Research Journal of Mathematics and Statistics 4(2): 30-38, 2012
ISSN: 2040-7505
© Maxwell Scientific Organization, 2012
Submitted: April 03, 2012
Accepted: April 30, 2012
Published: June 30, 2012
First-order Reactant in Homogeneous Dusty Fluid Turbulence Prior to the
Ultimate Phase of Decay for Four-Point Correlation
in a Rotating System
M.A. Bkar PK, M.A.K. Azad and M.S.A. Sarker
Department of Applied Mathematics, Rajshahi University, Rajshahi 6205, Bangladesh
Abstract: The aim of this study is to find the decay of energy in homogeneous dusty fluid turbulence at times
preceding to the ultimate phase for the concentration fluctuation of a dilute contaminant due to a first order
chamical reaction in a rotating system for the case of four-point correlation. Following Deissler’s approach,
two-, three-and four-point correlation equations have obtained and these equations are converted to spectral
form by their Fourier-transform, 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 law of homogeneous dusty fluid turbulence for the concentration fluctuations ahead of the
ultimate phase in a rotating system is obtained.
Keywords: Deissler’s method, dusty fluid, first-order reactant, rotating system, turbulent flow
Astrophysics and Stellar dynamics. It is also significant in
the Earth sciences, especially Meteorology, Physical
geology and Oceanography.
Deissler (1958, 1960) developed a theory ‘on the
decay of homogeneous turbulence before the final period.’
Using Deissler theory, Kumar and Patel (1974) studied
the first-order reactant in homogeneous turbulence before
the final period of decay for the case of multi-point and
single-time correlation. 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 dusty fluid MHD turbulence
before the final period in a rotating system. Sultan (2008)
studied the energy decay law of dusty fluid turbulent flow
in a rotating system. 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 et al. (2010) also extended their
previous problem in presence of dust particle.
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 dusty
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.
In geophysical flows, the system is usually rotation with
a constant velocity, such large Scale flows are generally
turbulent. Gustave Coriolis, French engineer and
mathematician, who showed that a force could be used to
allow the use of the ordinary laws of motion in a rotating
reference frame. The Coriolis Effect is caused by the
rotation of the Earth and the inertia of the mass
experiencing the effect. The most commonly encountered
rotating reference frame in the Earth. Because the Earth
completes only one rotation per day, this force causes
moving objects on the surface of the Earth to appear to
change direction to the right in the northern hemisphere
and to the left in the southern. This effect is responsible
for the rotation of large cyclones. The acceleration
entering the Coriolis force arises from two sources of
change in velocity that result from rotation: the first is the
change of the velocity of an object in time; the second is
the change of velocity in space. Different positions in a
rotating frame of reference have different velocities. The
nature and size of the Coriolis effects depend upon where
we are. The Coriolis Effect has great significance in
Corresponding Author: M.A. Bkar PK, Department of Applied Mathematics, Rajshahi University, Rajshahi 6205, Bangladesh
30
Res. J. Math. Stat., 4(2): 30-38, 2012
 XX '
 XX '
 2 XX '
 uk
D
 R XX '
t
 xk
 xk  xk
fluid turbulence prior to the ultimate phase of decay for
four-point correlation in a rotating system. Through this
study we would like to find the decay law of first order
reactant in homogeneous dusty fluid turbulence in a
rotating system. Later we shall try to show in this study
when the reaction rate R = 0 causes the concentration
fluctuation of decay more rapidly than for R…0, non pure
mixing dusty fluid turbulence.
where, X!j is the fluctuating concentration at the point
p!  x  and the point p! is at point a distance r from the
point p. The symbol + , is the ensemble average.
Putting the Fourier transforms:


Basic equation: The differential equation governing the
concentration of a dilute contaminant undergoing a firstorder chemical reaction in homogeneous dusty fluid
turbulence in a rotating system could be written as:
(5)

    
(6)
uk XX ' r     k K , t exp i K , r dK


    
u'k XX ' r     k K , t exp i K . r dK
ui
ui
 2 ui
 uk
D
 Rui  2 mki  m ui  f ui  vi  (1)
t
 xk
 xk  xk

(7)
into Eq. (4), one obtains:
where, ui  x  is a random function of position and time at a
point p, uk  x , t  is the turbulent velocity, R is the constant
reaction rate, D is the diffusivity, t is the time, gmki is the
alternating tensor, Sm is the angular velocity components,
kN
, dimension of frequency; N, constant number
f
   

 Dk 2  2 R   ik k  k K , t   K  K , t
t



(8)
Three-point correlation and spectral equations: The
same procedure can be used to find the three-point
correlation, i.e., by taking the Navier-Stokes equations for
a first-order chemical reaction in homogeneous dusty fluid
turbulence in a rotating system at p is:

density of dust particle, ms  4 Rs3s is the mass of single
3
spherical dust particle of radius Rs, Ds is the constant
density of the material in dust particle, ui is the turbulent
velocity component, vi is the dust particle velocity
component, xk is 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.
 ui
 ui
 2 ui
 uk
D
 Rui  2 mki  m ui  f ui  vi 
t
 xk
 xk  xk
and the fluctuation equations at p! & p" one can find the
three-point correlation equation as:
Two-point correlation and spectral equations: Under
the restrictions that the turbulence and the concentration
fields are homogeneous, the local mass transfers have no
effect on the velocity field and 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:
 ui X X    uk uk X X    ui uk X X    ui uk X X 



t
 xk
 x k
 x k

 2 ui X X 
1  pX X 
v

 xi
 xk  xk
 2
2 
 u X X   2 R ui X X   2 mki  m ui X X 
 D

  x k  x k  x k  x k  i
(2)

 f ui X X   vi X X 
Subtracting the mean of (2) from Eq. (2), we obtain:
X
X
 X
 uk
D
 RX
t
 xk
 xk  xk
    
XX ' r     K , t exp i K . r dK
MATERIALS AND METHODS


 2
 uk
D
 R
t
 xk
 xk  xk
(4)

(9)
Using the transformations:
2
 

  



,


,


 xk
  rk  rk   x k  x k  x k  rk
(3)
where, X  x , t  is the fluctuation of concentration about the
mean at a point p x and time t. The two-point correlation
for the fluctuating concentration can be written, as:
In order to convert Eq. (9) to spectral form, using sixdimensional Fourier transforms (Kumar and Patel, 1974)
and:
31
Res. J. Math. Stat., 4(2): 30-38, 2012

 
 
vi X X     i  K , K  , t .exp i K . r  K  . r  dKdK
 

 2 R ui uj X X   2mki  m ui uj X X 
 2
2 
 u u  X X 
 D

  x k x k  x k x k  i j
with the fact that:
ui uk X X   ui uk X X 
 f
one can write Eq. (9) in the form:


1



i k j  k j 
where,   K , K  , t  = L i  K , K  , t  and 1-L = Q , with L and Q
i
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:
1

(14)
In order to convert Eq. (14) to spectral form, using
nine-dimensional Fourier transforms (Kumar and Patel,
1974) and:
(11)
vi uj X X   r, r , r  

  
   ij  K , K  , K  , t 
  

  dK 
exp i K  . r  K  . r dKdK
In Fourier space it can be written as:
  k k  k k  ik  

 


 
,



 xk
 rk 
  rk  rk









,
,
 x k  x k  x k  r k  x k   rk 
(10)
2
 2 ui uk X X 
1  pX X 

 xi  x k
  xi  xi
 vi uj X X 
i j
Using the transformations:

2
 v K  K    D K  K    2 R  2 mki  m  Qf  j
t
  i k k  k k  jk  i k k  k k  jk  
 u u X X 
 ki  ki
and with the fact that:
(12)
ukui uj X X   r, r  , r    ukui uj X X   r, r  , r  
Substituting this into Eq. (10), we obtain
Equation (14) can be written as:


 j
2
 v K  K    D K  K    2 R  2 mki  m  Qf  j
t
(13)
  i k k  k k  jk

 gij
2
 v K  K   K    vK 2
t

 D K  K 
t


 ui uk u j X X 
 xk
 ui u j uk X X 
 x k


 



 k     K  K   K  , K  , K  


 j  K , K  , K  

1
 i k k  k k  hijk K , K  , K  
 ki  ki  ki 
j

(15)
i
where, ij  K , K  , K  , t  = M gij  K , K  , K  , t  and 1-M = S , with M
 x k
and S are an arbitrary constants. Following the same
procedure as was used in obtaining (12), we get:
 ui u j ukX X 
 x k 
 kk  kk  kk h K , K  , K 
 j K , K  , K    
 ki  ki  ki ijk








(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 in a rotating
 
 
 u u  X X 
 v

  x x k  x k x k  i j
2
 2 R  2mki  m  Sf  gij

  i  k k  k k  k k   hijk K , K  , K   ik k hijk  K  K   K  , K  , K  


 uiuk u j X X 

 p  ui X X 
1  pu j X X 
  


 xi
 x j

2

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:
 ui u j X X 

2
32
Res. J. Math. Stat., 4(2): 30-38, 2012
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:
3
2i k k  k k bi 2

3
3
D 2 1  N s  2
1


4 kD 2 F   
 2


1
1 
  t  t0  2
1  N s  2 


 Ns2  Ns k 2
 exp  D 
 N s kk   1  N s  k  2

 1  N s 
Sf   
 2 
  mki m 
   t  t0   exp  2 R t  t0 

D
D   

(20)

where,

 gij  K , K  , K  , t   gij  K , K  , K  , t 0 
 
exp  D 2 N s k  1  N s  k   k 
2
2
2

F    exp   2


 2 N s  k k k k  k k k k  k k k k  2 R  2mki  m  Sf   t  t0 
expression for
ui uk X X   r, r  

ij
0
0
(21)
i
Now, by taking r   0 in the expression for u X X  r, r 
is performed, we obtain and comparing the result with the
expression Xuk X   r :
:

ij
1/2
 g  K  K  , K  , K  , t   b
and comparing it with the
  g  K  K  , K  , K  , t dK 
ik K , K  
 t  t0 D 

s 

2
and
(17)
N s = v / D . A r e l a t i o n betwe e n
gij  K , K  , K  , t0  and ij  K , K   can be found by taking r   0 in the
ui uj X X   r, r  , r  

0
whe r e ,
expression for
 expn dn,  k  91  N

k  K    k  K . K  dk 
(18)

(22)

Substituting Eq. (21) into (8), we obtain:
Substituting this in to Eq. (13), we obtains:
G
 2 Dk 2  2 R G  W
t

 j
2 R 2mki  m Sf 


 D  1  N s  K 2  K  2  2 N s K . K  


t
D
D
D  j






  i  k k  k k   gij  K  K  , K  , K  , t dK 


(23)
where,
G = 2Bk22
which with the help of Eq. (17), gives:
and
 j
2 R 2mki  m Sf 

 D  1  N s  K 2  K  2  2 N s K . K  



D
D
D  j
t



  2i k  k   g   K  K  , K  , K  , t 
k

k
ij
a
D 3 / 2 1  N s 
0
3/ 2

0
 
  
 j K , K  , t0   j K , K  , t0
 


 exp  D 1  N s  k 2  k  2
 t  t0  3/ 2



Qf  
 2 
 2 N s K . K    mki m 
   t  t0  

D
D 

(19)
bi  K , K    bi  K , K  
1  N s  

2i 5/ 2

D
3/ 2
3/ 2
 2
4 kF  w D1/2 


1/ 2 
  t  t0 
1  N s 1/2 

 Ns 2  Ns k 2
 exp  D 
 2 N s kk  
 1  N s 
Now, the solution of Eq. (19) is:

 kk k  K , K    k   K , K  0

2 R 2 mki  m Qf  


  t  t0 
D
D
D  


3/ 2

 Ns2  Ns k 2
 exp  D 
 2 N s kk   1  N s  k  2
 1  N s 



W  2ik 2 exp  2 R t  t0 




1  N s  k  2   2mkiD m  SfD    t  t0  dk  
0

 exp  D 1  N s  K 2  K  2  2 N s K . K

 2 R 2mki  m Qf  



  t  t0  
 D
D
D  

(24)
In analogy with the turbulent energy spectrum
function, the quantity G, in Eq. (23) can be called
33
Res. J. Math. Stat., 4(2): 30-38, 2012
  N 2  

N s k 11
s

  7
  1  
  1  Ns 
  D1  N s  t  t0  


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:
2 2 ik k 0  K , K    0   K , K  

  0 k k   k k 
2
4
4
2

 21  N  2 
N 3s k 13
s
 
  5 
 2  1  Ns 
 1  N s  2



   
 N  2 
s

  1
 1  Ns 



  exp n2 dn exp  2 R t  t0 
0

(25)
0
It is very interesting that:
and

8
7
2
3

 
i b K , K   b  K , K 
3
D 2 1  N s  2

  1 k 4 k  6  k 6 k  4
 W dk = 0

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:

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:
W
0
2 t  t0 
3
2
1
2N
3
D2
 

 exp  2 Dk
5
2
1
1

11
9
3
2 t  t0  2 D 2 21  N s  2
 3 4  7 N s  6  6



 f1 8  2 f1 9 F  
3
 2 N s

  N 2  

N s k 10
x

  7
  1  
  D1  N s  t  t0  
  1  Ns 


1

  N  2  
 21  N  2
N 2 s k 12
s
s

 
  1 
  5 

 1  N s  2  t  t 0    1  N s 
 2  1  Ns 





0 2 N s

Qf  
 2 
 exp   1  2 N s   2   mki m 


D
D 


 Ns 
105k 6
15k 8


3
2
3 2
3
 8 D N s  t  t0   1  N s  4 D N s  t  t0 
21
0

Qf  
 2 
 exp   21  N s 2   mki m 


D
D 

7
2


  N s  2  mki  m Sf  

 exp   2 D  
k 
   t  t0  
 D
2 D  

  1  N s 

1
2N
(27)
2

 N 0 1  N s  
G  exp   2 R t  t0  

 D  t  t0 
1 2 N s
1  N s 
2

2
  N  3  N   
k6
s
s
 
 
 k8
N s D t  t0    1  N s   1  N s   

 
3
D2
2
where, J (k) = N0k is a constant of integration and can be
obtained as by Corrsin (1951). Thus, by integrating the
right-hand side of equation, we obtain:
2

 Ns    Ns 
15k 4
3


   5
  
2 
2 2
1
N
1
N
2








s
s


 4 D N s  t  t0 
 t  t0 
 
 2 R t  t  dt
 exp  2 Dk  2 R  t  t0   W
s
1  N s 
7
2

G  J  k  exp  2 Dk 2  2 R  t  t0 


  1  2 N s  2  2mki  m Qf  
 exp  D  

k 
   t  t0  



N
D
D
1



s




(26)


1 2
15
3
2 t  t0  D 2 1  N s  2
7
 105 6 5


11  24 N 2 s  8 N s  8  f 2 10
16
 32

s

  

Sf  
 2 
 f 3 12  2 f 3 14 exp  2 2 Ei 2 2  0.5772   mki m 


D
D  

3
 t  t0  2 D1  N s  4


  1  2 N s   2mki  m Sf  
 ex   D  


   t  t0  
D
D  


  1  N s  
9
7

 Ns 
105k
15k


2
3
3
3 2
 8 D N s  t  t0   1  N s  4 D N s  t  t0 

1
2
21 N s
15
3
2t  t1  D 2 21  N s  2
7
 
 2 mki  m Sf  
 exp   1  2 N s  2  2  



D
D  
 
34
Res. J. Math. Stat., 4(2): 30-38, 2012
 105   8 2 10 23 12 498 14





3
30
630
 8 N s  3


Sf

2



2
 exp   2 2   mki m 
  E 2  0.5772

D
D i


  

In a turbulent phenomenon, such as turbulent kinetic
energy, we associate the so-called concentration energy
with the fluctuating concentration, defined by the relation:


1
XX    G k , t dk
2

 31808
.
 10 2  14  2.7178  10 2  16
 13968
.
 10 2  18  4.6204  10 3  20
 11584
.
 10 3  22  2.3139  10 4  24
 5 26
 55108
.
 10 
 3 22
 2.3139  10 4  24
The substitution of Eq. (28) and subsequent
integration with respect to k leads to the result:
 4 28
 38589
.
 10 
 11584
.
 10 

1 2
X  exp  2 R t  t0 
2
 23
5

 A t  t0   B t  t0  exp  R2  t  t0 

15
1
 15
15

   C t  t0  2  D1 t  t0  2  t  t1  2  E  t  t0 

1
 29
15

2
2
   F  t  t0   G t  t0   t  t1   exp  R3  t  t0 


 7 30
 6.8898  10   



15 N 2 s  2 N s  1   10 5 12 14 14




4 Ns
6
15
 2
  

Sf  
 2 
2
 exp   2 2   mki m 
 E 2  0.5772

D
D   i



 8.6172  10 6  22  9.22057  10 7  24
 8.9477  10 8  26  7.9374  10 9  28
 6.4814  10 10  30  

A
12
s

4 14

3
  

 6.66926  10 
 7 26
 8.6642  10 
D
 8.0290  10 6  24
 8.4643  10 8  28
d

 7.5495  10 9  30  
 
  
 Ei 2 2  0.5772
 2.2654  10 
, C  cd , D1  ce,
N s 2 1  N s 
7


e  2 f3
h 
 1 16
 10679
.
 10 
7

D 2 I  2 N s 2 , 
1  N s 

1
2
1 1
2 2
37.123111 N s 2
, f 
9
D8 2
1  N s 
, g
105
2 Ns
1 6 75.90 2.52052  103 1 2
  2 
15
3 1
 1
1  N  2
s
 3 18
 3.3996  10   9.0694  10 3  20
 11099
.
 10 3  22  3.7350  10 4  24
9
i1  1118
. 1 2
 6 28
 8.2388  10 
 
 7 30
 8.47486  10   

1
1 N s 2

D

 I 1  2 N s   ,  2


j1  1633 31808
.
 10 2  4.0767  10 1 1 1  3561804
.
1 2
(28)



1
 2.2386  101 3  11786
.
 102 1 4  5.4148  102 1 5

 2.02576  103 1 6  8.7048  1 7  31561
.
 104 1 8  
where,
2  D
D6
9
 1 14
 4.444  10 
0 R1
2
f3N 2 s
105 35 11  24 N s  8 N s
63 f 2
693 f 3 1808 f 3 415910
.





1
32
64 N s
256
64 N 2 s 64 N 3s
1  N  2

 5 26
15
22
1

Sf  
 2 
 1  2 N s  N s   14 exp   2   mki m 


D
D

3
,B
151
8
 4.90805  10 
5 22
N0
3 7 1
D2 22  2
c
 4 20
 31572
.
 10 

F  cf   gi1  k1m  ht   , G  cf pr1 ,

 8.2505  1010  14  17495
.
 10 2  16
 3 18
(30)
E  cf   gh  gj1  k1l1  k1n  pq  ps  t vw ,

Sf  
 2 
2
 exp   2 2   mki m 
  E 2  0.5772

D
D i










This is the decay law of first order reactant in
homogeneous dusty fluid turbulence in a rotating system,
where,
 5.4047  10 4  18  7.2369  10 5  20
 11N 2 s  20 N s



 2.0260  10 2  14  35487
.
 10 3  16


 10 N   

(29)
 t  t1 k 2 , f  4 3 N 2  2 N  3

s
1  N s  1 3  s

k1 
 15  40 N 2 s  2 N s  1  4 N 2 s 11N 2 s  20 N s
f2  
2

161  N s  D


 15  20 4 N
f3  


2
s

 2 Ns  4 N
2
s
9 N
81  N s 
3
2
s

N s1
 



11
m  7.44691 2 
 20 N s  16 N s 



2
n
35
 , l   1  55  2.3510  10
135 6 N s 2  2 N s  1
163
12
1
2
1
 1 N s   2
D

61

11
 1631 2
215 1  N s 
I 1  2 N s   2 ,  2
2.0261  5.3230  10 2 1
3
15
2
Res. J. Math. Stat., 4(2): 30-38, 2012
RESULTS AND DISCUSSION
 13782
.
 10 1 12  35063
.
 10 1 1 3
 8.7676  10 1 1 4  2.57710 1 1 5

5.23471 6

12538
.
1 7
 2.9690 

In Eq. (30) we obtained the decay law of first order
reactant in homogeneous dusty fluid turbulence for fourpoint correlation in a rotating system after neglecting
quintuple correlation terms. The equation contains the

101 8  

p  396 11N s2  20 N s  10 N s12
4
q   1  3.0532  10 1
1  N s 
1
1 N s  2
r1  10638
.
1 2  t  t1  2
1
13
13
2
D

terms
15
2

 I 1  2 N s   2 ,  2

s  1311 8.2504  2.6243  101  1  8.0509  101  2


1 3
2.3780 
 6.78561 4  18789
.

1 6
2 7
5.0688  10 1  13370
.
 10 1

101 5

t    14219
.
N s 1 1  N s  2
1
 21
, t  t 0  , t  t 0 
5
 15/ 2
. Thus, the terms
1
11
2
3

1 2

5
X  exp  2 R t  t0    A t  t0  2  B t  t0  
2


1

u     31886
.
.
 103  14443
 102  t  t0  2

 D 2 1  N s 
 3/ 2
0
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 lower-order correlations
seems to be valid in our case. If the system is non-rotating
and the fluid is clean (i.e., Sm = 0, f = 0) the Eq. (30)
becomes:
 3.4583  102 1 8  
2
t  t 

I 1  2 N s   2 ,  2
15
15
1


15
 C t  t0  2  D1 t  t0  2  t  t1  2  E  t  t0 

29
1 

15
F  t  t0  2  G t  t0   t  t1  2  
 

v  18102
.
 102 1  2 N S  1 2
1
2

15

w  2.2653  10 1  160195
.
1 1
(31)
 8.6664  10 1   2  4.3941  101 3
 11293
.
 102 1 4  8.7405  102 1 5
which is obtained by Kumar and Patel (1974). Here A, B,
C, D1, E, F, G are an arbitrary constants. With R = 0 and
the contaminant replaced by the temperature, the results
agrees with the result obtained by Loeffler and Deissler
(1960) for the decay of temperature fluctuation in
homogeneous turbulence before the final period for threepoint 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, y6 at t0 = .3, t1
= .5; t0 = .8, t1 = 1 and t0 = 1.3, t1 = 1.5, respectively the
 2.6001  103 1 6  13013
.
 104 1 7

 38822
.
 104 1 8  
R1 

21  N S 1  2 N s 
5
2
 9 5 N s  7 N s  6
 
 16 161  2 N s 
f 1 N s 1  2 N s  2
105 f1 N s
 21
11
321  2 N s 
2 2 1  N s  2
5


135
. . ...2n  9




n
2n
n  0 n! 2 N s  1 2 1  N s  
Sf 
 2mki  m Qf 
 2 

 , R   mki m 


D
D 3 
D
D
1  21  2 N s  , R2  
Y4, y5, y6 for Eq.(31)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
t1 = 1 and t0 = 1.3, t1 = 1.5,
respectively
y6




I   ,    exp  
2
2
0
2

14
Decay of total energy of homo
2
.turbulence = X
and
   dk
exp  Ei 
2
2
The first term of Eq. (30) corresponds to the
concentration energy for two-point concentration; the
second term represents first order reactant in
homogeneous dusty fluid turbulence in a rotating system
for three-point correlation.
The expression exp[!R2(t!t0)] represents the dusty
fluid turbulence in a rotating system for three-point
correlation; the expression exp[!R3(t!t0)] and the
remainder are due to dusty fluid turbulence for four-point
correlation in a rotating system.
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
y4
y5
y1
y3
y2
Y1, y2, y3 for Eq.(30)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
t1 = 1 and t0 = 1.3, t1 = 1.5, respectively
1.8
2.0
2.2
2.4 2.6 2.8 3.0
Approximation of time = t
3.2
3.4
Fig. 1: Comparison between Eq. (30) and (31) if R = 0.025
36
Res. J. Math. Stat., 4(2): 30-38, 2012
CONCLUSION
Decay of total energy of homo
2
.turbulence = X
Y1, y2, y3 for Eq.(30)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
t1 = 1 and t0 = 1.3, t1 = 1.5, respectively
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
From the above figures and discussions, this study
shows that the chemical reaction rate increases than the
concentration fluctuation to decay more decreases and
vice versa. At the chemical reaction rate R = 0 causes the
concentration fluctuation of decay more rapidly than they
would for (R…0) non pure mixing homogeneous dusty
fluid turbulence. Also we conclude that due to the effect
of rotation of homogeneous dusty fluid turbulence in the
flow field of the first order chemical reaction for fourpoint correlation the turbulent energy decays more slowly
than the energy decay for the first order reactant in
homogeneous turbulence.
Y4, y5, y6 for Eq.(31)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
y6 t1 = 1 and t0 = 1.3, t1 = 1.5,
respectively
y5
y3
y4
y2
y1
0
1.8
2.2
2.0
2.4
2.6
3.0
2.8
Approximation of time = t
3.4
3.2
Fig. 2: omparison between Eq. (30) and (31) if R = 0.5
ACKNOWLEDGMENT
Decay of total energy of homo
2
.turbulence = X
Y1, y2, y3 for Eq.(30)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
t1 = 1 and t0 = 1.3, t1 = 1.5, respectively
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
y5
y4
y3
This study is supported by the research grant under
the ‘National Science and Information and
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 preparing this study.
Y4, y5, y6 for Eq.(31)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
y6 t1 = 1 and t0 = 1.3, t1 = 1.5,
respectively
y2
REFERENCES
y1
1.8
2.0
2.2
2.4 2.6 2.8 3.0
Approximation of time = t
3.2
Aziz, M.A., M.A.K. Azad and M.S.A. Sarker, 2009. First
order reactant in Magneto-hydrodynamic turbulence
before the final period of decay for the case of multipoint and multi-time in a rotating system. Res. J.
Math. Stat., 1(2): 35-46.
Aziz, M.A., M.A.K Azad and M.S.A. Sarker, 2010. First
oder reactant in MHD turbulence before the final
period of decay for the case of multi-point and multitime in a rotating system in presence of dust particle.
Res. J. Math. Stat., 2(2): 56-68.
Batchelor, G.K., 1953. The Theory of Homogeneous
Turbulence, Cambridge University Press, Cambridge,
pp: 79.
Chandrasekhar, S., 1951. The invariant theory of isotropic
turbulence in magneto-hydrodynamics. Proc. Roy.
Soc., London, A204: 435-449.
Corrsin, S., 1951. On the spectrum of isotropic
temperature fluctuations in isotropic turbulence. J.
Apll. Phys., 22: 469-473.
Deissler, R.G., 1958. On the decay of homogeneous
turbulence before the final period. Phys. Fluid, 1:
111-121.
Deissler, R.G., 1960. A theory of decaying homogeneous
turbulence, Phys. Fluid, 3: 176-187.
Kumar, P. and S.R. Patel, 1974. First order reactant in
homogeneous turbulence before the final period of
decay. Phys. Fluids, 17: 1362-1368.
3.4
Fig. 3: Comparison between Eq. (30) and (31) if R =8
Decay of total energy of homo
2
.turbulence = X
Y4, y5, y6 for Eq.(31)at t0 = 0.3, t1 = 0.5, t0 = 0.8
t1 = 1 and t0 = 1.3, t1 = 1.5, respectively
5.0
y6
4.5
y4
y5
4.0
y3
3.5
3.0
y2
y1
2.5
2.0
1.5
1.0 Y1, y2, y3 for Eq.(30)
at t0 = 0.3, t1 = 0.5, t0 = 0.8
0.5
t1 = 1 and t0 = 1.3, t1 = 1.5, respectively
0
1.8 2.0 2.2 2.4 2.6 2.8 3.0
Approximation of time = t
3.2
3.4
Fig. 4: Comparison between Eq. (30) and (31) if R = 0
snergy decays more slowly. For R2 = 0.25 and R3 = 0.5,
the comparison between the Eq. (30) and (31) are shown
in Fig. 1, 2, 3 and 4 corresponding to the values R =
0.025, 0.5, 0.8 and 0, respectively, the energy decays
slowly than non rotating clean system which indicates in
the figures clearly.
37
Res. J. Math. Stat., 4(2): 30-38, 2012
Sarker, M.S.A. and M.A. Islam, 2001. Decay of dusty
fluid MHD turbulence before the final period in a
rotating system. J. Math and Math Sci., 16: 35-48.
Sultan, S., 2008. Energy decay law of dusty fluid
turbulent flow in a rotating system. J. Eng. Appl.
Sci., 3(2): 166-174.
Kumar, P. and S.R. Patel, 1975. First order reactant in
homogeneous turbulence before the final period for
the case of multi-point and multi-time. Int. J. Eng.
Sci., 13: 305-315.
Loeffler, A.L and R.G. Deissler, 1961. Decay of
temperature fluctuations in homogeneous turbulence
before the final period. Int. J. Heat Mass Transfer, 1:
312-324.
Sarker, M.S.A and N. Kishore, 1991. Decay of MHD
turbulence before the final period. Int. J. Engr. Sci.,
29: 1479-1485.
38