous media (n = 2), B = 1.0,
γ = 1.0, C = 0.52, Da = 0.01, and G = + 10.0, 0.0 and
–10.0, the solutions of Equation (1) have been compared
with the implicit finite-difference solution of Equation (8)
in Tables 1, 2 and 3, for Couette flow, pressure driven
flow and generalized Couette flow respectively.
Table 1. B = 1.0, γ = 1.0, Da = 0.01, G = 0.0 & C = 0.52.
y ANALYTICAL SOLUT ION NUMERICAL SOLUTION
(IMPLICIT FINITE-DIFFERENCE SOLUTION)
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
Table 2. B = 0.0, γ = 1.0, Da = 0.01 & C = 0.52.
G = 10.0 y ANALYTICAL S OLUT ION NUMERICAL SOLUTION
(IMPLICIT FINITE-DIFFERENCE SOLUTION)
0.0 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

B. K. JHA ET AL.1435
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. B = 1.0, γ = 1.0, Da = 0.01 & C = 0.52.
G = 10.0 y ANALYTICAL SOLUTION NUMERICAL SOLUTION
(IMPLICIT FINIT E-DIFFERENCE SOLU TION)
0.0 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

B. K. JHA ET AL. 1436
From the results presented in Tables, it can be noticed
that the approximate analytical solutions presented in this
work, though it is simple, it gives good and accurate re-
sults, and hence it can be efficiently used to solve this
class of nonlinear differential equation models.
6. References
[1] W. J. Minkowycz, P. Cheng and R. N. Hirschberg, “Non-
similar Boundary Layer Analysis of Mixed Convection
about a Horizontal Heated Surface in a Fluid—Saturated
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doi:10.1016/0735-1933(84)90017-4
[2] M. P. Cheng and C. H. Chang, “Mixed Convection about
a Nonisothermal Cylinder and Sphere in a Porous Me-
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[3] H. C. Brinkman, “A Calculation of the Viscous Force
Exerted by a Flowing Fluid on a Dence Swarm of Parti-
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pp. 727-734.
[4] D. A. Nield and A. Bejan, “Convection in Porous Me-
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[5] K. Vafai and S. J. Kim, “Forced Convection in a Channel
filled with a Porous Medium: An Exact Solution,” Tran-
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[6] D. A. Nield, S. L. M. Junqueira and J. L. Lage, “Forced
Convection in a Fluid-Saturated Porous-Medium Channel
with Isothermal or Iso-Flux Boundaries,” Journal of
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doi:10.1017/S0022112096002765
[7] M. A. Al-Nimr and T. K. Aldoss, “The Effect of the Mac-
roscopic Local Inertial Term on the Non-Newtonian Fluid
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[8] B. A. Abu-Hijleh and M. A. Al-Nimr, “The Effect of the
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Nomenclature
d
d
p
x
axial pressure gradient Darcy number, 2
K
H



Da

dimensionless pressu re gra die nt, 3
2
d
d
H
p
x





u


velocity of the fluid
G
dimensionless velocity of the fluid, uH





u

H
total width of the channel
ydimensional co-ordinate t


dimensional time
dimensionless co-ordinate, y
H




dimensionless tim e , 2
t
H





yt

nindex Greek symbols
eff


effective kinematics viscosity of porous medium
Cinertia coefficient
C = dimensionless inertia coefficient,
3
3
22
n
n
CH 
















kinematics viscosity of fluid


ratio of kinematics viscosity
G = dimensionless axial pressure gradient
0
Umotion of the channel wall at 0y
Bdimensionless m otion of the channel wall at 0y


Kpermeability of the porous medium