Approximate Analytical Solutions for the Nonlinear Brinkman-Forchheimer-Extended Darcy Flow Model

doi:10.4236/am.2011.212202

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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

E-mail: *[email protected]

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

-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

uu C

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



(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







uu CuG

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





 (4)

where 4

Da Da







3.2. Pressure Driven Flow [B = 0 and G ≠ 0.0]

  



SinhSinh 1.1.

Sinh























(5)

3.3. Generalized Couette Flow [B = 1.0 and G ≠

0.0]

 



Sinh 1.

Sinh

Sinh 1.Sinh 1.

Sinh Sinh





















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







in terms of value of



at the iteration . It

should be noted here that i



is function of



which

really stands for





u. Thus Equation s (4 ) to (6) can be

written in the following algorithmic form







uyFyuy





(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

240

uuuC

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







, 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

yuB

yu 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)



















 



,,1 1,2,,

uij uijuijuij uij

Cui j

uij G

Da Da











 



(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

ij ij





 (12)

with respect to velocity field.

Here ,ij

represents the velocity field,

is the

number of interior grid points and max

A is the maxi-

mum absolute value of ,ij

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



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

Porous Medium,” International Communications in Heat

and Mass Transfer, Vol. 11, 1984, pp. 127-141.

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-

dium,” Numerical Heat and Mass Transfer, Vol. 8, 1985,

pp. 349-359.

[3] H. C. Brinkman, “A Calculation of the Viscous Force

Exerted by a Flowing Fluid on a Dence Swarm of Parti-

cles,” Applied Scientific Research, Vol. 1, No. 1, 1949,

pp. 727-734.

[4] D. A. Nield and A. Bejan, “Convection in Porous Me-

dia,” Springer-Verlag, New York, 1992.

[5] K. Vafai and S. J. Kim, “Forced Convection in a Channel

filled with a Porous Medium: An Exact Solution,” Tran-

sactions of the ASME (American Society of Mechanical

Engineers) Series C: Journal of Heat Transfer, Vol. 111,

No. 4, 1989, pp. 1103-1106. doi:10.1115/1.3250779

[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

Fluid Mechanics, Vol. 322, 1996, pp. 201-214.

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

Flow in Channels Filled with Porous Medium,” Interna-

tional Journal of Heat and Mass Transfer, Vol. 47, 2004,

pp. 125-133. doi:10.1016/S0017-9310(03)00382-X

[8] B. A. Abu-Hijleh and M. A. Al-Nimr, “The Effect of the

Local Inertial Term on the Fluid Flow in Channels Par-

tially Filled with Porous Material,” International Journal

of Heat Mass Transfer, Vol. 44, No. 8, 2001, pp. 1565-

1572. doi:10.1016/S0017-9310(00)00207-6

[9] A. F. Khadrawi and 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,”

Journal of Porous Media, Vol. 6, No. 1, 2003, pp. 1-12.

doi:10.1615/JPorMedia.v6.i1.40

[10] M. L. Kaurangini and B. K. Jha, “Effect of Inertial on

Generalized Couette Flow in Composite Parallel Plates

with Uniform Porous Medium in the Presence of Suction

and Injection,” International Journal of Modelling, Simu-

lation and Control (AMSE), 2011, Article in Press.

Nomenclature

axial pressure gradient Darcy number, 2









dimensionless pressu re gra die nt, 3















velocity of the fluid

G

dimensionless velocity of the fluid, uH













total width of the channel

ydimensional co-ordinate t





dimensional time

dimensionless co-ordinate, y









dimensionless tim e , 2











yt



nindex Greek symbols

eff





effective kinematics viscosity of porous medium

Cinertia coefficient

C = dimensionless inertia coefficient,































kinematics viscosity of fluid





ratio of kinematics viscosity

G = dimensionless axial pressure gradient

Umotion of the channel wall at 0y

Bdimensionless m otion of the channel wall at 0y





Kpermeability of the porous medium