Research Journal of Mathematics and Statistics 2(2): 37-55, 2010
ISSN: 2040-7505
© M axwell Scientific Organization, 2010
Submitted Date: November 20, 2009
Accepted Date: January 25, 2010
Published Date: June 25, 2010
Statistical Theory of Distribution Functions in Magneto-hydrodynamic
Turbulence in a Rotating System Undergoing a First Order
Reaction in Presence of Dust Particles
M.A. Aziz, M.A.K. Azad and M.S.A. Sarker
Department of Applied Mathem atics, University of Rajshahi, Bangladesh
Abstract: In this research, a hierarchy of equations for evolution of one and two-point bivariate distribution
for simultaneous velocity, magnetic, temperature, concentration fields and reaction in MHD turbulent flow in
a rotating system undergoing a first order reaction in presence of dust particles have been derived. Various
properties of con structed distributions such as reduc tion, sep aration, coincidenc e, symmetric and
incom pressibility conditions have been discussed. Finally, a comparison of the equation for one-point
distribution functions in the case of zero viscosity and negligible diffusivity is made with the first equation of
BBG KY hierarchy in the kinetic theory of gases.
Key w ords: Concentration, distribution functions, first order reaction, MHD turbulence, rotating system
Kishore (1991) discussed the distribution functions in the
INTRODUCTION
statistical theory of convective MH D turbulence of an
incompressible fluid. Also Sarker and Kishore (1999)
In the past, several authors discussed the distribution
studied the distribution functions in the statistical theory
functions in the statistical theory of turbulence. The
of convective MHD turbulence of mixture of a miscib le
kinetic theory of gases and the statistical theo ry of fluid
incom pressible fluid. Sarker and Islam (2002) studied the
mechanics are the two major and distinct areas of
Distribution functions in the statistical theory of
investigations in statistical mechanics. Lundgren (1967)
convective MH D turbulence of an incomp ressible fluid in
derived a hierarchy of coupled equations for multi-point
a rotating system. Islam and Sarker (2007) studied
turbulence velocity distribution functions, which resemble
Distribution functions in the statistical theory of MHD
with BBG KY hierarchy of equations of Ta-You (1966) in
turbulence for velocity and concentration undergoing a
the kinetic theory of gasses. Kishore (1978) studied the
first order reaction. Azad and Sarker (2004) also sdudied
distributions functions in the statistical theory of MHD
the statistical theory of certain distribution functions in
turbulence of an incom pressible fluid. Pope (1981)
MHD turbulence in a rotating system in presence of dust
derived the transport equa tion for the joint probability
particles.
density function of ve locity and scalars in turbulent flow.
In present resea rch, w e hav e studied the statistical
Kishore and Singh (1984) derived the transport equation
theory of certain distribution function for simultaneous
for the bivariate joint distribution function of velocity and
velocity, magnetic, temperature, concentration fields and
temperature in turbulent flow. Also Kishore and
reaction in M HD turbulence in a rotating system in
Singh (1985) have been derived the transport equation for
presence of dust particles. Finally, the transport equations
the joint distribution function of velocity, temperature and
for evolution of distribution functions have been derived
concen tration in convective turbulent flow. Dixit and
and variou s prop erties of the distribution function have
Upadhyay (1989) considered the distribution functions in
been discussed. The resulting one-point equation is
the statistical theory of MH D turbulence of an
compared with the first equation of BBGKY hierarchy of
incom pressible fluid in the presence of the coriolis force.
equations and the closure difficulty is to be removed as in
Kollman and Janicka (1982) derived the transport
the case of ordinary turbulence.
equation for the probability density function of a scalar in
turbulent shear flow and considered a closure model
MATERIALS AND METHODS
based on gradient - flux mo del.B ut at this stage, on e is
met with the difficulty that the N-point distribution
Basic equations: The equations of motion and continuity
function depends upon the N+1-point distribution function
for viscous incompressible dusty fluid MHD turbulent
and thus result is an unclosed system. This soca lled
flow, the diffusion equations for the temperature and
“closer problem” is enco untere d in turbulenc e, kinetic
concentration undergoing a first order chemical reaction
theory and other non-linear system. Sarker and
in a rotating system are given by;
Corresponding Author: M.A. Aziz, Department of Applied Mathematics, University of Rajshahi, Bangladesh
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Res. J. Math. Stat., 2(2): 37-55, 2010
In a conducting infinite fluid only the particular solution
of the E q. (6) is related, so that
(1)
(7)
(2)
Hence E q. (1)-(4) becomes
(3)
(4)
with
(8)
(5)
where;
u " (x, t), " - component of turbulen t velocity; h " (x, t), " component of magn etic field; 2(x, t), temperature
fluctuation; C, concen tration of contamina nts; v a , dust
particle velocity; R, constant reaction rate; , m " $
(9)
alternating tensor;
, dimension of frequency;
N, constant numbe r of den sity of the dust particle;
( 1 0)
, total pressure;
( 1 1)
, hydrodynam ic pressure; D, fluid density; S,
angular velocity of a uniform rotation; v, Kinetic
visco sity;
,
magnetic
Formulation of the problem: We consider the
turbulence and the concentration fields are homogeneou s,
the chemical reaction and the local mass transfer have no
effect on the velocity field and the reaction rate and the
diffusivity are constant. We also consider a large
ensemble of identical fluids in which each member is an
infinite incompressible reacting and h eat conducting fluid
in turbulent state..The fluid velocity u, Alfven velocity h,
temperature 2 and concentration C, are rando mly
distributed functions of position and tim e and satisfy their
field. Different members of ensemble are subjected to
different initial conditions and our aim is to find out a way
by which we can determine the ensemble averages at the
initial time. C ertain m icroscopic properties of conducting
fluids, such as total energy, total pressure, stress tensor
which are nothing but ensemble averages at a particular
time, can be determine d with the he lp of the bivariate
distribution functions (defined as the averaged distribution
functions with the help of Dirac delta-functions). Our
present aim is to construct the distribution functions,
study its prope rties and derive an equation for its
evolution of this distribu tion fun ctions.
diffusiv ity;
, therm al diffusiv ity; c p , specific heat at
constant pressu re; k T , thermal conductivity; F, electrical
conductivity; :, magnetic permeability; D, diffusive coefficient for contaminants
The repeated suffices are assumed over the values 1,
2 and 3 and unrepeated suffices may take any of these
values. Here u, h and x are vector quantities in the w hole
process.
The total pressure w which, occu rs in Eq. (1) may be
eliminated with the help of the equation obtained by
taking the divergence of Eq. (1)
(6)
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Res. J. Math. Stat., 2(2): 37-55, 2010
Distribution function in MH D turbulence an d their properties: In MHD turbulence, we may consider the fluid
veloc ity u, Alfven velocity h, temperature ., concentration C and constant reaction rate R at each point of the flow field.
Lundgren (1967) has studied the flow field on the basis of one variable character only (namely the fluid u), but we can
study it for two or more variable characters as well. The corresponding to each point of the flow field, we have four
mea surab le characteristics. We represent the four variables by v, g, N and Q and denote the pairs of these variables at
the points
as:
at a fixed instant
of time.It is possible that the same pair may be occur more than once; herefore, we simplify the problem by an
assumption that the distribution is discrete (in the sense that no pairs occur more than once). Symbolically we can express
the distribution as:
Instead of considering discrete
points in the flow field, if we consider the continuous distribution of the variables
field, statistically behaviour of the fluid may be described by the distribution function
and Q over the entire flow
, which is
norm alized so that:
whe re the integration ranges over all the possible values of v, g, N and Q W e shall make use of the same normalization
condition for the discrete distributions also.Th e distribution functions of the above quantities can be defined in terms
of Dirac delta functions.
The one-point distribution function
define d so that
is the probab ility that the fluid velocity, Alfven veloc ity, temperature and concentration field at a time t are in the element
dv ( 1 ) abou t v ( 1 ), dg( 1 ) abou t g ( 1 ), dN ( 1 ) abou t N ( 1 ), dQ ( 1 ) abou t Q ( 1 ), respectively and is given by:
(12)
where d is the Dirac delta-function defined as:
Two-point distribution function is given by:
(13)
and three point distribution function is given by
(14)
Similarly, we can define an infinite numbers of multi-point distribution functions F 4 ( 1 ,2 , 3 ,4 ) , F 5 ( 1 ,2 , 3 ,4 , 5 ) and so on.
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Res. J. Math. Stat., 2(2): 37-55, 2010
The distribution functions so constructed have the follow ing pro perties:
Reduction properties: Integration w ith respect to pair of variables at one-point, lowers the order of distribution function
by one. Fo r exam ple,
Also the integration with respect to any one of the variables, reduces the number of eltafunctions from the distribution
function by one as:
and
Separation properties: The pairs of variables at the two points are statistically indepen dent of each othe r if these points
are far apart from each other in the flow field i.e.,
and similarly,
etc.
Co-incidence property: When two points coincide in the flow field, the components at these p oints sh ould b e obv iously
the same that is F 2 ( 1 ,2 ) must be zero. Thus
and
have the property.
and hence it follows that
Similarly,
40
but F 1 ( 1 ,2 ) must also
Res. J. Math. Stat., 2(2): 37-55, 2010
Sym metric conditions:
Incom pressibility conditions:
(i)
(ii)
Con tinuity equation in terms of distribution functions: An infinite number of continuity equations can be derived
for the convective MH D turbulent flow and the continuity equations can be easily expressed in terms of distribution
functions and are obtained directly by div u = 0. Taking ensem ble average of Eq. (5)
(15)
and similarly,
(16)
which are the first order continuity equations in which only one point distribution function is involved. For second-order
continuity equations, if we multiply the continuity equation by:
and if we take the ensemble average, we obtain:
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Res. J. Math. Stat., 2(2): 37-55, 2010
(17)
and similarly,
(18)
The Nth – order continuity equations are
(19)
and
(20)
The continuity equation s are sy mm etric in their argum ents i.e.;
(21)
Since the divergence property is an important property and it is easily verified by the use of the property of distribution
function as:
(22)
and all the pro perties o f the distribution fu nction obtained in section (*) can also be verified.
Equations for evolution of distribution functions: The Eq. (8)-(11) w ill be used to convert these into a set of equations
for the variation of the distribution function with time. This, in fact, is done by making use of the definitions of the
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Res. J. Math. Stat., 2(2): 37-55, 2010
constructed distribution functions, differentiating them pa rtially with respe ct to time, making some suitable operations
on the right-hand side of the equation so obtained and lastly replacing the time de rivative of v, h, 2 and C from the Eq.
(8)- (11). Differe ntiating Eq. (12), and then using E q. (8)-(11 ) we get,
(23)
Using Eq. (8) - (11 ) in the Eq. (23), w e get:
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Res. J. Math. Stat., 2(2): 37-55, 2010
44
Res. J. Math. Stat., 2(2): 37-55, 2010
(24)
Various terms in the Eq. (24) can be simplified as that they may be expressed in terms of one point and two point
distribution fun ctions.
The 1st term on the right hand side of the abov e equ ation is simplified as follow s:
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Res. J. Math. Stat., 2(2): 37-55, 2010
(25)
Similarly, 7th, 10th and 12th terms of right hand-side of Eq. (24) can be simplified as follows;
(26)
10th term,
(27)
and 12th term
(28)
Adding (25) - (28), we get
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Res. J. Math. Stat., 2(2): 37-55, 2010
[ Using the properties of distribution functions ]
(29)
Similarly, 2nd and 8th terms on the right hand-side of the Eq. (24) can be simplified as:
(30)
and
(31)
4th term can be reduced as
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Res. J. Math. Stat., 2(2): 37-55, 2010
(32)
9th, 11th and 13th terms of the right hand side o f Eq. (24)
(33)
(34)
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Res. J. Math. Stat., 2(2): 37-55, 2010
(35)
Now, we reduce the 3rd term of right hand side of Eq. (24)
(36)
5th and 6th terms of right hand side of Eq. (24)
(37)
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Res. J. Math. Stat., 2(2): 37-55, 2010
and
(38)
And, the last term of the Eq . (24) red uces to
(39)
Substituting the results (25)-(39) in Eq. (24) we get the transport equation for one point distribution function
, in MHD turbulence for con centration un dergoing a first order reaction in a rotating system in
presence of dust particles as:
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Res. J. Math. Stat., 2(2): 37-55, 2010
(40)
Similarly, an equation for two-point distribution function
in MHD dusty fluid turbulence for concentration
undergoing a first order reaction in a rotating sy stem can be derived by differentiating Eq. (13) and using Eq. (2), (3),
(4), (8) and simplifying in the same man ner, which is:
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Res. J. Math. Stat., 2(2): 37-55, 2010
(41)
Following this way, w e can derive the equations for evolution of
and so on. Logically, it is possible
to have an equation for every Fn (n is an integer) but the system of equations so obtained is not closed. It seems that
certain appro xima tions w ill be requ ired thus obtained.
RESULTS AND DISCUSSION
If the reaction constant R = 0 , then the transport equation for one point distribution function in MH D turbulent flow (40)
becomes:
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Res. J. Math. Stat., 2(2): 37-55, 2010
(42)
W hich w as obtained earlier by Azad an d Sarker (20 04).
If the fluid is clean and the system is non rotating then f=0 and .m=0, the transport equation for one point distribution
function in MHD turbulent flow (40) becomes:
(43)
W hich was obtained earlier by Islam and Sarker (200 7).
W e can exhibit an analogy of this equation with the first equation in BBG KY hierarchy in the kinetic theory of
gases. The first equation of BBGK Y hierarchy is given as
(44)
where
is the inter m olecu lar potential.
If we drop the viscous, magnetic and thermal diffusive, concentration terms and constant reaction terms from the
one point evolution Eq. (43), we have:
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Res. J. Math. Stat., 2(2): 37-55, 2010
(44)
The existence of the term:
can be explained on the basis that tw o cha racteristics of the flow field are related to each other and describe the
interaction between the two mod es (velocity an d ma gnetic) at a sing le point x(1).
In order to close the system of equations for the distribution functions, some approximations are required. If we
consider the collection of ionized particles, i.e., in plasma turbulence case, it can be provided closure form easily by
decomposing
as
. But such type of approximations can be possible if there is no interaction or
correlation between two particles. If we decompose
as
and
whe re , is the correlation coefficient between the particles. If there is no correlation between the particles, , will be zero
and distribution function can b e decomp osed in usual wa y. Here w e are considering such type of appro ximation on ly
to prov ide closed from of the equ ation i.e., to appro ximate two-point equation as one point equation.
The transport equation for distribution function of velocity, magnetic, temperature, concentration and reaction have
been show n here to prov ide an adva ntage ous b asis for modeling the turbulent flows in a rotating system in presence of
dust particles. Here we have made an attempt for the modeling of various terms such as fluctu ating pressure, visco sity
and diffusivity in order to close the equation for distribution function of velocity, magnetic, temperature, concentration
and reaction. It is also possible to construct such type of distribution functions in variable density follows.
Kishore, N., 1978. Distribution functions in the statistical
theory of MHD turbulence of an incom pressible fluid
J. Sci. R es., BHU , 28(2): 163.
Kishore, N. and S.R. Singh, 1984. The effect of Coriolis
force on acceleration covariance in turbulent flow
with rotational sym metry . Astrophys. Space Sci.,
104: 121-1 25.
Kishore, N. and S.R. Singh, 1985. The transport equation
for the joint distribution function of velocity,
temperature and concentration in convective
turbulent flow . Prog. Maths, B HU , 19(1-2 ): 13.
Kollman, W. and J. Janica, 1982. The transp ort equation
for the probability density function of a scalar in
turbulent she ar flow . Phys. Fluid., 25: 1955.
Lundgren, T.S., 1967. Hierarchy of coupled equations for
multi-point turbulence velocity distribution functions.
Phys. Fluid., 10: 967.
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