International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Heat Transfer in Adjacent Flow of Two
Immiscible Fluids Bounded by Two Parallel
Porous Beds− Darcy Model
Roger Bostic1, Sreedhara Rao Gunakala1, SambasivaRao Baragada2
1
Dept. of Mathematics and Statistics, The University of West Indies, St. Augustine, Trinidad and Tobago.
2
Dept. of Computer Science, Govt. Degree College, Khairatabad, AP, India-500004
ABSTRACT
The Darcy model of flow of two immiscible fluids bounded by two parallel porous beds is used along with the Beavers and Joseph
slip condition. The exact solution is found and compared with the two, four and eight element solutions obtained using the finite
element method. The effects of the permeability parameter and slip parameter on the exact solution are investigated and presented
in the paper.
Keywords: Finite elements, permeability, finite element method, incompressible fluid flow, slip condition, temperature
distribution
1. INTRODUCTION
The study of heat transfer in two immiscible fluids between two parallel porous plates is important in several applications
of Engineering[15]. Such specific model has been earlier investigated by S. Ramakrishna, D. Nagaveni and Rao [1] where
the velocity field was found exactly using finite element method subjected to the Darcy model of laminar flow.
Malashetty, Umavathi and Kumar [5] investigated the nature of steady flow between plates for an inclined plane. K. Sai
[6] examined the unsteady flow of two immiscible viscous fluids over a naturally permeably bed. Investigations were
carried by K. Vajravelu, Arunachalam, S. Sreenadh [2] to examine the nature of unsteady flow of two immiscible
conducting fluids between two permeable beds. Both Sai and Vajravelu et. al. [6, 7] used the Beavers and Joseph’s slip
condition [4] at the boundary and Darcy’s law in the porous region. A study on the flow of a magnetohydrodynamic
viscous fluid was done by S. Ganesh and S. Krishnambal [8].
Rao, Bhatt and Comissiong [3] have presented the nature of heat transfer in differentially heated plates. They have
evaluated velocity and temperature fields along with the varying range of shear stress and Nusselt numbers using finite
element method. The non-dimensionalization and heat equations used to get an expression for ϴ presented by Rao. et. al.
are considered for this work.
2. FORMULATION
Figure 1: Schematic diagram
Consider the flow of two incompressible fluids between two porous plates as shown in figure 1. The flow is caused by a
constant pressure gradient in the z – direction. The upper and lower porous plates are held at constant temperatures, TU
and TL, respectively. The flow is in the channel is governed by the Navier – Stokes equations. The flow in the porous
layer follows Darcy’s law. This particular problem is an extension of a problem of Finite Element Analysis of an Adjacent
Flow of two Immiscible Fluids between two Porous Beds done by D. Nagaveni et al [1]. In their paper, an expression for
the velocity field, V, was found exactly by using the finite element method. Their work also presents the stress and Darcy
velocities. The exact solution is used for the computation of velocity. The assumptions made were:
 The two fluids are Newtonian and incompressible.
 The lower and upper beds are homogeneous.
 The flow is fully developed and all properties are functions of x only.
 The pressure gradient is constant.
 There is steady heat transfer throughout liquids one and two.
 There is no internal heat source or sink.
Volume 2, Issue 9, September 2013
Page 158
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
The governing equations for velocity and temperature are:
2.1
2.2
2.3
,
2.4
2.5
2.6
The boundary conditions are
2.7
2.1 NONDIMENSIONALIZATION OF QUANTITIES
The quantities are redefined into dimensionless quantities.
For convenience the ‘*’ have been dropped for the rest of the analysis. The resulting non-dimensional equations are:
2.8
2.9
with boundary conditions,
Volume 2, Issue 9, September 2013
Page 159
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
where
and
, here
are the thermal conductivities of the lower and upper fluid respectively and
are the viscosities of the lower and upper fluid respectively.
2.2 EXACT SOLUTION
An exact solution for the velocity and the temperature fields are now computed. Some substitutions are made to equation
2.9 for feasible calculation.
The exact velocity field is:
2.10
where
2.11
2.12
for the temperature field
Prandtl Number.,
2.13
Eckert Number.,
2.14
2.15
Now combine the product
to B1 and B2 for the respective fluids. Substituting 2.10, 2.13 and 2.14 into 2.9 gives,
2.16
Volume 2, Issue 9, September 2013
Page 160
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
After integrating,
2.17
2.18
Again, integrating 2.17 and 2.18,
2.19
2.20
Now using the boundary conditions at x=0,
2.21
where
,
And recalling the boundary conditions at x=1 and x=–1
2.22
where
and
Combining equations 2.19 to 2.22 gives the exact solution for
This solution is shown in the graph below.
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
Figure 2: Exact solution to temperature distribution
3
.
2.3 FINITE ELEMENT SOLUTIONS
According to [9, 10, 11, 13, and 14], the underlying mathematical basis of the finite element method first lies with the
classical Rayleigh-Ritz and Variational calculus procedures. These theories provided the reasons why the finite element
method worked well for the class of problems in which Variational statements could be obtained (e.g., linear diffusion
type problems). However, as interest expanded in applying the finite element method to more types of problems, the use
of classical theory to describe such problems became limited and could not be applied, e.g., fluid-related problems.
Basically, the method requires the governing differential equation to be multiplied by a set of predetermined weights and
the resulting product integrated over space; this integral is required to vanish. Technically, Galerkin’ s method is a part of
the general MWR procedure, since various types of weights can be utilized; in the case of Galerkin’ s method, the weights
are chosen to be the same as the functions used to define the unknown variables. The finite element method now employs
Galerkin’s method to establish the approximations to the governing equations
We now apply the Galerkin finite element method to the differential equations (2.16) and dividing the flow region into
line elements. The following procedure leads to the finite element equations for velocity and temperature in each zone.
Volume 2, Issue 9, September 2013
Page 161
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
k
If V s
and
 sk are approximation functions of Vs and  s in the element ek
refers to the zone-I & II
respectively. We define the errors (residual) for zone- I &II
E sk 
k
and
 dV sk

 dx


  Pi


2
 d sk 
 dVsk 
2.24




 dx   Bs  dx 




are linear combinations of Lagrange polynomials in terms of the respective local nodal values
 sk
Vsk   U kj  kj ,
 sk   t kj  kj ( s  1.2)
j
Where
2.23
d
dx
E sTk 
Where V s
d
dx
j
k
j
 ’s are shape functions which are given in appendix I
These errors are orthogonal to the weight function over the domain e k . Under the Galerkin method, we choose
the approximation functions as the weight functions. Multiply both sides of (2.23 ) and (2.24 ) by the weight functions
and integrate over the domain ek we obtain
E
k
s
 Jk dx  0
ek
E
k
sT
 Jk dx  0
ek
On integrating by parts these line integrals we obtain
k
 dV sk d kj

k
k dV s

P

dx


dx


s
j
j
 dx dx

dx

eK 
eK

 d k d k
s
J
e  dx dx  Bs
K
k
Substituting V s
 dVsk

 dx

2


dVsk
  kj dx    kj
dx

dx


eK
 k d kj d ik 
dx   Ps   kj dx  Q skj
U i 

dx
dx
i 1 
 ek

ek
 d kj d k 
 dV k
i
t ik  
dx   B s   kj  s

i 1

 dx
ek
 ek dx dx
n
sk
dVsk
dx ;
dx
k
 j
ek
2.26
 sk in (2.25) and (2.26), respectively, we get
and
n
where Q j 
2.25
2.27
2

 dx  Q skjT


Q skj    kj
ek
2.28
d sk
dx
dx
We make use of linear polynomial approximations and divide each zone into
matrices for V1 & 1 in terms of the n 1+1 respective global nodal values UJ1 & J1
to zone-1 are given by











a 111
a
1
21
a 112
a
1
22
0
a
2
11
a
0 . . .
0
0 . . .
0
0 . . .
0
2
12
0
a 222
a 222
.
.......... ...
.
.
.
.
.
.
0
0
0
0
.......... ...
n1
0 a n221 -1  a11n 1 a 12
.
.
0
a n211
a n221
Volume 2, Issue 9, September 2013
  U 11   f 1  Q 11 
 1   1


2
  U 2  f 2  f 1   0 
 .
  .  . 
 
 

 
 .  .
  .


  . .  .

.

 
 

 .
  .   0 
 1  
  1n 
n
 U n1 1   f 2 1  Q2 1 
elements. The global
with reference
2.29
Page 162
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
and











where
b111
b
1
21
b112
0
1
22
0 . . . .
0
2
12
0 . . .
.
0
b 222
0 . . .
.
0
.
.
.
2
11
b b b
b 222
0
.
.
.............
.
.............
0
0
0
0
x k 1
a ijk 

xy
  11   f11  Q11 
1
 1   1


2 
   2  f 2  f 1  0 
 .   .
 . 
 
 
 

 .  .
  . 





. 
 .   .
 

  .   . .  .0 
 1  
  1n 
n
  n1 1   f 21  Q 21 
0 b n221 -1  b11n1 b12n1
.
.
d kj  kj
dx dx
xK 1

xK
b n221
x K 1
dx
 dV
Q11k   1
 dx
bijk 
b n211
0
2.30
fi k 


 x  xk
k
d ik  j
dx
dx dx
 d
Q1k   1
 dx

P1k  j dx
xk


 x xk
 dV
Q21k   1
 dx
x K 1
f ik 

xK


 x  xk 1
 dV k
B1  1
 dx
 d
Q2k   1
 dx
2
 k
  i dx



 x  xk 1
Similar global matrices can be obtained for the zone-II.
These global matrices corresponding to the velocity and temperature in each zone can be assembled to obtain for the
velocity and temperature global matrices for the entire flow region, making use of the interface and boundary conditions
with reference to the velocity and temperature.
For computational purposes, we choose four elements in each zone. The corresponding global matrix with reference to the
velocity and temperature in zone I are given by
 1 -1
 1 2

4  0 1

 0 0
 0 0
1
11
0 0 0   U1 
1  Q 1 


1

2  0 
- 1 0 0   U 2 
 
 1  - P1    
2 - 1 0  U 3  
2  0 
8    

- 1 2 - 1   U14 
2
0
 
   
1  Q214 
0 - 1 1   U1 

 5
2.31
and
 1 -1
 1 2

4  0 1

 0 0
 0 0
where
1
11
0 0 0  1 
2 Q1 



1



- 1 0 0   2 
4 0 
1
 1  B1bi   

2 - 1 0   3  
4  0 
8   
  1
-1 2 -1   4
4
0 
 
  

2 Q214 
0 - 1 1   1 
5
 
 dV
Q11   1
 dx

 dV

Q51   1
 x  1
 dx
 d 
Q11   1 
 dx  x  1
2.32


 x 0
 d 
Q51   1 
 dx  x 0
Similarly the global matrix equation for the velocity and temperature may be obtained in the Zones - II. The expressions
for the velocity & temperature in each zone may be represented in term of the respective global nodal values.
Making use of the boundary conditions, interfacial conditions in terms of nodal values and the balance of the secondary
variables with respect to the velocity and temperature respectively given by
The boundary conditions:
U11 = U1 = VB1 , U5 2 = U9 = VB2
~
11 =
~
 1 =  (constant) & 52 =  9 = u (constant)
The interfacial continuity conditions:
Volume 2, Issue 9, September 2013
Page 163
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
~
U51 = U1 2 = U5 and
5 1 = 12 =
5
The balance of the secondary variables:
Q 21  Q31  0; Q31  Q41  0; Q 41  Q51  0 ; Q51  Q12  0; Q 22  Q32  0; Q32  Q 42  0; Q42  Q52  0
Q 21  Q31  0; Q31  Q41  0; Q 41  Q51  0 ; Q51  Q12  0; Q22  Q32  0; Q32  Q42  0; Q42  Q52  0
1
2
1
2
and Q1 ,Q5 , Q1 , Q5  0
Assembling the matrix equations of two zones, the global matrix for
reduces to a 9  9 matrix equations with respect to the
.
Solving these 9  9 matrix equations we obtain the solution for
The finite Element solutions for the velocity of two zones;
2.33
U 111 (x)  U 2 12 (x)

2
2
U 21 (x)  U 3 2 (x)
V1  
3
3
U 31 (x)  U 4 2 (x)
U  4 (x)  U  4 (x)
 4 1
5 2
U 515 (x)  U 6 25 (x)

6
6
U 61 (x)  U 7  2 (y)
V2  
7
7
U 71 (x)  U 8 2 (x)
U  8 (x)  U  8 (x)
9 2
 8 1
- 1  x  - 1/4
- 1/4  x  - 1/2
- 1 / 2  x  - 3/4
- 3/ 4  x  0
0  x  1/4
1/4  x  1/2
1 / 2  x  3/4
3/ 4  x 1
The finite Element solutions for the temperature of two zones;
~ 1 ~ 1
1 1   2  2 ;
~ 2 ~ 2
 2 1   3  2 ;
1   ~
~
 3 13   4  23 ;
~
  4  ~  4 ;
 4 1
5
2
- 1  y  - 3/4
2.34
- 3/4  y  - 1/2
- 1/2  y  - 3/4
-3/ 4  y 0
~ 5 ~ 5
 5 1   6  2 ;
~ 6 ~ 6
    7  2 ;
 2   ~6 1
~
 7 17   8  27 ;
~
  8  ~  8 ;
 8 1
9
2
0  y   1/4
1/4  y  1/2
1 / 2  y  3/4
3/ 4  y 1
are temperature and velocity at the nodal points the details of which are provided in appendix II .
2.3.1 A Two Element Solution
The nature of this problem dictates that we use a minimum of two elements. In this solution the distance between the
elements is 1(i.e. h=1) and we have two elements. We will divide our domain into two equal pieces each of length h. This
means we have two regions representing the layers for each fluid. We define our entire region as Ω, Ω1 will represent
elements in the first fluid layer and Ω2 will represent elements in the second fluid layer. In our case n=2 and h=1, the
equation 2.34 becomes as two element solution, this solution is seen below on the Fig.3
1
0.8
0.6
0.4
- Exact solution
-- 2 element solution
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Figure 3: Exact solution of ϴ1, ϴ2 vs 2-element solution
Volume 2, Issue 9, September 2013
Page 164
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
2.3.2 A Four Element Solution
A four element solution should give a better approximation of our exact solution than the two elements. We will divide
our domain into two equal pieces each of length h. This means we have two regions representing the layers for each fluid.
We define our entire region as Ω, Ω1,Ω2will represent elements in the first fluid layer and Ω3 ,Ω4 will represent elements
in the second fluid layer. Since this is a one-dimensional model we will use a linear line element. In this case, we use n=4
and h= ½ in equation (2.34), this solution is seen in the figure 4. .
.
1
0.8
0.6
0.4
- exact solution
-- 4 element
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Figure 4: Exact solution of ϴ1, ϴ2 vs 2-element solution
2.3.3: Eight Element Solution
An eight element solution is similar to our exact solution. This is the basis of the finite element method in that the more
elements that can be used (a more refined mesh) the better the approximation should be. In this case, we use
and
in equation (2.34), The exact solution and eight element solution is plotted in the figure 5 below.
1
0.8
0.6
0.4
- exact solution
--- 8 element solution
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Figure 5: Exact solution to temperature distribution vs element solution
2.4 DISCUSSION
The approach considers five aspects of the solution:
(i) The effect of different porosity parameter σ.
(ii) The effect of different slip parameter α.
(iii) The effect of varying Prandtl number and Eckert number.
(iv) A comparison of the exact solution and the three element solutions.
(v) The effect of the rate of heat transfer with varying σ and α.
2.4.1 Effect of different permeability parameter
The graph presented in figure 6 shows the exact solution varies with sigma at 10, 20 and 30.
1
0.8
σ:
- 10
-- 20
…. 30
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Figure 6: Exact temperature distribution (σ= 10, 20 and 30)
Volume 2, Issue 9, September 2013
Page 165
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
It is observed that as σ increases the temperature field decreases. This would lead to conclude that the more porous the
parallel plates are, the faster temperature decreases in both fluids.
2.4.2 Effect of different slip parameter
Figure 7 shows how the exact solution varies with different values of slip parameter. Clearly it is observed as the slip
parameter increases the temperature field decreases. It appears the effect is equal in both fluids.
1
0.8
0.6
α: - .1-.05….
.15
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Figure 7: Exact temperature distribution (σ= 0.1, 0.05 and 0.15)
2.4.3 Effect of Prandtl and Eckert numbers.
The graphs in figure 8 show that as both the Prandtl number and Eckert number increase there is an increase in the
temperature field. As the Prandtl number increases the temperature distribution appears to become more linear especially
in the more viscous fluid.
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
0
100
200
300
400
500
-1
-200
600
0
200
400
600
800
1000
1200
1400
1600
1800
a)
b)
Figure 8: Exact temperature distribution a) Pr= 3, 1.5, 0.7, b) Ec=3, 2, and 0.5
2.4.5 Comparison of Solutions
The graph in figure 9 shows how the exact solution varies with different elemental solutions. This gives a good indication
of how the finite element method approximates to the exact solution as the number of elements increases.
It can be clearly seen that the number of elements increases the approximation to the exact solution gets progressively
better. The eight element solution is almost equivalent to the exact solution especially for the upper fluid. This leads to
conclude that the finite element method can be a very powerful and effective tool to provide approximate solutions to
differential equations particularly when the exact solution is difficult to evaluate otherwise.
1
- Exact Solution
** 2 Element Solution
- - - 4 Element Solution
. . . 8 Element Solution
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Figure 9: Exact Temperature Distribution vs 2,4 and 8 Element Solutions
Volume 2, Issue 9, September 2013
Page 166
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
2.4.6 NUSSELT NUMBERS
The rate of heat transfer at the boundary, the Nusselt Number, Nu, was calculated for various values of the permeability
parameter, σ (Table 2.1), and slip parameter, α (Table 2.2).
 d 
Nu  

 dx  x 1,1
The values in the tables indicate the rate of heat transfer increases with increasing σ and increasing α.
Table 2.1 Nusselt Numbers for
σ
At x=1
10
-1.65
20
-1.203
30
-1.0211
σ= 10,20,30
At x = -1
-2.65
-2.203
-2.0211
Table 2.2 Nusselt Numbers forα=.05,0.1,0.15
α
At x =1
At x = -1
0.05
-2.1655
-3.1655
0.1
-1.65
-2.65
0.15
-1.3848
-2.3848
APPENDIX-I
11 =
 12
13
 14
15
16
 17
18
 12 =
 22 =
 23 =
 24 =
=
=
=
 25 =
=
 26 =
=
 27 =
=
 28 =
=
APPENDIX-II
~
x K 1
k
i
i  f 

xK
 dV k
B s1  s
 dx
2
 k
  i dx , for

are;
~
1  f11 =0 (due to boundary condition);
~
 2  f 21  f 22 
;
~
 3  f 32  f 33 
Volume 2, Issue 9, September 2013
;
~
 4  f 43  f 44 
Page 167
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
~
~
 5  f 54  f 55 
;
 6  f 65  f 66 
~
;
~
 7  f 76  f 77 
;
 8  f 87  f 88 
and
~
 9  f 98  1 (due to boundary condition)
where
A1 =
(refers to fluid 1); A2 =
)(-B1) ;
(refers to fluid 2)
)(-B1)
)(-B1) ;
)(-B2)
)(-B2) ;
)(-B2)
REFERENCES
[1] S. Ramakrishna, D. Nagaveni and G. Sridhara Rao Finite Element Analysis of an Adjacent Flow of Two Immiscible
Fluids Bounded by Porous Beds – Darcy Model, Acta Ciencia, Vol. XXXII No. 2, 883 (2006).
[2] K. Vajravelu, Arunachalam, S. Sreenadh, Unsteady Flow of two Immiscible Conducting Fluids between two
Permeable Beds, Journal of Mathematical Analysis and Applications 196, pp. 1105 – 1116, 1995.
[3] S. Rao Gunakala, B. Bhatt, D. M. G Comissiong Journal of Mathematics Research Vol.3, No. 2; May 2011.
[4] G. S. Beavers and D. D Joseph, Journal of Fluid Mechanics 30 (1967), 197.
[5] M. S. Malashetty, J. C. Umavathi, J. Prathap Kumar Two Fluid Flow and Heat in an Inclinde Channel Containing
Porous and Fluid Layer (2004).
[6] J. C. Umavathi, Ali J. Chamkha, Abdul Mateen Ali Al-Mudhaf, Heat Mass Transfer 42:81 – 90 (2005)
[7] K. S. Sai Def. Sci. Journal, Vol 40, No. 2; 183 – 189 April 1990.
[8] S. Ganesh, S. Krishnambal Journal of Applied Sciences 6(11): 2420 – 2425 (2006).
[9] David Burnett, Finite Elements Analysis from Concepts To Applications, Addison-Wesley Publishing Company
(1988).
[10] Young W. Kwon, Hyochoong Bang, The Finite Element Method using Matlab, CRC Press (2000).
[11] J. N. Reddy, An Introduction To the Finite Element Method Third Edition, Mc Graw Hill (2006).
[12] Daryl L. Logan, A First Course in the Finite Element Method 4th Edition, Thomson(2007).
[13] William B. Bickford, A First Course in The Finite Element Method 2nd , Irwin(1994).
[14] Z.U.A Warsi, Fluid Dynamics Theoretical and Computational Approaches, Second Edition, CRC Press (1999).
[15] M. Kaviany, Principles of heat transfer in porous media, Springer, NY, (1996).
AUTHORS
Roger Bostic received his bachelors degree in Mathematics, Computing and Statistics from the University of
London. Ccurrently he is pursuing his masters degree in Mathematics from the University of the West Indies,
Trinidad and Tobago. His research insterests include fluid and thermo dynanics.
Dr Sreedhara Rao Gunakala received his Ph.D. in Mathematics in the year 2004 from Sri Venkateswara
University, Tirupati, India. He is former Head & Associate Professor of Mathematics, Department of
Mathematics, Haramaya University, Ethiopia. At present he is working in the Dept. of Mathematics and
Statistics, The University of West Indies, St. Augustine, Trinidad and Tobago. He has more than 20 years of
teaching experience and published numerous articles in various national and International Journals.
SambasivaRao Baragada received his M.Sc and M.Phil in Computer Science in 2001 and 2006 respectively. He
is awarded his Ph.D in Computer Science from Sri Venkateswara University, Tirupati in the year 2011. Earlier he
was associated as Scientist at Satellite Data Acquisition & Processing System (SDAPS), Data and Information
Management Group (DMG), Indian National Center for Ocean Information Services (INCOIS), Ministry of Earth
Sciences, Govt. of India, Hyderabad. At present he is working as Lecturer in Computer Science in Govt. Degree College,
Khairatabad, Hyderabad. He has published numerous articles in various Journals and Conferences.
Volume 2, Issue 9, September 2013
Page 168