Research Journal of Applied Sciences, Engineering and Technology 5(2): 585-595, 2013
ISSN: 2040-7459; E-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: May 19, 2012
Accepted: June 06, 2012
Published: January 11, 2013
First-Order Reactant in Homogeneous Turbulence Prior to the Ultimate Phase of Decay
for Four-Point Correlation in Presence of Dust Particle
M.A. Bkar PK, M.A.K. Azad and M.S. Alam Sarker
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
Abstract: In this study, following Deissler’s approach the decay in homogeneous turbulence at times preceding to
the ultimate phase for the concentration fluctuation of a dilute contaminant undergoing a first-order chemical
reaction for the case of four-point correlation in presence of dust particle is studied. Two, three and four-point
correlation equations have been obtained and the correlation equations are converted to spectral form by their
Fourier-transforms. The terms containing quintuple correlations are neglected in comparison to the third and fourth
order correlation terms. Finally, integrating the energy spectrum over all wave numbers, the energy decay of
homogeneous dusty fluid turbulent flow for the concentration fluctuations ahead of the ultimate phase for four-point
correlation has been obtained and is shown graphically.
Keywords: Deissler’s method, dust particle, fourier-transform, navier-stokes equation, turbulent flow
et al. (2010) also extended their previous problem in
presence of dust particle. Corrsin (1951) obtained on
the spectrum of isotropic temperature fluctuations in
isotropic turbulence. Azad et al. (2011) studied the
statistical theory of certain distribution functions in
MHD turbulent flow for velocity and concentration
undergoing a first order reaction in a rotating system. It
is noted that in all above cases, the researcher had
considered three-point correlations and through their
study they obtained the energy decay law ahead of the
ultimate phase.
By analyzing the above theories we have studied
the first-order reactant in homogeneous fluid turbulence
prior to the ultimate phase of decay for four-point
correlation in presence of dust particle. Through this
study we would like to find the decay law of first order
reactant in homogeneous fluid turbulence in presence of
dust particle. Presently we shall try to show in this
study the chemical reaction rate R = 0 causes the
concentration fluctuation of decay in presence of dust
particle is more rapidly than for the reaction rate R  0
in the absence of dust particle f = 0. The comparison
between the effects of the chemical reaction in the
homogeneous fluid turbulence and the dust particles are
graphically discussed. It is observed that the chemical
reaction rate increases than the concentration
fluctuation to decay more decreases and vice versa.
INTRODUCTION
In recent year, the motion of dusty viscous fluids in
a rotating system has developed rapidly. The motion of
dusty fluid occurs in the movement of dust-laden air, in
problems of fluidization, in the use of dust in a gas
cooling system and in the sedimentation problem of
tidal rivers. The behavior of dust particles in a turbulent
flow depends on the concentrations of the particles and
the size of the particles with respect to the scale of
turbulent fluid.
Deissler (1958, 1960) developed a theory ‘on the
decay of homogeneous turbulence before the final
period.’ Using Deissler’s theory, Kumar and Patel
(1974) studied the first-order reactant in homogeneous
turbulence before the final period of decay. Kumar and
Patel (1975) extended their problem for the case of
multi-point and multi-time concentration correlation.
Loeffler and Deissler (1961) studied the decay of
temperature fluctuations in homogeneous turbulence
before the final period. Batchelor (1953) studied the
theory of homogeneous turbulence. Chandrasekhar
(1951) obtained the invariant theory of isotropic
turbulence in magneto-hydrodynamics. Sarker and
Kishore (1991) studied the decay of MHD turbulence
before the final period. Sarker and Islam (2001)
obtained the decay of MHD turbulence before the final
period for the case of multi-point and multi-time. Islam
and Sarker (2001) also obtained the first order reactant
in MHD turbulence before the final period of decay for
the case of multi-point and multi-time. Aziz et al.
(2009) studied the first order reactant in MHD
turbulence before the final period of decay for the case
of multi-point and multi-time in a rotating system. Aziz
MATERIALS AND METHODS
Basic equation: The differential equation governing the
concentration of a dilute contaminant undergoing a
first-order chemical reaction in homogeneous dusty
fluid turbulence could be written as:
Corresponding Author: M.A. Bkar PK, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205,
Bangladesh
585
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
u i
u i
 2ui
 uk
D
 Ru i  f (ui vi )
t
x k
x k x k
(1)

ˆ
ˆ
ˆ
XX  (r)    ( K , t ) exp[ i ( K . rˆ )] d K
(5)

where,
ui ( ) = A random function of position and time at a
point p
uk ( , t) = The turbulent velocity
R
= The constant reaction rate
D
= The diffusivity
t
= The time
ƒ
= kN/p, dimension of frequency N, Constant
number density of dust particle
ms
= (4/3) Rs3 ps is the mass of single spherical
dust particle of radius Rs
ps
= The constant density of the material in dust
particle
ui
= The turbulent velocity component
vi
= The dust particle velocity component
xk
= The space coordinate and repeated subscript
in a term indicates a summation of terms,
with the subscripts successively taking on
the values 1, 2 and 3.
Two-point correlation and spectral equations: Under
the restrictions that the turbulence and the concentration
fields are homogeneous, the local mass transfer has no
effect on the velocity field; the reaction rate and the
diffusivity are constant. The differential equation of a
dilute contaminant undergoing a first-order chemical
reaction in homogeneous system could be written as:


 2
 uk
D
 R
t
x k
x k x k
 XX 
t
 uk
 XX 
x k
 D
 2 XX 
x k x k

(4)
where, X′j is the fluctuating concentration at the point
p′ ( ) and the point p′ is at point a distance r from the
point p. The symbol ⟨ ⟩ is the ensemble average.
Putting the Fourier transforms:
(6)
(7)

into Eq. (4), one obtains:

ˆ
 ( Dk 2  2 R )  ik k [  k ( Kˆ , t ) ‐  K ( K , t ) ] (8)
t
Three-point correlation and spectral equations:
Taking the Navier-Stokes equations for a first-order
chemical reaction in homogeneous dusty fluid
turbulence at p is:
u i
 2ui
u i
 uk
D
 Ru i  f (u i  vi )
t
x k
x k x k
and the fluctuation equations at p′&p′′one can find the
three-point correlation Equation as:
 ui X X 
t


 u k u k X X 
x k

 ui u k X X   ui u kX X 

xk
xk
 2 u i X X 
1  p X X 


xi
 x k x k
 2
2
 D 

 x k  x k
 x k x k
(3)
 R XX 
ˆ
ˆ
ˆ
 k ( K , t ) exp[ i ( K .rˆ )] d K
ˆ
ˆ ˆ
ˆ
u k XX  (r)    k ( K , t ) exp[ i ( K .r )] d K
Subtracting the mean of (2) from Eq. (2), we
obtain:
where, X( , t) is the fluctuation of concentration about
the mean at a point p( ) and time t. The two-point
correlation for the fluctuating concentration can be
written, as:


(2)
X
X
2X
 uk
 D
 RX
t
x k
x k x k

u k XX  (r) 
 f
u
i

 u i X X   2 R u i X X 

X X   v i X X 

(9)
Using the transformations:
 




  
,


  

,




r
r

x

x

x

rk
xk


k 
k
k
k
 k
In order to convert Eq. (9) to spectral form, using
six-dimensional Fourier transforms (Kumar and Patel,
1974) and:
v i X X  
 


586 i
( Kˆ , Kˆ , t ). exp[ i ( Kˆ .rˆ  Kˆ .rˆ ) d Kˆ d Kˆ ]
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
with the fact that u i u k X X  = ui uk X X 
 2
2 
 u i u j X X 
 D 

  x k x k  x k x k 
one can write Eq. (9) in the form:
 j
t




 2 R u i u j X  X  
2
 [ Kˆ  Kˆ   D Kˆ  Kˆ   2 R  Qf ] j
  i ( k k  k k )
jk
 i ( k k  k k )
jk 

1


xi xk
 


 

 



xk
rk
 rk rk


 
 



,
,







xk xk xk rk xk rk
2
1  pX X 
 xi xi
(11)
and nine-dimensional Fourier transforms (Kumar and
Patel, 1974) and:
In Fourier space it can be written as:
 ( k k  k k ) ik  
1

(14)
In order to convert Eq. (14) to spectral form, using
the transformations:
and Q arbitrary constants. If the momentum Eq. (3) at p
is multiplied by X′X′′ and divergence of the time
averages is taken, the resulting equation will be:


 f u i u j X X   v i u j X X  i ( k j  k j ) (10)
where, i ( Kˆ , Kˆ , t ) = L  j ( Kˆ , Kˆ , t ) and 1-L = Q ,with L
 2 u i u k X X 
v i u j X X  ( rˆ , rˆ , rˆ  )
(ki  ki)
  
(12)

 
ij
( Kˆ , Kˆ , Kˆ , t )

exp[ i ( Kˆ .rˆ  Kˆ .rˆ   Kˆ .rˆ  ) d Kˆ d Kˆ d Kˆ 
Substituting this into Eq. (10), we obtain:
 j
t



and with the fact that:

2
 [ Kˆ  Kˆ   D Kˆ  Kˆ   2 R  Qf ] j
 i ( k k  k k ) jk
u ku i u j X  X  ( rˆ , rˆ  , rˆ  )
Four-point correlation and spectral equations:
Again by taking the Navier-Stokes equations at p and p′
and the concentration equations at p′′p′′′ and following
the same procedure as before, one can get the four-point
correlations as:
 u i u j X X 
t

 u ku i u j X X  ( rˆ , rˆ  , rˆ  )
(13)
 u i u k u j X X 
x k

Eq. (14) can be written as:

 g ij

2
 [ Kˆ  Kˆ   Kˆ    Kˆ 2
t

 D Kˆ   Kˆ 

2
 2 R  Sf ] g ij
 u j u ku i X X 
  i ( k k  k k  k k )[ h ijk ( Kˆ , Kˆ , Kˆ  )
 x k
 ik k [ h ijk (  Kˆ  Kˆ   Kˆ , Kˆ , Kˆ  )]

 u i u j u k  X  X  
 x k

  X  
 u i u j u k  X
 i ( k k  k k) h ijk ( Kˆ , Kˆ , Kˆ  )
 x k 

 p u i X X 
1  p u j X X 
 [

]

xi
 x j
 2
2
  

 x k x k
 xk xk

 u i u j X X 

1

[  ( k i  k i  k i)  j ( Kˆ , Kˆ , Kˆ  )
 k j i ( Kˆ  Kˆ   Kˆ , Kˆ , Kˆ )]
(15)
where, ij ( , ′, ′′′, t) = M gij( , ′, ′′′, t) and
1-M = S, with M and S are an arbitrary constants.
587 Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
Following the same procedure as was used in obtaining
(12), we get:
 j ( Kˆ , Kˆ , Kˆ  )
( k  k k  k k )
  k
hijk ( Kˆ , Kˆ , Kˆ )
( k i  k i  k i)

a 3 / 2
(1  N s ) 3 / 2 (t  t 0 ) 3 / 2


N s (2  N s )k 2
 2 N s kk 
 D[


(1  N s )
 exp 
(t  t 0 )
2
R
Qf
  (1  N ) k  2 


]
s


D
D
(16)
Equation (15) and (16) are the spectral equations
corresponding to the four-point correlations. To get a
better picture of the of the first-order reactant of
homogeneous dusty fluid turbulence decay system from
its initial period to its final period, four-point
correlations are to be considered. The same ideology
could be applied to the concentration phenomenon.
Here, we neglect the quintuple correlations since the
decay faster than the lower-order correlations. As
pointed out by Deissler (1958, 1960) when the
quintuple correlations are neglected, the corresponding
pressure-force terms which are related to them are also
neglected. Under these assumptions, Eq. (15) and (16)
give the solution as:
D
3/ 2
Now, the solution of Eq. (19) is:

 j ( Kˆ , Kˆ , t 0 )   j ( Kˆ , Kˆ , t 0 )


 exp 


2 i ( k k  k k ) b i  3 / 2
D 3 / 2 (1  N s ) 3 / 2
[
4 kD 1 / 2 F (  )
 2
]

1/ 2
(t  t0 )
(1  N s ) 1 / 2
ƒ/|
exp
1
2 ′
(20)
where,

F ( )  exp(  2 )  exp( n 2 ) dn
0
 (t  t 0 ) D 
  k

 91  N s ) 
with the expression for ui u k X X (rˆ, rˆ) :
(18)
2
2
where, Ns = v/D. A relation between gij( , ′, ′′′, t0)
and ′ij( , ′) can be found by taking ′′ = 0 in the
expression for u i u j X X (rˆ, rˆ , rˆ ) and comparing it

0

 [ g ij ( Kˆ , Kˆ , Kˆ , t 0 ) exp{ D[2 N s k 2  (1  N s )(k  2  k  2 )
ik ( Kˆ , Kˆ )   g ij (Kˆ  Kˆ , Kˆ , Kˆ , t )dKˆ 

 K  2 )  2 N s Kˆ . Kˆ  


Qf 
 2R


 ]( t  t 0 )

D 
 D

D [( 1  N s )( K
g ij ( Kˆ , Kˆ , Kˆ , t )
 2 N s ( k k k k  k k k k  k k k k)]  2 R  Sf }( t  t 0 ) (17)
(19)
1/ 2
and

[ g ij (  Kˆ  Kˆ , Kˆ , Kˆ , t0 )]0  bi
(21)
Substituting this in to Eq. (13), we obtains:
 j
2 R Sf
] j
 D[(1  N s )( K  K  )  2 N s Kˆ .Kˆ  

t
D
D
2
2

  i ( k k  k k )  g ij (  Kˆ  Kˆ , Kˆ , Kˆ , t ) d Kˆ ,
Now, by taking rˆ   0 in the expression for
u X X ( rˆ, rˆ  is performed, we obtain and comparing
the result with the expression X u k X (rˆ ) :

 k ( Kˆ ) 
which with the help of Eq. (17), gives:


k
( Kˆ . Kˆ  ) d k 
(22)

Substituting Eq. (21) into (8), we obtain:
 j
2 R Sf
 D[(1  N s )( K 2  K  2 )  2 N s Kˆ .Kˆ  
 ] j
D
D
t
ˆ
ˆ
ˆ
ˆ
 2i ( k k  k k )[ g ij (  K  K , K , K , t 0 )] 0
G
 ( 2 Dk
t
588 2
 2 R )G  W
(23)
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
where, G  2  k 2  and:
1  2N s
 exp{  D [ 
 1 Ns
W  2  ik 2 exp[  2 R ( t  t 0 )]
{


k
k
[  k ( Kˆ , Kˆ )   k (  Kˆ ,  Kˆ ) ]0
 exp{  D [(1  N s )( k 2  k  2 )
 2 N s Kˆ . Kˆ   (  Qf / D )]( t  t 0 )}

2i 
D 3 / 2 (1  N s ) 3 / 2
[


 [b ( Kˆ , Kˆ ) b ( Kˆ , Kˆ )]
i

( 2 ) 2 ik k  0 ( Kˆ , Kˆ )   0 (  Kˆ ,  Kˆ )
[



105 k 7
8 D N 2 s (t  t 0 ) 3

15 k 9
 
3
2
 4 D N s (t  t 0 )
 Ns
 [ 7 
1 N s
[
  1/ 2 N s
2 (t  t 0 )


Sf
  ( 
)]( t  t 0 ) 
D


3
 Ns

1 N s
where,  0 and 1 are arbitrary constants depending on
the initial conditions. If these results are used in Eq.
(24) and the integration is performed, we obtain:
0
3/2
2

  1]}

0
  1 ( k 4 k  6  k 6 k  4 ),
W 
2
 Ns

N 2 s k 12
  5 ] 
[ 
2


1
(
)
N
t
t


1 N s

0
s
2  1 1 / 2 N s
( t  t 0 ) 3 / 2 D (1  N s ) 4
{
And
D
2

1  2N s
 exp   D [ 
 1 Ns

0
2
8 7 / 2
i b ( Kˆ , Kˆ )  b (  Kˆ ,  Kˆ )
(1  N s ) 3 / 2

15 k 8
 
3
2
 4 D N s (t  t 0 )
21  N s

2  1  N s

 2
Sf
 k  ( 
)]( t  t 0 )}
2
D




N s k 10

  1]  



1
(
)


D
N
t
t
s
0 


 Ns
 [ 7 
1 N s
  0 ( k k   k k  )
3/2
 8
 ]k }

105 k 6
8 D 3 N 2 s (t  t 0 ) 3
 Ns

1 Ns
(24)
In analogy with the turbulent energy spectrum
function, the quantity G, in Eq. (23) can be called
concentration energy spectrum function and W, the
energy transfer function, is responsible for the transfer
of concentration from one wave number to another. In
order to find solution completely and following
Deissler (1958, 1960) we assume that:
4
2
(t  t 0 ) 7 / 2 D 3 / 2 (1  N s ) 7 / 2
 Ns
 exp{  2 D [
1 N s
N s (2  N s )k 2
 2 N s kk  
(1  N s )
4
3
  Ns
  
 1 Ns



 1 1 / 2 N s
{
(1  N s ) k  2  (  Sf / D )]( t  t 0 ) d k }
2

 Ns
  [5

1 N s
i

2
4 kF ( w ) D 1 / 2

]
(t  t 0 ) 1 / 2
(1  N s ) 1 / 2
 exp{  D [
 Ns

1 N s
 Ns
3
k6
 ]
 [
2 N s D (t  t 0 )  1  N s

5/2
15 k 4
2
4 D N 2 s (t  t 0 ) 2
 2
 k  (  Qf / D )]( t  t 0 )}

21
2

D 3 / 2 (1  N s ) 5 / 2

0
589  Ns

1 N s
 exp(
2

  1] 

2


N s k 11




1
(
)


D
N
t
t
s
0



N 3 s k 13
  5 ] 
1  N s 2

 Ns
[ 
1 N s
n 2 ) dn } exp[  2 R ( t  t 0 )]
2

  1]

(25)
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013

 exp  [(1  2 N s ) 
It is very interesting that:

W
dk = 0
 
This indicates that the expression for W satisfies
the condition of continuity and homogeneity. Physically
it was to be expected, since W is the measure of transfer
of energy and the total energy transformed to all wave
numbers is to be zero, which is what Eq. (26) gives.
The linear Eq. (23) can be solved to give:


 exp  ( 2 Dk
W
2

2
 2 R )( t  t 0 )
 2 R )( t  t 0 )
 exp  ( 2 Dk
2

 exp[(  2  2 )  ( Sf / D )][ E i ( 2  2 )  0 . 5772 ]
 3 . 1808  10  2  14  2 . 7178  10  2  16
 1 .3968  10  2  18  4 .6204  10 3  20
 (27)
 1.1584  10 3  22  2.3139  10 4  24

 3.8589  10 5  26  5.5108  10 4  28
 2 R )( t  t 0 ) dt
 1.1584  10 3  22  2.3139  10 4  24
2
where, J (k)  N 0 k is a constant of integration and

 6.8898 10 7  30  ...
can be obtained as (Corrsin, 1951).
Thus, by integrating the right-hand side of
equation, we obtain:

2
G = exp[  2 R (t  t )  N 0 (1  N s )
0 
 D  (t  t 0 )
 exp[  2 (1  N s )

2
 (
 2  ( Sf / D ) ]
 105   8
2  10
23  12
498  14





3
30
630
8N s  3
(26)
G  J ( k ) exp  ( 2 Dk
2
15( N 2 S  2 N s  1)   10 5 12 14 14  exp



4N s
6
15
 2
[(  2  2 )  ( Sf / D )][ E i ( 2  2 )  0 . 5772 ]
 2.0261  10 2  14  3.5487  10 3  16
Qf
)]
D
 5.4047  10 4  18  7.2369  10 5  20
 0 1 / 2 N s
2 (t  t 0 )
D 11 / 2 2 (1  N s ) 3 / 2
 8.6172  10 6  22  9.22057  10 7  24
9/2
 8.9477  10 8  26  7.9374  10 9  28
 exp[  (1  2 N s ) 
2
Qf
 (
)]
D
 6.4814  10 10  30  ...
 3 4 7 N s  6  6



 f 1 8  2 f 1 9 F ( ) 
3
 2N s


[( 2 2 )  ( Sf / D )][ E i ( 2 2 )  0.5772 ]
 1 1 / 2

7
2 ( t  t 0 ) D 15 / 2 (1  N s ) 3 / 2
 8.2505  10 2  14  1.7495  10 2  16
105 6
5

 (11  24 N 2 s  8 N s ) 8  f 2  10
32
16

 f 3
 2 f 3
12
14
 3.1572  10 3  18  4.90805  10 4  20
 6.66926  10 5  22  8.0290  10 6  24
exp(  2  )
2
[ E i ( 2  )  0 . 5772 ]  ( Sf / D ) 
 8.6642  10 7  26  8.4643  10 8  28
 7.5495  10 9  30  ...
 (1  2 N s ) N 3 s  14 exp[(   2 )  ( Sf / D )] 
2


4

 11 N 2 s  20 N s  10 N s    12   14  exp
3

2  1
Ns
15 / 2
2 (t  t1 ) D
2 (1  N s ) 3 / 2
1/ 2
7
590 Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
[ E i ( 2 2 )  0.5772 ]
E  cf ( gh  gj1  k1l1  k1n  pq  ps  t vw )
 2.2654  10 1  14  1.0679  10 1  16
F  cf ( gi1  k1 m  h t ), G  cf pr1
 3.3996  10 3  18  9.0694  10 3  20
 1.1099  10 
3
22
 3.7350  10 
4
d

Ns
7/2
(1  N s )
105 35(11  24 N 2 s  8 N s ) 63 f 2



32
64 N s
8N 2 s
693 f 3 1808 f 3 41 .5910 f 3 N


1
256
64 N 3 s
(1  N s ) 2
4
(t  t1 ) k 2
2
, f1  (3N s  2 N s  3)
3
(1  N s )
 15  40 N s (2 N s  1)  4 N s (11N
f 2  
16(1  N s ) 2 D

2
D 2
15 / 2
(28)
where,
2  D
8
24
 4.444  10 5  26  8.2388  10 6  28
 8.47486  107  30  ...
15  1
c 
2
2
s
7
2
S
1
e  2 f3
 20 N s ) 


D 2 I ( 2 N s  2 ,  ) 2
1
(1  N s ) 2 
1
2
1
37.12311 1 N s  2 , g = 105
f
9
2Ns
D 8  2 (1  N s )
 15  20(4 N 2 s  2 N s )  4 N 2 s (9 N 2 s  20 N s )  16 N 2 s 

f 3  
8(1  N s )3


9
In a turbulent phenomenon, such as turbulent
kinetic energy, we associate the so-called concentration
energy with the fluctuating concentration, defined by
the relation:
h=
1 6 75 .90 2.52052  10  3  1 2
 
 2 
15
3 1
 1
(1  N s ) 2
1
1
2
XX  

 G ( k , t ) dk
i1  1 .118  1
(29)
(1  N s ) 2
 I [(1  2 N s ) ,  2 ]
D
9
2

j1 
The substitution of Eq. (28) and subsequent
integration with respect to k leads to the result:
163
1
3
1
(3.1808 10  2  4.0767 10 1  1  3.561804 1
3
1
X
2
 At  t  32  B t  t 5 exp[ R (t  t )]

0
0
2
0


15
1
15
 


15
2
2
2














C
t
t
D
t
t
t
t
E
t
t
 

0
1
0
1
0
 

29
1

15

  F t  t0  2  G t  t0  t  t1 2  exp[ R3 (t  t0 )]

 

6
 2.02576  103 1  8.7048  1
N0
3
7
1
D2 22 2
8
135(6 N s  2 N s  1) ,
1 55
l1   
N s1
2 6 1
2
k1 
11

2 . 3510  10  3  163  1 2
2 15 (1  N s )
, B   0 R1 , C  cd , D1  ce,
D
7
 3.1561  10 4 1  ...)
(30)
This is the decay law of first order reactant in
homogeneous turbulence in presence of dust particle,
where,
A
4
 2.2386  10 1  1.1786  10 2  1  5.4148  10 2  1
= exp[  2 R ( t  t 0 )
2
11
m = 7 .4469  1 2
6
591 15
2
2
5
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
(1  N s )

D

1
2
I [(1  2 N s ) 2 ,  2 ]
4
 1.1293  10 2  1  8.7405  10 2  1
 2.6001  10 3  1
n = 163 (2.0261  5.3230  10  2  1
1
 12
6
5
 1.3013  10 4  1
7
8
 3.8822104 1  ...)
 1.3782  10 1  1  3.5063  10 1  1
2
 8.7676  10  1
1
 5.2347  1
6
4
 2.57710  1
1
 1.2538 1
7
3
R1 
5
 2.9690  10 1
8
 ...)

p = 396 (11 N s 2  20 N s  10 ) N s  1 2
13
2
q =  1  3 .0532  10  1 (1  N s )
4
r1  1 .0638  1
13
2
(1  N s )
(t  t1 )
D
1
2

 9 5N s (7 N s  6)

 
2(1  N S )(1  2N s ) 5 / 2 16 16(1  2N s )
f N (1  2 N s ) 5 / 2
105 f1 N s
 121 / s2
32(1  2 N s ) 2 (1  N s )11 / 2
1.3.5...( 2 n  9)
] 1  2(1  2 N s )
2n
(1  N s ) n
n  0 n! ( 2 N s  1) 2


 15
2
R 2  (Qf / D ) R3  (Sf / D)
1
2
f = constant number density dust particle and
 I [(1  2 N s )  ,  ]
2
2
I [  2 ,  2 ] 
S = 13 1 1 8.2504
3
4
 1.3370  10 2  1
8
 ...)
 3.4583  10 2  1
11
2
t    1 .4219 N s  1 (1  N s )
2
7
1
2
u =  3.1886  10 3  1.4443  10 2 (t  t 0 ) 2
1
1
 D 2 (1  N s )

1
2
I [(1  2 N s ) 2 ,  2 ]}
1
2
v = 1.8102  10 2 (1  2 N S )  1
15

2
w = ( 2.2653  10 1  1.60195  1 1
 8.6664  10 1  2  4.3941  10 1
3
14
exp(  2 ) E i (  2 )] dk
RESULTS AND DISCUSSION
 2.3780   1  6.7856 1  1.8789  10 1
6
exp(  2 )
 [
0
 2.6243  10 1  1  8.0509  10 1  2
 5.0688  10 1  1

5
The first term of Eq. (30) corresponds to the
concentration energy for two-point concentration; the
second term represents first order reactant in
homogeneous fluid turbulence in presence of dust
particle for three-point correlation. The expression
exp[ R2 (t  t 0 )] represents the dusty fluid turbulence for
three-point correlation; the expression exp[R3 (t  t 0 )]
and the remainder are due to fluid turbulence in
presence of dust particle for four-point correlation. In
Eq. (30) we obtained the decay law of first order
reactant in homogeneous dusty fluid turbulence for
four-point correlation after neglecting quintuple
correlation terms. The equation contains the terms
(t  t 0 ) 3 / 2 , (t  t 0 ) 5 , (t  t 0 ) 15 / 2 . Thus, the terms
associated with the higher-order correlation die out
faster than those associated with the lower-order ones.
Therefore, the assumption that the higher-order
correlations can be neglected in comparison with lowerorder correlations seems to be valid in our case. If the
fluid is clean (i.e., f  0) the Eq. (30) becomes:
592 Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
5
Decay of total energy of homo.turbulence=X2
4.5
y6
4
y5
3.5
y2
3
y4
y1
2.5
2
1.5
y1,y2,y3 of Eq.(30) at
t0=.3,t1=.5; t0=.8,t1=1
and t0=1.3,t1=1.5 respectively
1
y4,y5,y6 of Eq.(31) at
t0=.3,t1=.5; t0=.8,t1=1
and t0=1.3,t1=1.5 respectively
0.5
0
1.8
2
2.2
2.4
2.6
2.8
Approximation of time=t
Fig. 1: Comparison between Eq. (30) and (31) if R = 0.25
g
p
5
q(
4.5
Decay of total energy of homo.turbulence=X2
y3
)
3
q(
3.2
3.4
)
y3
y6
4
y1,y2,y3 of Eq.(30) at
t0=.3,t1=.5; t0=.8,t1=1
and t0=1.3,t1=1.5 respectively
3.5
y2
3
y5
y4,y5,y6 of Eq.(31) at
t0=.3,t1=.5; t0=.8,t1=1
and t0=1.3,t1=1.5 respectively
2.5
y1
2 y4
1.5
1
0.5
0
1.8
2
2.2
2.4
2.6
2.8
Approximation of time=t
Fig. 2: Comparison between Eq. (30) and (31) if R = 0.5
593 3
3.2
3.4
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
5
y3
Decay of total energy of homo.turbulence=X2
4.5
y6
4
y2
3.5
y5
3
y1
2.5
y4
2
1.5
1
0.5
0
y4,y5,y6 of Eq.(31) at
t0=.3,t1=.5; t0=.8,t1=1
and t0=1.3,t1=1.5 respectively
y1,y2,y3 of Eq.(30) at
t0=.3,t1=.5; t0=.8,t1=1
and t0=1.3,t1=1.5 respectively
1.8
2
2.2
2.4
2.6
2.8
Approximation of time=t
3
3.2
3.4
Fig. 3: Comparison between Eq. (30) and (31) if R = 0
1
X 2 = exp[  2 R ( t  t 0 ) ]
2
 [ At  t 0 

3
2
 B t  t 0 
5
turbulence than clean fluid which indicates in the
figures clearly.
CONCLUSION

15
15
1


C t  t 0  2  D1 t  t 0  2 t  t1  2 
29
1 ]

15

15

 E t  t 0   F t  t 0  2  G t  t 0  t  t1  2 


(31)
this is obtained by Kumar and Patel (1974). Here A, B,
C, D1, E, F, G are constants which can be determined.
With R = 0 and the contaminant replaced by the
temperature, the results show complete argument with
the result obtained by Loeffler and Deissler (1961) for
the decay of temperature fluctuation in homogeneous
turbulence before the final period up to three-point
correlations. For large times, the last terms become
negligible and give the-3/2 power decay law for the
final period. In figures, Eq. (30) represented by the
curves y1, y2, y3 and (31) by y4, y5 and y6,
respectively. For R2  0.25 and R3  0.25 , the
comparison between the Eq. (30) and (31) are shown in
Fig. 1, 2 and 3 corresponding to the values R = 0.025,
0.5 and 0, respectively, the energy decays rapidly in
presence of dust particle of homogeneous fluid
From the figures and discussions, this study shows
that the terms associated with the higher-order
correlations die out faster than those associated with the
lower-order ones and if the chemical reaction rate
increases than the concentration fluctuation to decay
more decreases and vice versa. At the chemical reaction
rate R = 0 of homogeneous fluid turbulence in presence
of dust particle causes the concentration fluctuation of
decay more rapidly than they would for the chemical
reaction rate R  0 and f = 0. Also we conclude that
due to the effect of homogeneous turbulence in the flow
field of the first order chemical reaction for four-point
correlation in presence of dust particle prior to the
ultimate phase of decay, the turbulent energy decays
more rapidly than the energy decay for the first order
reactant in homogeneous turbulence before the final
period.
ACKNOWLEDGMENT
the
594 This study is supported by the research grant under
‘National Science and Information and
Res. J. Appl. Sci. Eng. Technol., 5(2): 585-595, 2013
Communication Technology’ (N.S.I.C.T.), Bangladesh
and the authors (Bkar, Azad) acknowledge the support.
The authors also express their gratitude to the
Department of Applied Mathematics, University of
Rajshahi for providing us all facilities regarding this
study.
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