First Order Reactant in the Statistical Theory of …...First Order Reactant in the Statistical Theory of Three- Point Distribution Functions in Dusty Fluid MHD Turbulent Flow for

First Order Reactant in the Statistical Theory of Three- Point Distribution Functions in Dusty Fluid MHD Turbulent Flow for Velocity, Magnetic Temperature and Concentration in Presence of Coriolis Force

By M. A. K. Azad, M. Mamun Miah, Mst. Mumtahinah, Abdul Malek & M. Masidur Rahman

University of Rajshahi, Bangladesh Abstract- In this paper, an attempt is made to study the three-point distribution function for simultaneous velocity, magnetic, temperature and concentration fields in dusty fluid MHD turbulence in presence of coriolis force under going a first order reaction. The various properties of constructed distribution functions have been sufficiently discussed. In this study, the transport equation for three-point distribution function under going a first order reaction has been obtained. The resulting equation is compared with the first equation of BBGKY hierarchy of equations and the closure difficulty is to be removed as in the case of ordinary turbulence.

Keywords: dust particles, coriolis force, magnetic temperature, concentration, three-point distribution functions, MHD turbulent flow, first order reactant.

GJSFR-A Classification : FOR Code: 010506p, 040403

FirstOrderReactantintheStatisticalTheoryofThreePointDistributionFunctionsinDustyFluidMHDTurbulentFlowforVelocityMagneticTemperatureandConcentrationinPresenceofCoriolisForce

Strictly as per the compliance and regulations of :

© 2016. M. A. K. Azad, M. Mamun Miah, Mst. Mumtahinah, Abdul Malek & M. Masidur Rahman. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Global Journal of Science Frontier Research: Physics and Space ScienceVolume 16 Issue 1 Version 1.0 Year 2016 Type : Double Blind Peer Reviewed International Research JournalPublisher: Global Journals Inc. (USA)

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Online ISSN: 2249-4626 & Print ISSN: 0975-5896

First Order Reactant in the Statistical Theory of Three- Point Distribution Functions in Dusty

Fluid MHD Turbulent Flow for Velocity, Magnetic Temperature and Concentration in Presence of

Coriolis Force M. A. K. Azad α, M. Mamun Miah σ, Mst. Mumtahinah ρ, Abdul Malek Ѡ & M. Masidur Rahman ¥

Abstract- In this paper, an attempt is made to study the three-point distribution function for simultaneous velocity, magnetic, temperature and concentration fields in dusty fluid MHD turbulence in presence of coriolis force under going a first order reaction. The various properties of constructed distribution functions have been sufficiently discussed. In this study, the transport equation for three-point distribution function under going a first order reaction has been obtained. The resulting equation is compared with the first equation of BBGKY hierarchy of equations and the closure difficulty is to be removed as in the case of ordinary turbulence. Keywords: dust particles, coriolis force, magnetic temperature, concentration, three-point distribution functions, MHD turbulent flow, first order reactant.

I. Introduction

t present, two major and distinct areas of investigations in non-equilibrium statistical mechanics are the kinetic theory of gases and

statistical theory of fluid mechanics. In molecular kinetic theory in physics, a particle's distribution function is a function of several variables. Particle distribution functions are used in plasma physics to describe wave-particle interactions and velocity-space instabilities. Distribution functions are also used in fluid mechanics, statistical mechanics and nuclear physics. Various analytical theories in the statistical theory of turbulence have been discussed in the past by Hopf (1952) Kraichanan (1959), Edward (1964) and Herring (1965). Author α: Associate Professor, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh. e-mail: [email protected] Author σ: Research Fellow, Department of Applied Mathemtics, University of Rajshahi, Rajshahi-6205, Bangladesh, e-mail: [email protected] Author ρ: Lecturer, Department of Business Administration Ibais University, Dhanmondi-16, Dhaka, Bangladesh. e-mail: [email protected]

Author Ѡ: Research Fellow, Department of Applied Mathemtics, University of Rajshahi, Rajshahi-6205, Bangladesh. e-mail: [email protected] Author ¥: Research Fellow, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh. e-mail: [email protected]

Lundgren (1967) derived a hierarchy of coupled equations for multi-point turbulence velocity distribution functions, which resemble with BBGKY hierarchy of equations of Ta-You (1966) in the kinetic theory of gasses.

Kishore (1978) studied the Distributions functions in the statistical theory of MHD turbulence of an incompressible fluid. Pope (1979) studied the statistical theory of turbulence flames. Pope (1981) derived the transport equation for the joint probability density function of velocity and scalars in turbulent flow. Kollman and Janicka (1982) derived the transport equation for the probability density function of a scalar in turbulent shear flow and considered a closure model based on gradient – flux model. Kishore and Singh (1984) derived the transport equation for the bivariate joint distribution function of velocity and temperature in turbulent flow. Also Kishore and Singh (1985) have been derived the transport equation for the joint distribution function of velocity, temperature and concentration in convective turbulent flow. On a rotating earth the Coriolis force acts to change the direction of a moving body to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. This deflection is not only instrumental in the large-scale atmospheric circulation, the development of storms, and the sea-breeze circulation Also a first-order reaction is defined a reaction that proceeds at a rate that depends linearly only on one reactant concentration. Later, the following some researchers included coriolis force and first order reaction rate in their works.

Dixit and Upadhyay (1989) considered the distribution functions in the statistical theory of MHD turbulence of an incompressible fluid in the presence of the coriolis force. Sarker and Kishore (1991) discussed the distribution functions in the statistical theory of convective MHD turbulence of an incompressible fluid. Also Sarker and Kishore (1999) studied the distribution functions in the statistical theory of convective MHD turbulence of mixture of a miscible incompressible fluid. Sarker and Islam (2002) studied the Distribution

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functions in the statistical theory of convective MHD turbulence of an incompressible fluid in a rotating system. In the continuation of the above researcher, Azad and Sarker(2003) considered the decay of MHD turbulence before the final period for the case of multi-point and multi-time in presence of dust particle. Azad and Sarker (2004a) discussed statistical theory of certain distribution functions in MHD turbulence in a rotating system in presence of dust particles. Sarker and Azad (2004b) studied the decay of MHD turbulence before the final period for the case of multi-point and multi-time in a rotating system. Sarker and Azad(2006), Islam and Sarker (2007) studied distribution functions in the statistical theory of MHD turbulence for velocity and concentration undergoing a first order reaction. Azad and Sarker(2008) deliberated the decay of temperature fluctuations in homogeneous turbulence before the final period for the case of multi- point and multi- time in a rotating system and dust particles. Azad and Sarker(2009a) had measured the decay of temperature fluctuations in MHD turbulence before the final period in a rotating system. Azad, M. A. Aziz and Sarker (2009b, 2009c) studied the first order reactant in Magneto-hydrodynamic turbulence before the final Period of decay with dust particles and rotating System. Aziz et al (2009d, 2010c) discussed the first order reactant in Magneto- hydrodynamic turbulence before the final period of decay for the case of multi-point and multi-time taking rotating system and dust particles. Aziz et al (2010a, 2010b) studied the statistical theory of certain Distribution Functions in MHD turbulent flow undergoing a first order reaction in presence of dust particles and rotating system separately. 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. Azad et al (2012) derived the transport equatoin for the joint distribution function of velocity, temperature and concentration in convective tubulent flow in presence of dust particles. Molla et al (2012) studied the decay of temperature fluctuations in homogeneous turbulenc before the final period in a rotating system. Bkar Pk. et al (2012) studed the First-order reactant in homogeneou dusty fluid turbulence prior to the ultimate phase of decay for four-point correlation in a rotating system. Azad and Mumtahinah(2013) studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence prior to final period. Bkar Pk. et al (2013a,2013b) discussed the first-order reactant in homogeneous turbulence prior to the ultimate phase of decay for four-point correlation with dust particle and rotating system. Bkar Pk.et al (2013,2013c, 2013d)

studied the decay of MHD turbulence before the final period for four- point correlation in a rotating system and dust particles. Very recent Azad et al (2014a) derived the transport equations of three point distribution functions in MHD turbulent flow for velocity, magnetic temperature and concentration, Azad and Nazmul (2014b) considered the transport equations of three point distribution functions in MHD turbulent flow for velocity, magnetic temperature and concentration in a rotating system, Nazmul and Azad (2014) studied the transport equations of three-point distribution functions in MHD turbulent flow for velocity, magnetic temperature and concentration in presence of dust particles. Azad and Mumtahinah (2014) further has been studied the transport equatoin for the joint distribution functions in convective tubulent flow in presence of dust particles undergoing a first order reaction.Very recently, Bkar Pk.et al (2015) considering the effects of first-order reactant on MHD turbulence at four-point correlation. Azad et al (2015) derived a transport equation for the joint distribution functions of certain variables in convective dusty fluid turbulent flow in a rotating system under going a first order reaction. Bkar Pk et al (2015), Azad et al (2015a, 2015b, 2015c, and 2015d) have done their research on MHD turbulent flow considering 1st order chemical reaction for three- point distribution function. Also Bkar Pk. (2015a) studied 4-point correlations of dusty fluid MHD turbulent flow in a 1st order reaction. Bkar Pk (2015b) extended the above problem consiodering Coriolis force.

In present paper, the main purpose is to study the statistical theory of three-point distribution function for simultaneous velocity, magnetic, temperature, concentration fields in MHD turbulence in a rotating system in presence of dust particles Under going a first order reaction. Through out the study, the transport equations for evolution of distribution functions have been derived and various properties of the distribution function have been discussed. The obtained three-point transport equation is compared with the first equation of BBGKY hierarchy of equations and the closure difficulty is to be removed as in the case of ordinary turbulence.

II. Material and Methods

Basic Equations The equations of motion and continuity for

viscous incompressible dusty fluid MHD turbulent flow in a rotating system with constant reaction rate, the diffusion equations for the temperature and concentration are given by

( ) ( )ααααβαα

βαβαβ

α ν vufuuxwhhuu

xtu

mm −+Ω∈−∇+∂∂

−=−∂∂

+∂∂ 22 (1)

First Order Reactant in the Statistical Theory of Three- Point Distribution Functions in Dusty Fluid MHDTurbulent Flow for Velocity, Magnetic Temperature and Concentration in Presence of Coriolis Force


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)

( ) αβαβαβ

α λ hhuuhxt

h 2∇=−∂∂

+∂∂

, (2)

θγθθ

ββ

2∇=∂∂

+∂∂

xu

t, (3)

RccDxcu

tc

−∇=∂∂

+∂∂ 2

ββ

(4)

with 0=∂∂

=∂∂

=∂∂

α

α

α

α

α

α

xh

xv

xu

, (5)

where

( )txu ,α , α – component of turbulent velocity,

( )txh ,α ; α – component of magnetic field; ( )tx,θ ,

temperature fluctuation; c, concentration of

contaminants; vα, dust particle velocity; αβm∈ ,

alternating tensor; ,ρ

KNf = dimension of frequency; N,

constant number of density of the dust particle;

( ) ,ˆˆ21

21,ˆ

22xhPtxw ×Ω++=

ρ total pressure;

( )txP ,ˆ , hydrodynamic pressure; ρ, fluid density; Ω,

angular velocity of a uniform rotation; ν, Kinematic

viscosity; ( ) 14 −= πµσλ , magnetic diffusivity; p

T

ckρ

γ = ,

thermal diffusivity; cp, specific heat at constant pressure; kT, thermal conductivity; σ, electrical conductivity; µ, magnetic permeability; D, diffusive co-efficient for contaminants; R, constant reaction rate.

The repeated suffices are assumed over the values 1, 2 and 3 and unrepeated suffices may take any of these values. In the whole process u, h and x are the vector quantities.

The total pressure w which, occurs in equation (1) may be eliminated with the help of the equation obtained by taking the divergence of equation (1)

( ) [ ]α

β

β

α

α

β

β

αβαβα

βα xh

xh

xu

xuhhuu

xxw

∂∂

∂∂

−∂∂

∂∂

−=−∂∂∂

−=∇2

2 (6)

In a conducting infinite fluid only the particular solution of the Equation (6) is related, so that

[ ]xx

xxh

xh

xu

xuw

−′′∂

′∂

′∂′∂′∂

−′∂

′∂′∂′∂

= ∫α

β

β

α

α

β

β

α

π41

(7)

Hence equation (1) – (4) becomes

( ) [ ] αα

β

β

α

α

β

β

α

αβαβα

β

α νπ

uxx

xdxh

xh

xu

xu

xhhuu

xtu 2

41

∇+−′′

′∂

′∂′∂′∂

−′∂

′∂′∂′∂

∂∂

−=−∂∂

+∂∂

∫

( )ααααβ vufumm −+Ω∈− 2 , (8)

( ) αβαβαβ

α λ hhuuhxt

h 2∇=−∂∂

+∂∂

, (9)

θγθθ

ββ

2∇=∂∂

+∂∂

xu

t, (10)

RccDxcu

tc

−∇=∂∂

+∂∂ 2

ββ , (11)


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Formulation of the Problem It has considered that the turbulence and the concentration fields are homogeneous, the chemical reaction and the local mass transfer have no effect on the velocity field and the reaction rate and the diffusivity are constant. It is also considered a large ensemble of identical fluids in which each member is an infinite incompressible reacting and heat conducting fluid in turbulent state. The fluid velocity u, Alfven velocity h, temperature θ and concentration c are randomly distributed functions of position and time and satisfy their field. Different members of ensemble are subjected to different initial conditions and the aim is to find out a way by which we can determine the ensemble averages at the initial time. Certain microscopic properties of conducting fluids such as total energy, total pressure, stress tensor which are nothing but ensemble averages at a particular time can be determined with the help of the bivariate distribution functions (defined as the averaged distribution functions with the help of Dirac delta-functions). The present aim is to construct a joint distribution function for its evolution of three-point distribution functions in dusty fluid MHD turbulent flow in

a rotating system under going first order reaction, study its properties and derive a transport equation for the joint distribution function of velocity, temperature and concentration in dusty fluid MHD turbulent flow in a rotating system under going a first order reaction.

Distribution Function in MHD Turbulence and Their Properties

In MHD turbulence, it is considered that the fluid velocity u, Alfven velocity h, temperature θ and concentration c at each point of the flow field. Corresponding to each point of the flow field, there are four measurable characteristics represent by the four variables by v, g, φ and ψ and denote the pairs of these

variables at the points )()2()1( ,,, nxxx −−−−− as( ),,,, )1()1()1()1( ψφgv ( ),,,, )2()2()2()2( ψφgv --- ( ))()()()( ,,, nnnn gv ψφ at a fixed instant of time.

It is possible that the same pair may be occurred more than once; therefore, it simplifies 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 bivariate distribution as

( );,,, )1()1()1()1( ψφgv ( ) ( ) )()()()()2()2()2()2( ,,,;,,, nnnn gvgv ψφψφ −−−−−−

Instead of considering discrete points in the flow field, if it is considered the continuous distribution of the variables φ,, gv and ψ over the entire flow field, statistically behavior of the fluid may be described by the distribution function ( )ψφ ,,, gvF which is normalized so that

( ) 1,,, =∫ ψφψφ ddgdvdgvF

where the integration ranges over all the possible values of v, g,φ and ψ. We shall make use of the same normalization condition for the discrete distributions also.

The distribution functions of the above quantities can be defined in terms of Dirac delta function.

The one-point distribution function ( ))1()1()1()1()1(

1 ,,, ψφgvF , defined so that

( ) )1()1()1()1()1()1()1()1()1(1 ,,, ψφψφ dddgdvgvF is the probability

that the fluid velocity, Alfven velocity, temperature and concentration at a time t are in the element dv(1) about

v(1), dg(1) about g(1), d )1(φ about )1(φ and dψ(1) about ψ(1) respectively and is given by

( ) ( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1()1()1()1()1(1 ,,, ψδφθδδδψφ −−−−= cghvugvF (12)

where δ is the Dirac delta-function defined as

( ) ∫ =− 10vdvuδ

vupoint at theelsewhere

=

Two-point distribution function is given by

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1()2,1(2 ghvucghvuF −−−−−−= δδψδφθδδδ ( ) ( ))2()2()2()2( ψδφθδ −− c (13)

and three point distribution function is given by

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1()3,2,1(3 ghvucghvuF −−−−−−= δδψδφθδδδ

( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2( ψδφθδδδψδφθδ −−−−−−× cghvuc (14)



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Similarly, we can define an infinite numbers of multi-point distribution functions F4

(1,2,3,4), F5(1,2,3,4,5) and so

on. The following properties of the constructed distribution functions can be deduced from the above definitions:

a) Reduction Properties Integration with respect to pair of variables at

one-point lowers the order of distribution function by one. For example,

∫ =1)1()1()1()1()1(1 ψφ dddgdvF ,

∫ = )1(1

)2()2()2()2()2,1(2 FdddgdvF ψφ ,

∫ = )2,1(2

)3()3()3()3()3,2,1(3 FdddgdvF ψφ

And so on. Also the integration with respect to any one of the variables, reduces the number of Delta-functions from the distribution function by one as

( ) ( ) ( ))1()1()1()1()1()1()1()1(1 ψδφθδδ −−−=∫ cghdvF ,

( ) ( ) ( ))1()1()1()1()1()1()1()1(1 ψδφθδδ −−−=∫ cvudgF ,

( ) ( ) ( ))1()1()1()1()1()1()1()1(1 ψδδδφ −−−=∫ cghvudF ,

and

( ) ( ) ( ) ( ) ( ))2()2()1()1()1()1()1()1()1()1()2()2,1(2 ghcghvudvF −−−−−=∫ δψδφθδδδ

( ) ( ))2()2()2()2( ψδφθδ −− c

b) Separation Properties If two points are far apart from each other in the flow field, the pairs of variables at these points are

statistically independent of each other i.e.,

and similarly,

etc.

c) Co-incidence Properties When two points coincide in the flow field, the components at these points should be obviously the same

that is F2(1,2) must be zero.

Thus ,)1()2( vv = ,)1()2( gg = )1()2( φφ = and )1()2( ψψ = , but F2(1,2) must also have the property.

∫ = )1(1

)2()2()2()2()2,1(2 FdddgdvF ψφ

and hence it follows that

Similarly,

etc.


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lim

∞→− )1()2( xx )2(1

)1(1

)2,1(2 FFF =

lim

∞→− )2()3( xx )3(1

)2,1(2

)3,2,1(3 FFF =

lim

∞→− )1()2( xx ( ) ( ) ( ) ( ))1()2()1()2()1()2()1()2()1(1

)2,1(2 ψψδφφδδδ −−−−=∫ ggvvFF

∞→− )2()3( xx ( ) ( ) ( ) ( ))1()3()1()3()1()3()1()3()2,1(2

)3,2,1(3 ψψδφφδδδ −−−−=∫ ggvvFF

lim

d) Symmetric Conditions

),,,2,1(),,,2,1( nrsn

nsrn FF −−−−−−−−−−−−−−−−−−−−−− = .

e) Incompressibility Conditions

(i) ∫ =∂

∂ −−−

0)()()()(

),2,1(rrr

r

nn hdvdv

xF

αα

(ii) ∫ =∂

∂ −−−

0)()()()(

),2,1(rrr

r

nn hdvdh

xF

αα

Continuity Equation in Terms of Distribution Functions The continuity equations can be easily expressed in terms of distribution functions. An infinite number of

continuity equations can be derived for the convective MHD turbulent flow and are obtained directly by using div0=u

Taking ensemble average of equation (5), we get

∫∂∂

=∂∂

= )1()1()1()1()1(1

)1()1()1(

)1(0 ψφα

αα

α dddgdvFuxx

u

∫∂∂

= )1()1()1()1()1(1

)1()1( ψφα

αdddgdvFu

x

)1()1()1()1()1(1

)1()1( ψφα

αdddgdvFu

x ∫∂∂

=

)1()1()1()1()1(1

)1()1( ψφα

αdddgdvFv

x ∫∂∂

=

)1()1()1()1()1()1(

)1(1 ψφαα

dddgdvvxF∂

∂= ∫ (15)

and similarly,

)1()1()1()1()1()1(

)1(10 ψφαα

dddgdvgxF∂∂

= ∫ (16)

Equation (15) and (16) 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

( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2( ψδφθδδδ −−−− cghvu

and if we take the ensemble average, we obtain

( ) ( ) ( ) ( ) )1(

)1()2()2()2()2()2()2()2()2(

α

αψδφθδδδxucghvuo∂∂

−−−−=

( ) ( ) ( ) ( ) )1()2()2()2()2()2()2()2()2()1( α

αψδφθδδδ ucghvu

x−−−−

∂∂

=

[ ( ) ( ) ( ) ( )∫ −−−−∂∂

= )1()1()1()1()1()1()1()1()1()1( ψδφθδδδα

αcghvuu

x



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( ) ( ) ( ) ( ) ])1()1()1()1()2()2()2()2()2()2()2()2( ψφψδφθδδδ dddgdvcghvu −−−−×

∫∂∂

= )1()1()1()1()2,1(2

)1()1( ψφα

αdddgdvFv

x (17)

and similarly,

∫∂∂

= )1()1()1()1()2,1(2

)1()1( ψφα

αdddgdvFg

xo (18)

The Nth – order continuity equations are

)1()1()1()1(),,2,1()1(

)1( ψφαα

dddgdvFvx

o NN∫ −−−

∂

∂= (19)

and

∫∂

∂= )1()1()1()1(),,.........2,1()1(

)1(ψφα

α

dddgdvFgx

o NN (20)

The continuity equations are symmetric in their arguments i.e.

( ) )()()()(),.....,,.....2,1()()(

)()()()()..;,.........2,1()()(

ssssNsrN

ss

rrrrNrN

rr dddgdvFv

xdddgdvFv

xψφψφ α

αα

α∫∂

∂=

∂∂

(21)

Since the divergence property is an important property and it is easily verified by the use of the property of distribution function as

oxu

ux

dddgdvFvx

=∂

∂=

∂

∂

∂

∂∫ )1(

)1()1(

)1()1()1()1()1()1(

1)1(

)1(α

αα

αα

α

ψφ (22)

and all the properties of the distribution function obtained in section (4) can also be verified.

Equations for evolution of one –point distribution function )1(1F

It shall make use of equations (8) - (11) 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 constructed distribution functions, differentiating them partially with respect to time, making some suitable operations on the right-hand side of the equation so obtained and lastly replacing the time derivative of θ,,hu and c from the equations (8) - (11). Differentiating equation (12) with respect to time and using equation (8) - (11), we get,

( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1(

1 ψδφθδδδ −−−−∂∂

=∂

∂cghvu

ttF

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) )1()1()1()1()1()1()1()1(

)1()1()1()1()1()1()1()1(

ght

cvu

vut

cgh

−∂∂

−−−+

−∂∂

−−−=

δψδφθδδ

δψδφθδδ

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1(

)1()1()1()1()1()1()1()1(

ψδφθδδδ

φθδψδδδ

−∂∂

−−−+

−∂∂

−−−+

ct

ghvu

tcghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( vu

vtucgh −

∂

∂∂

∂−−−−= δψδφθδδ


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( A)

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( gh

gthcvu −

∂

∂∂

∂−−−−+ δψδφθδδ

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( φθδ

φθψδδδ −

∂∂

∂∂

−−−−+t

cghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( ψδ

ψφθδδδ −

∂

∂∂

∂−−−−+ c

tcghvu (23)

Using equations (8) to (11) in the equation (23), we get

( ) ( ) ( ) ( ))1()1()1()1()1(

)1()1()1()1()1()1()1(

1βαβα

βψδφθδδ hhuu

xcgh

tF

−∂∂

−−−−−=∂

∂

[ ]

( ) ( ))1()1()1(

)1()1(

)1()1(2)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1( 241

vuv

vuf

uuxx

xdx

h

xh

x

u

xu

xmm

−∂

∂×−+

Ω∈−∇+−′′

∂

∂

∂

∂−

∂

∂

∂

∂

∂

∂− ∫

δ

νπ

ααα

ααβαα

β

β

α

α

β

β

α

α

( ) ( ) ( ) ( ) )1(2)1()1()1()1()1(

)1()1()1()1()1()1(αβαβα

βλψδφθδδ hhuuh

xcvu ∇+−

∂∂

−−−−−+

( ) ( ) ( ) ( ) )1(2)1(

)1()1()1()1()1()1()1()1()1()1(

)1( θγθψδδδδβ

βα

∇+∂∂

−−−−−+−∂∂

×x

ucghvughg

( ) ( ) ( ) ( ) 12)1(

)1()1()1()1()1()1()1()1()1()1(

)1(RccD

xcughvu −∇+∂

∂−−−−−+−

∂

∂×

ββφθδδδφθδ

φ

( ))1()1()1( ψδ

ψ−

∂∂

× c

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( vu

vx

uucgh −

∂∂

∂

∂−−−= δψδφθδδ

αβ

βα

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( vu

vxhh

cgh −∂∂

∂

∂−−−−+ δψδφθδδ

αβ

βα

( ) ( ) ( ) [ ])1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()1()1()1()1()1()1(

41

α

β

β

α

α

β

β

α

απψδφθδδ

x

h

xh

x

u

xu

xcgh

∂

∂

∂∂

−∂

∂

∂∂

∂∂

−−−+ ∫

( )

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1(

)1()1()1(

vuv

ucgh

vuvxx

xd

−∂∂

∇×−−−−+

−∂∂

−′′

×

δνψδφθδδ

δ

αα

α

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( 2 vuv

ucgh mm −∂∂

Ω∈×−−−+ δψδφθδδα

ααβ



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)

( ) ( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1()1( vuv

vufcgh −∂∂

−×−−−−+ δψδφθδδα

αα

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( gh

gx

uhcvu −

∂∂

∂

∂×−−−+ δψδφθδδ

αβ

βα

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( gh

gx

hucvu −

∂∂

∂

∂×−−−−+ δψδφθδδ

αβ

βα

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1( ghg

hcvu −∂∂

∇×−−−−+ δλψδφθδδα

α

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( φθδ

φθψδδδβ

β −∂∂

∂∂

×−−−+x

ucghvu

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1( φθδφ

θγψδδδ −∂∂

∇×−−−−+ cghvu

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( ψδ

ψφθδδδ

ββ −

∂

∂

∂

∂×−−−+ c

xcughvu

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1( ψδψ

φθδδδ −∂

∂∇×−−−−+ ccDghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( ψδψ

φθδδδ −∂

∂×−−−+ cRcghvu (24)

Various terms in the above equation can be simplified as that they may be expressed in terms of one point and two point distribution functions. The 1st term in the above equation is simplified as follows:

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( vu

vx

uucgh −

∂∂

∂

∂−−− δψδφθδδ

αβ

βα

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( vu

vxucghu −

∂∂

∂∂

−−−= δψδφθδδαβ

αβ

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( vu

xxucghu −

∂∂

∂∂

−−−−= δψδφθδδββ

αβ

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( vux

cghu −∂∂

−−−−= δψδφθδδβ

β ;(since 1)1(

)1(=

∂∂

α

α

vu )

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( vux

ucgh −∂∂

−−−−= δψδφθδδβ

β (25)

Similarly, 7th, 10th and 12th terms of right hand-side of equation (24) can be simplified as follows;

7th term

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( gh

gx

uhcvu −

∂∂

∂

∂−−− δψδφθδδ

αβ

βα (26)

10th term,


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( A)

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( φθδ

φθψδδδβ

β −∂∂

∂∂

−−−x

ucghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( φθδψδδδβ

β −∂∂

−−−−=x

ucghvu (27)

and 12th term

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( ψδ

ψφθδδδ

ββ −

∂∂

∂∂

−−− cxcughvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( ψδφθδδδβ

β −∂∂

−−−−= cx

ughvu (28)

Adding these equations from (25) to (28), we get

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( vux

ucgh −∂∂

−−−− δψδφθδδβ

β

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( ghx

ucvu −∂∂

−−−−+ δψδφθδδβ

β

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( φθδψδδδβ

β −∂∂

−−−−+x

ucghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( ψδφθδδδβ

β −∂∂

−−−−+ cx

ughvu

( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1()1( ψδφθδδδβ

β−−−−−

∂∂

−= cghvuux

)1(1

)1()1( Fv

x ββ∂

∂−= [Applying the properties of distribution function]

)1(

)1(1)1(

ββ x

Fv

∂

∂−= (29)

Similarly second and eighth terms on the right hand-side of the equation (24) can be simplified as

( ) ( ) ( ) ( ))1()1()1()1(

)1()1()1()1()1()1()1()1( vu

vx

hhcgh −

∂∂

∂

∂−−−− δψδφθδδ

αβ

βα

)1(1)1()1(

)1()1( F

xvg

gβα

αβ

∂

∂

∂

∂−= (30)

and

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1()1( gh

gx

hucvu −

∂∂

∂

∂−−−− δψδφθδδ

αβ

βα

)1(1)1()1(

)1()1( F

xgv

gβα

αβ

∂

∂

∂

∂−= (31)



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4th term can be reduced as

( ) ( ) ( ) ( )⟩−∂

∂−−−∇− )1()1(

)1()1()1()1()1()1()1()1(2 vu

vcghu δψδφθδδν

αα

[ ( ) ( ) ( ) ( ) ])1()1()1()1()1()1()1()1()1(2)1(

ψδφθδδδν αα

−−−−∇∂

∂−= cghvuu

v

[ ( ) ( ) ( ) ( ) ])1()1()1()1()1()1()1()1()1()1()1(

2

)1(ψδφθδδδν α

ββα

−−−−∂∂

∂

∂

∂−= cghvuu

xxv

[ ( ) ( ) ( ) ( ) ])1()1()1()1()1()1()1()1()2()2()2(

2

)1(

lim

)1()2(ψδφθδδδν α

ββα

−−−−∂∂

∂

→∂

∂−= cghvuu

xxxx

v

( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2()2()2()2(

2

)1(

lim


ββα

−−−−∂∂

∂

→∂

∂−= ∫ cghvuu

xxxx

v

( ) ( ) ( ) ( ) ⟩−−−−× )2()2()2()2()1()1()1()1()1()1()1()1( ψφψδφθδδδ dddgdvcghvu

)2()2()2()2()2,1(

2)2(

)2()2(

2

)1(

lim

)1()2(ψφν α

ββα

dddgdvFvxx

xxv ∫

∂∂

∂

→∂

∂−= (32)

9th, 11th

and 13th

terms of the right hand side of equation (24)

9th

term,

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1( ghg

hcvu −∂∂

∇−−−− δλψδφθδδα

α

( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1(

)1(2 ψδφθδδδλα

α −−−−∂∂

∇−= cghvug

h

)2()2()2()2()2,1(2

)2()2()2(

2

)1(

lim

)1()2(ψφλ α

ββαdddgdvFg

xxxx

g ∫∂∂∂

→∂∂

−= (33)

11th

term,

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1( φθδφ

θγψδδδ −∂∂

∇−−−− cghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1(2 φθδφ

ψδδδθγ −∂∂

−−−∇−= cghvu (34)

13th

term,

( ) ( ) ( ) ( ))1()1()1(

)1(2)1()1()1()1()1()1( ψδψ

φθδδδ −∂∂

∇−−−− ccDghvu

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1(2 φθδφ

ψδδδ −∂∂

−−−∇−= cghvucD


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( A)

)2()2()2()2()2,1(2

)2()2()2(

2

)1(

lim

)1()2(ψφψ

ψ ββ

dddgdvFxx

xxD ∫∂∂

∂

→∂∂

−= (35)

We reduce the 3rd term of right hand side of equation (24), we get

( ) ( ) ( ) [ ] ( ))1()1()1()1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()1()1()1()1()1()1(

41 vu

vxxxd

xh

xh

xu

xu

xcgh −

∂∂

−′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

−−− ∫ δπ

ψδφθδδαα

β

β

α

α

β

β

α

α

[ ( )( ) 141 )2()2()2()2()2()2,1(

2)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)1()2()1()1( ψφπ α

β

β

α

α

β

β

α

αα

dddgdvdxFxg

xg

xv

xv

xxxv ∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

= ∫ (36)

5th and 6th terms of right hand side of equation (24)

( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1( 2 vuv

ucgh mm −∂∂

Ω∈×−−− δψδφθδδα

ααβ

[ ( ) ( ) ( ) ( ) ])1()1()1()1()1()1()1()1()1(

)1(2 ψδφθδδδα

ααβ −−−−∂

∂Ω∈= cghvu

vumm

( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1()1(

2 ψδφθδδδαα

αβ −−−−∂

∂Ω∈= cghvuu

vmm

( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1(

)1(2 ψδφθδδδ

α

ααβ −−−−

∂

∂Ω∈= cghvu

vu

mm

)1(12 Fmm Ω∈= αβ (37)

and ( ) ( ) ( ) ( ) ( ))1()1()1(

)1()1()1()1()1()1()1()1( vuv

vufcgh −∂

∂−−−−− δψδφθδδ

ααα

( ) [ ( ) ( ) ( ) ( ) ])1()1()1()1()1()1()1()1()1(

)1()1( ψδφθδδδα

αα −−−−∂∂

−−= cghvuv

vuf

( ) ( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()1(

)1()1( ψδφθδδδα

αα −−−−∂

∂−−= cghvu

vvuf

( ) )1(1)1(

)1()1( Fv

vufα

αα∂

∂−−=

(38)

and the last term,

( ) ( ) ( ) ( ) ( )( )

( )111

1)1()1()1(

)1()1()1()1()1()1()1( FRcRcghvuψ

ψψδψ

φθδδδ∂∂

=−∂∂

×−−−

(39)

Substituting the results (25) to (39) in equation (24) we get the transport equation for one point distribution function in MHD turbulent flow in a rotating system in presence of dust particles under going a first order reaction as

( ) [ ( ))1()2()1()1()1(

)1(1

)1(

)1(

)1(

)1()1(

)1(

)1(1)1(

)1(1 1

41

xxxvxF

gv

vgg

xFv

tF

−∂∂

∂∂

−∂∂

∂∂

+∂∂

+∂∂

+∂

∂∫

ααβα

α

α

αβ

ββ π

( ) )2()2()2()2()2()2,1(2)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(ψφ

α

β

β

α

α

β

β

α dddgdvdxFx

g

x

gx

v

x

v∂

∂

∂

∂−

∂

∂

∂

∂×



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)

)2()2()2()2()2,1(2

)2()2()2(

2

)1(

lim

)1()2(ψφν α

ββαdddgdvFv

xxxx

v ∫∂∂

∂

→∂

∂+

)2()2()2()2()2,1(2

)2()2()2(

2

)1(

lim

)1()2(ψφλ α

ββα

dddgdvFgxx

xxg ∫

∂∂

∂

→∂

∂+

)2()2()2()2()2,1(2

)2()2()2(

2

)1(

lim

)1()2(ψφφ

φγ

ββdddgdvF

xxxx

∫∂∂∂

→∂∂

+

)2()2()2()2()2,1(2

)2()2()2(

2

)1(

lim

)1()2(ψφψ

ψ ββ

dddgdvFxx

xxD ∫∂∂

∂

→∂∂

+

+ ( ) ( )( )

( ) 02 111

1)1(1)1(

)1()1()1(1 =

∂

∂−

∂

∂−+Ω∈ FRF

vvufFmm

ψψ

ααααβ

Equations for two-point distribution function )2,1(2F

Differentiating equation (13) with respect to time, we get,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ))2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()2,1(

2

ψδφθδδ

δψδφθδδδ

−−−

−−−−−∂∂

=∂

∂

cgh

vucghvutt

F

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))1()1()2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ght

cghvu

cvuvut

c

ghvucgh

−∂∂

−−−−

−−−+−∂∂

−

−−−−−−=

δψδφθδδδ

ψδφθδδδψδ

φθδδδψδφθδδ

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδδδφθδψδ

φθδδδψδδδ

−−−+−∂∂

−

−−−−−−+

ghvut

c

ghvucghvu

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2()1()1(

)1()1()1()1()1()1()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2()2()2()2()2()2()1()1(

)1()1()1()1()1()1()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)1()1()2()2()2()2()2()2()2()2(

ψδφθδδδψδ

φθδδδφθδψδ

δδψδφθδδδ

δψδφθδδψδ

φθδδδδψδ

φθδδψδφθδδδ

ψδψδφθδδδ

−∂∂

−−−−

−−−+−∂∂

−

−−−−−−+

−∂∂

−−−−

−−−+−∂∂

−

−−−−−−+

−∂∂

−−−−

ct

ghvuc

ghvut

c

ghvucghvu

ght

cvuc

ghvuvut

c

ghcghvu

ct

cghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuvt

uc

ghvucgh

−∂∂

∂∂

−

−−−−−−−=

δψδ




(39)

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13

( A)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghgt

hc

ghvucvu

−∂∂

∂∂

−

−−−−−−−+

δψδ


( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θψδ

φθδδδψδδδ

−∂∂

∂∂

−

−−−−−−−+

tc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghgt

hc

vucghvu

vuvt

uc

ghcghvu

ct

cc

ghvughvu

−∂∂

∂∂

−

−−−−−−−+

−∂∂

∂∂

−

−−−−−−−+

−∂∂

∂∂

−

−−−−−−−+

δψδ


δψδ


ψδψ

ψδ

φθδδδφθδδδ

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

φθδφ

θψδ

δδψδφθδδδ

−∂∂

∂∂

−

−−−−−−−+

−∂∂

∂∂

−

−−−−−−−+

ct

c

ghvucghvu

tc

ghvucghvu

.

Using equations (8) to (11) we get,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) [ ])1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()1()1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()2,1(

2

41

α

β

β

α

α

β

β

α

αβαβα

βπ

ψδ


x

h

xh

x

u

xu

xhhuu

xc

ghvucght

F

∂

∂

∂

∂−

∂

∂

∂

∂

∂

∂−−

∂

∂−−

−−−−−−−==∂

∂

∫

( ) ( ))1()1()1(

)1()1()1()1(2 2 vuv

vufuuxx

xdmm −

∂∂

×−+Ω∈−∇+−′′′′

× δνα

ααααβα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))1()1(

)1()1(2)1()1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghg

hhuuhx

c

ghvucvu

−∂∂

×∇+−∂∂

−−

−−−−−−−+

δλψδ


ααβαβα

β

( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2()1()1()1()1()1()1( ψδφθδδδψδδδ −−−−−−−−+ cghvucghvu

( ) ( ) ( ) ( ) ( ))2()2()1()1()1()1()1()1()1()1()1(

)1(2)1(

)1()1( vughvu

xu −−−−−+−

∂∂

×∇+∂∂

− δφθδδδφθδφ

θγθ

ββ

( ) ( ) ( ) ( ) ( ) ( ))1()1()1(

112)1(

)1()1()2()2()2()2()2()2( ψδ

ψψδφθδδ

ββ −

∂

∂−∇+

∂

∂−−−− cRccD

xcucgh



14

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IV

I( A

)

( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()1()1()1()1()1()1()1()1( ψδφθδδψδφθδδδ −−−−−−−−+ cghcghvu

( ) [ ]xx

xdxh

xh

xu

xu

xhhuu

x ′−′′′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

−−∂∂

− ∫ )2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2()2()2()2()2(

)2( 41

α

β

β

α

α

β

β

α

αβαβα

β π

( ) ( ))2()2()2(

)2()2()2()2(2 2 vuv

vufuu mm −∂∂

×−+Ω∈−∇+ δνα

ααααβα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ))2()2()2(

)2(2)2()2()2()2()2(

)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hhuuhx

c

vucghvu

−∂∂

×∇+−∂∂

−−

−−−−−−−+

δλψδ


ααβαβα

β

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1( ghvucghvu −−−−−−−+ δδψδφθδδδ

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2(2)2(

)2()2()2()2(

ghvucghvu

xuc

−−−−−−−+

−∂∂

×∇+∂∂

−−

δδψδφθδδδ

φθδφ

θγθψδβ

β

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

uuc

ghvucgh

−∂∂

∂

∂×−

−−−−−−=

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

hhc

ghvucgh

−∂∂

∂

∂×−

−−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) [ ] ( ))1()1()1()1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

41 vu

vxxxd

xh

xh

xu

xu

xc

ghvucgh

−∂∂

−′′′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−

−−−−−−+

∫ δπ

ψδ


αα

β

β

α

α

β

β

α

α

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ))1()1()1(

)1(2)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1(

vuv

uc

ghvucgh

−∂∂

∇×−−

−−−−−−+

δνψδφθδ

δδψδφθδδ

αα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

2 vuv

uc

ghvucgh

mm −∂∂

Ω∈×−

−−−−−−+

δψδ


αααβ

( ) ( ) ( ))2()2()2(

2)2(2)2(

)2()2()2()2( ψδ

ψφθδ

ββ −

∂

∂×−∇+

∂

∂−− cRccD

xcu


1

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IV

IYea

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15

( A)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))1()1(

)1()1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuv

vufc

ghvucgh

−∂∂

−×−

−−−−−−−+

δψδ


ααα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

uhc

ghvucvu

−∂∂

∂

∂×−

−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

huc

ghvucvu

−∂∂

∂

∂×−

−−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghg

hc

ghvucvu

−∂∂

∇×−

−−−−−−−+

δλψδ


αα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θψδ

φθδδδψδδδ

ββ −

∂∂

∂∂

×−

−−−−−−+

xuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θγψδ

φθδδδψδδδ

−∂∂

∇×−

−−−−−−−+

c

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδ


ββ −

∂∂

∂∂

×−

−−−−−−+

cxcuc

ghvughvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδ


−∂

∂∇×−

−−−−−−−+

ccDc

ghvughvu.

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδ


−∂

∂×−

−−−−−−+

cRcc

ghvughvu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

uuc

ghcghvu

−∂∂

∂

∂×−

−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

hhc

ghcghvu

−∂

∂

∂

∂×−

−−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) [ ] ( ))2()2()2()2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

41 vu

vxxxd

xh

xh

xu

xu

xc

ghcghvu

−∂∂

′−′′′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−

−−−−−−+

∫ δπ

ψδ


αα

β

β

α

α

β

β

α

α



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I( A

)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuv

uc

ghcghvu

−∂∂

∇×−

−−−−−−−+

δνψδ


αα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

2 vuv

uc

ghcghvu

mm −∂∂

Ω∈×−

−−−−−−+

δψδ


αααβ

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))2()2(

)2()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuv

vufc

ghcghvu

−∂∂

−×−

−−−−−−−+

δψδ


ααα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

uhc

vucghvu

−∂∂

∂

∂×−

−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

huc

vucghvu

−∂∂

∂

∂×−

−−−−−−−+

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hc

vucghvu

−∂∂

∇×−

−−−−−−−+

δλψδ


αα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδ

δδψδφθδδδ

ββ −

∂∂

∂∂

×−

−−−−−−+

xuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2(

)2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θγψδ

δδψδφθδδδ

−∂∂

∇×−

−−−−−−−+

c

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

ββ −

∂∂

∂∂

×−

−−−−−−+

cxcu

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

−∂

∂∇×−

−−−−−−−+

ccD

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

−∂

∂×−

−−−−−−−+

cRc

ghvucghvu

(41)


1

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IV

IYea

r20

16


17

( A)

Various terms in the above equation can be simplified as that they may be expressed in terms of one point, two- point and three-point distribution functions. The 1st term in the above equation is simplified as follows:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

uuc

ghvucgh

−∂∂

∂

∂×−

−−−−−−

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1(

vuvx

uc

ghvucghu

−∂∂

∂∂

×−

−−−−−−=

δψδ


αβ

α

β

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) )1 (since; )1(

)1()1()1(

)1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1(

=∂∂

−∂∂

∂∂

×−

−−−−−−−=

α

α

βα

α

β

δψδ


vuvu

xvuc

ghvucghu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vux

uc

ghvucgh

−∂∂

×−

−−−−−−−=

δψδ


ββ

(42)

Similarly, 7th, 10th

and 12th

terms of right hand-side of equation (41) can be simplified as follows;

7th

term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

uhc

ghvucvu

−∂∂

∂

∂×−

−−−−−−

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghx

uc

ghvucvu

−∂∂

×−

−−−−−−−=

δψδ


ββ

(43)

10th

term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θψδ

φθδδδψδδδ

ββ −

∂∂

∂∂

×−

−−−−−−

xuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδψδ

φθδδδψδδδ

ββ −∂∂

×−

−−−−−−−=

xuc

ghvucghvu

(44)

and 12th

term

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδ


ββ −

∂∂

∂∂

×−

−−−−−−

cxcuc

ghvughvu



18

Globa

lJo

urna

lof

Scienc

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ontie

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rch

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eYea

r20

16XVI

Iss u

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IV

I( A

)

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) .)1()1(

)1()1()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1(

ψδψδφθδ

δδφθδδδ

ββ −∂∂

×−−

−−−−−−=

cx

uc

ghvughvu

(45)


( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vux

uc

ghvucgh

−∂∂

×−

−−−−−−−

δψδ


ββ

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghx

uc

ghvucvu

−∂∂

×−

−−−−−−−+

δψδ


ββ

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδψδ

φθδδδψδδδ

ββ −∂∂

×−

−−−−−−−+

xuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψδ


ββ −∂∂

×−

−−−−−−−+

cx

uc

ghvughvu

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ))2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()1()1(

ψδφθδδ

δψδφθδδδββ

−−−

−−−−−−∂∂

−=

cgh

vucghvuux

)2,1(2

)1()1( Fv

x ββ∂∂

−= [Applying the properties of distribution functions]

)1(

)2,1(2)1(

ββ x

Fv∂∂

−= (46)

Similarly, 15th, 21st, 24th

and 26th

terms of right hand-side of equation (41) can be simplified as follows; 15th

term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

uuc

ghcghvu

−∂∂

∂

∂×−

−−−−−−

δψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vux

uc

ghcghvu

−∂∂

×−

−−−−−−−=

δψδ


ββ

(47)

21st term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

uhc

vucghvu

−∂∂

∂

∂×−

−−−−−−

δψδ


αβ

βα


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19

( A)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghx

uc

vucghvu

−∂∂

×−

−−−−−−−=

δψδ


ββ

(48)

24th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδ

δδψδφθδδδ

ββ −

∂∂

∂∂

×−

−−−−−−

xuc

ghvucghvu (49)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

φθδψδ

δδψδφθδδδ

ββ −

∂∂

×−

−−−−−−−=

xuc

ghvucghvu

and 26th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

ββ −

∂∂

∂∂

×−

−−−−−−

cxcu

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδφθδ

δδψδφθδδδ

ββ −

∂∂

×−

−−−−−−−=

cx

u

ghvucghvu (50)


( ) ( ) ( ) ( ) ( )

( ) ( ) ( ))2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()2()2(

ψδφθδδ

δψδφθδδδββ

−−−

−−−−−∂∂

−

cgh

vucghvuux

)2(

)2,1(2)2(

ββ x

Fv∂∂

−= (51)

Similarly, the terms 2nd, 8th, 16th and 22nd of right hand-side of equation (41) can be simplified as follows; 2nd term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1(

)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

hhc

ghvucgh

−∂∂

∂

∂×−

−−−−−−−

δψδ


αβ

βα

= - )1(βg )1(

)1(

α

α

vg∂∂

)1(

)2,1(2

βxF∂∂

(52)

8th term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

huc

ghvucvu

−∂∂

∂

∂×−

−−−−−−−

δψδ


αβ

βα



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I( A

)

= - )1(βg )1(

)1(

α

α

gv∂∂

(53)

16th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

hhc

ghcghvu

−∂∂

∂

∂×−

−−−−−−−

δψδ


αβ

βα

= - )2(βg )2(

)2(

α

α

vg∂∂

)2(

)2,1(2

βxF∂∂

(54)

and 22nd term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

huc

vucghvu

−∂∂

∂

∂×−

−−−−−−−

δψδ


αβ

βα

= - )2(βg )2(

)2(

α

α

gv∂∂

)2(

)2,1(2

βx

F

∂

∂.

Fourth term can be reduced as

( ) ( ) ( ) ( ) ( ) ( )( ) ( )⟩−

∂

∂∇×−

−−−−−−−

)1()1()1(

)1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuv

uc

ghvucgh

δνψδ


αα

[ ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ])2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1(2)1(

ψδφθδδδ

ψδφθδδδν αα

−−−−

−−−−∇∂

∂−=

cghvu

cghvuuv

[ ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ])2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1()1()1(

2

)1(

ψδφθδδδ

ψδφθδδδν αββα

−−−−

−−−−∂∂

∂

∂

∂−=

cghvu

cghvuuxxv

[ ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ])2()2()2()2()2()2()2()2()1()1(

)1()1()1()1()1()1()3()3()3(

2

)1(

lim

)1()3(

ψδφθδδδψδ

φθδδδν αββα

−−−−−

−−−∂∂

∂

→∂

∂−=

cghvuc

ghvuuxx

xxv

( ) ( ) ( ))3()3()3()3()3()3()3()3()3(

2

)1(

lim

)1()3(φθδδδν α

ββα

−−−∂∂∂

→∂∂

−= ∫ ghvuuxx

xxv


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( A)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) )3()3()3()3()1()1()1()1()1()1(

)1()1()2()2()2()2()2()2()2()2()3()3(

ψφψδφθδδ

δψδφθδδδψδ

dddgdvcgh

vucghvuc

−−−

−−−−−−

)3()3()3()3()3,2,1(3

)3()3()3(

2

)1(

lim

)1()3(ψφν α

ββαdddgdvFv

xxxx

v ∫∂∂

∂

→∂

∂−= (56)

Similarly, 9th, 11th, 13th, 18th, 23rd, 25th and 27th terms of right hand-side of equation (41) can be simplified as follows; 9th term,

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

)1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ghg

hc

ghvucvu

−∂

∂∇×−

−−−−−−−

δλψδ


αα

)3()3()3()3()3,2,1(3

)3()3()3(

2

)1(

lim

)1()3(ψφλ α

ββα

dddgdvFgxx

xxg ∫

∂∂

∂

→∂

∂−= (57)

11th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θγψδ

φθδδδψδδδ

−∂

∂∇×−

−−−−−−−

c

ghvucghvu

)3()3()3()3()3,2,1(3

)3()3()3(

2

)1(

lim

)1()3(ψφφ

φγ

ββ

dddgdvFxx

xx∫

∂∂

∂

→∂

∂−= (58)

13th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1(2)2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδ


−∂

∂∇×−

−−−−−−−

ccDc

ghvughvu

)3()3()3()3()3,2,1(3

)3()3()3(

2

)1(

lim

)1()3(ψφψ

ψ ββ

dddgdvFxx

xxD ∫

∂∂

∂

→∂

∂−= (59)

18th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuv

uc

ghcghvu

−∂

∂∇×−

−−−−−−−

δνψδ


αα

)3()3()3()3()3,2,1(3

)3()3()3(

2

)2(

lim

)2()3(ψφν α

ββα

dddgdvFvxx

xxv ∫

∂∂

∂

→∂

∂−= (60)

23rd

term,



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)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hc

vucghvu

−∂

∂∇×−

−−−−−−−

δλψδ


αα

)3()3()3()3()3,2,1(3

)3()3()3(

2

)2(

lim

)2()3(ψφλ α

ββαdddgdvFg

xxxx

g ∫∂∂∂

→∂∂

−= (61)

25th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θγψδ

δδψδφθδδδ

−∂∂

∇×−

−−−−−−−

c

ghvucghvu

)3()3()3()3()3,2,1(3

)3()3()3(

2

)2(

lim

)2()3(ψφφ

φγ

ββ

dddgdvFxx

xx∫∂∂

∂

→∂∂

−= (62)

27th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2(2)2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

−∂∂

∇×−

−−−−−−−

ccD

ghvucghvu

)3()3()3()3()3,2,1(3

)3()3()3(

2

)2(

lim

)2()3(ψφψ

ψ ββdddgdvF

xxxx

D ∫∂∂∂

→∂∂

−= (63)

We reduce the third term of right - hand side of equation (41),

( ) ( ) ( ) ( ) ( ) ( )

( ) [ ] ( ))1()1()1()1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

41 vu

vxxxd

x

h

xh

x

u

xu

xc

ghvucgh

−∂∂

−′′′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−

−−−−−−

∫ δπ

ψδ


αα

β

β

α

α

β

β

α

α

[ ( )( ) ]

141 )3()3()3()3()3()3,2,1(

3)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)1()3()1()1( ψφπ α

β

β

α

α

β

β

α

ααdddgdvdxF

x

g

xg

x

v

xv

xxxv×

∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

= ∫

(64)

Similarly, term 17th,

( ) ( ) ( ) ( ) ( ) ( )

( ) [ ] ( ))2()2()2()2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

41 vu

vxxxd

xh

xh

xu

xu

xc

ghcghvu

−∂∂

′−′′′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−

−−−−−−

∫ δπ

ψδ


αα

β

β

α

α

β

β

α

α

[ ( )( ) ]

141 )3()3()3()3()3()3,2,1(

3)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)2()3()2()2( ψφπ α

β

β

α

α

β

β

α

αα

dddgdvdxFxg

xg

xv

xv

xxxv×

∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

= ∫ (65)

5th

and 6th

terms of right hand side of equation (41), we get

5th

term,


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23

( A)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

2 vuv

uc

ghvucgh

mm −∂

∂Ω∈×−

−−−−−−

δψδ


αααβ

[ ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ])2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1(

)1(2

ψδφθδδδ

ψδφθδδδα

ααβ

−−−−

−−−−∂

∂Ω∈=

cghvu

cghvuv

umm

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1()1(

2

ψδφθδδδ

ψδφθδδδαα

αβ

−−−−

−−−−∂

∂Ω∈=

cghvu

cghvuuv

mm

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1(

)1(

2

ψδφθδδδ

ψδφθδδδα

ααβ

−−−−

−−−−∂∂

Ω∈=

cghvu

cghvuvu

mm

)2,1(22 Fmm Ω∈= αβ (66)

and 6th term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))1()1(

)1()1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuv

vufc

ghvucgh

−∂

∂−×−

−−−−−−−

δψδ


ααα

( ) [ ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )])2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1(

)1()1(

ψδφθδδδ

ψδφθδδδα

αα

−−−−

−−−−∂

∂−−=

cghvu

cghvuv

vuf

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ))2()2()2()2()2()2()2()2(

)1()1()1()1()1()1()1()1()1(

)1()1(

ψδφθδδδ

ψδφθδδδα

αα

−−−−

−−−−∂

∂−−=

cghvu

cghvuv

vuf

( ) )2,1(2)1(

)1()1( Fv

vufα

αα∂∂

−−= (

(67)

Similarly, 19th

and 20th

terms of right hand side of equation (41),

19th

term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

2 vuv

uc

ghcghvu

mm −∂∂

Ω∈×−

−−−−−−

δψδ


αααβ

= )2,1(22 Fmm Ω∈ αβ

(68)

and 20th term,



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)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))2()2(

)2()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vuv

vufc

ghcghvu

−∂

∂−×−

−−−−−−−

δψδ


ααα

( ) )2,1(2)2(

)2()2( Fv

vufα

αα∂

∂−−= (69)

14th and 28th terms of equation (41)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(

)1()1()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδ


−∂

∂×−

−−−−−−

cRcc

ghvughvu

( )( )

( )2,121

1 FRψ

ψ∂

∂= (70)

And 28th term

( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(

)2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδ

δδψδφθδδδ

−∂

∂×−

−−−−−−−

cRc

ghvucghvu

( )( )

( )2,122

2 FRψ

ψ∂

∂= (71)

Substituting the results (42) – (71) in equation (41) we get the transport equation for two point distribution

function ),,,()2,1(2 ψφgvF in MHD turbulent flow in a rotating system in presence of dust particles as

( ) ( ) )2,1(2)1()1(

)1(

)1(

)1()1()2,1(

2)2()2(

)1()1(

)2,1(2 F

xgv

vggF

xv

xv

tF

βα

α

α

αβ

ββ

ββ ∂

∂∂∂

+∂∂

+∂∂

+∂∂

+∂

∂

( ) [ ( ))1()3()1()1(

)2,1(2)2()2(

)2(

)2(

)2()2( 1

41

xxxvF

xgv

vgg

−∂∂

∂∂

−∂∂

∂∂

+∂∂

+ ∫ααβα

α

α

αβ π

( ) ])3)3()3()3()3()3,2,1(3)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

ψφα

β

β

α

α

β

β

α dddgdvdxFxg

xg

xv

xv

∂∂

∂∂

−∂∂

∂∂

×

[ ( ) ( ))3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)2()3()2()2( 141

α

β

β

α

α

β

β

α

αα π xg

xg

xv

xv

xxxv ∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

− ∫

] )3)3()3()3()3()3,2,1(3 ψφ dddgdvdxF×

( ) )3()3()3()3()3,2,1(3

)3()3()3(

2

)2()1(

lim

)2()3(

lim

)1()3(ψφν α

ββαα

dddgdvFvxx

xxv

xxv ∫∂∂

∂

→∂∂

+

→∂∂

+

( ) )3()3()3()3()3,2,1(3

)3()3()3(

2

)2()1(

lim

)2()3(

lim

)1()3(ψφλ α

ββαα

dddgdvFgxx

xxg

xxg ∫∂∂

∂

→∂∂

+

→∂∂

+


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( A)

( ) )3()3()3()3()3,2,1(3

)3()3()3(

2

)2()1(

lim

)2()3(

lim

)1()3(ψφφ

φφγ

ββ

dddgdvFxx

xxxx∫∂∂

∂

→∂∂

+

→∂∂

+

( ) )3()3()3()3()3,2,1(3

)3()3()3(

2

)2()1(

lim

)2()3(

lim

)1()3(ψφψ

ψψ ββ

dddgdvFxx

xxxxD ∫∂∂

∂

→∂∂

+

→∂∂

+ (72)

4 )2,1(

2Fmm Ω∈+ αβ [ ( ) ( ) ] )2,1(2)2(

)2()2()1(

)1()1( Fv

vufv

vufα

ααα

αα∂

∂−+

∂

∂−+

( )( )

( ) ( )( )

( ) 02,122

22,121

1 =∂

∂−

∂

∂− FRFR

ψψ

ψψ

Equations for three-point distribution function )3.2,1(3F :

Differentiating equation (14) with respect to time, we get

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()3,2,1(

3

ψδφθδδδψδφθδ

δδψδφθδδδ

−−−−−−

−−−−−−∂∂

=∂

∂

cghvuc

ghvucghvutt

F

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

)1()1()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

)1()1()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ght

cghvuc

ghvucvu

vut

cghvuc

ghvucgh

−∂∂

−−−−−

−−−−−−+

−∂∂

−−−−−

−−−−−−=





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))1()1()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

φθδψδφθδδδψδ

φθδδδψδδδ

−∂∂

−−−−−

−−−−−−+

tcghvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1(

)1()1()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(


ψδψδφθδδδψδ


−−−−−−+

−∂∂

−−−−−

−−−−−−+

ghcghvu

ct

cghvuc

ghvughvu

( ) ( ) ( ) ( ) ( ) ( ))2()2()3()3()3()3()3()3()3()3()2()2( vut

cghvuc −∂∂

−−−−− δψδφθδδδψδ

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ghvucghvu

ght

cghvuc

vucghvu

−−−−−−+

−∂∂

−−−−−

−−−−−−+

δδψδφθδδδ





26

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lJo

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sion

IV

I( A

)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))2()2()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()3()3()3()3()3()3()3()3()2()2(

ψδψδφθδδδφθδ

δδψδφθδδδ

φθδψδφθδδδψδ

−∂∂

−−−−−

−−−−−−+

−∂∂

−−−−−

ct

cghvu

ghvucghvut

cghvuc

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vut

cghc

ghvucghvu

−∂∂

−−−−−

−−−−−−+

δψδφθδδψδφθδ

δδψδφθδδδ

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(


δδψδφθδδδ

φθδψδδδψδφθδ

δδψδφθδδδ

δψδφθδδψδφθδ

δδψδφθδδδ

−∂∂

−−−−−

−−−−−−+

−∂∂

−−−−−

−−−−−−+

−∂∂

−−−−−

−−−−−−+

ct

ghvuc

ghvucghvut

cghvuc

ghvucghvu

ght

cvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))1()1(

)1(

)1()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

vuvt

ucghvuc

ghvucgh

−∂∂

∂∂

−−−−−

−−−−−−−=



( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghgt

hcghvu

cghvucvu

−∂

∂∂

∂−−−−

−−−−−−−−+

δψδφθδδδ

ψδφθδδδψδφθδδ

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θψδφθδδδ

ψδφθδδδψδδδ

−∂

∂∂

∂−−−−

−−−−−−−−+

tcghvu

cghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)1()1()1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδφθδδψδφθδδδ

δψδφθδδδ


ψδψ

ψδφθδδδ

ψδφθδδδφθδδδ

−−−−−−−−+

−∂

∂∂

∂−−−−

−−−−−−−−+

−∂

∂∂

∂−−−−

−−−−−−−−+

cvucghvu

vuvt

ucghvu

cghcghvu

ct

ccghvu

cghvughvu

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2()3()3()3()3()3()3()3()3(

ψδδδψδφθδδδ

δψδφθδδδ

−−−−−−−−+

−∂

∂∂

∂−−−−

cghvucghvu

ghgt

hcghvu


1

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27

( A)

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2(

)2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2()3()3()3()3()3()3()3()3(

ψδψ

ψδφθδδδ

φθδδδψδφθδδδ

φθδφ

θψδφθδδδ

−∂

∂∂

∂−−−−

−−−−−−−−+

−∂

∂∂

∂−−−−

ct

ccghvu

ghvucghvu

tcghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3(

)3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)3()3()3(

)3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)3()3()3(

)3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδδδψδ


δψδφθδδψδ


δψδφθδδψδ


−∂

∂∂

∂−−−−

−−−−−−−−+

−∂

∂∂

∂−−−−

−−−−−−−−+

−∂

∂∂

∂−−−−

−−−−−−−−+

tcghvuc

ghvucghvu

ghgt

hcvuc

ghvucghvu

vuvt

ucghc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3(

)3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδψδ


−∂∂

∂∂

−−−−

−−−−−−−−+

ct

cghvuc

ghvucghvu

Using equations (8) to (11), we get from the above equation

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()3,2,1(

3

ψδφθδδδψδ


−−−−−

−−−−−−−=∂

∂

cghvuc

ghvucght

F

( ) [ ]

( ) ( ))1()1()1(

)1()1()1()1(2

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()1()1()1()1(

)1(

2

41

vuv

vufuu

xx

xdx

h

xh

x

u

xu

xhhuu

x

mm

n

−∂

∂×−+Ω∈−∇+

−′′′

′′′

∂

∂

∂

∂−

∂

∂

∂

∂

∂

∂−−

∂

∂− ∫

δν

π

αααααβα

α

β

β

α

α

β

β

α

αβαβα

β

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδφθδδδψδ


−−−−−

−−−−−−−+

cghvuc

ghvucvu

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

)1()1()1(

)1(2)1()1()1()1()1(

ψδφθδδδψδ

φθδδδψδδδ

δλα

αβαβαβ

−−−−−

−−−−−−−+

−∂

∂×∇+−

∂

∂−

cghvuc

ghvucghvu

ghg

hhuuhx

( ))1()1()1(

)1(2)1(

)1()1( φθδ

φθγθ

ββ −

∂

∂×∇+

∂

∂−×

xu



28

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IV

I( A

)

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ))1()1()1(

1)1(2)1(

)1()1(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδφθδδδψδ


ββ −

∂

∂×−∇+

∂

∂−

−−−−−

−−−−−−−+

cRccDx

cu

cghvuc

ghvughvu

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1( φθδδψδφθδδδ −−−−−−− ghcghvu

( ) ( ) ( ) ( ) ( ) ( ) [ ]

xxxd

x

h

xh

x

u

xu

xhhuu

x

cghvuc

′−′′′′′′

∂

∂

∂

∂−

∂

∂

∂

∂

∂

∂−−

∂

∂−

−−−−−

∫ )2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2()2()2()2()2(

)2(

)3()3()3()3()3()3()3()3()2()2(

41

α

β

β

α

α

β

β

α

αβαβα

βπ

ψδφθδδδψδ

( ) ( ))2()2()2(

)2()2()2()2(2 2 vuv

vufuu mm −∂∂

×−+Ω∈−∇+ δνα

ααααβα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδφθδδδψδ


−−−−−

−−−−−−−+

cghvuc

vucghvu

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ))2()2()2(

2)2(2)2(

)2()2(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

ψδφθδδδφθδ

δδψδφθδδδ

ββ −

∂

∂−∇+

∂

∂−

−−−−−

−−−−−−−+

cRccDxcu

cghvu

ghvucghvu

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))2()2(

)2()2(2

)2(

)2()2(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(

)2(2)2()2()2()2()2(

φθδφ

θγθ

ψδφθδδδψδ

δδψδφθδδδ

δλ

ββ

ααβαβα

β

−∂∂

×∇+∂∂

−×

−−−−−

−−−−−−−+

−∂∂

∇+−∂∂

−

xu

cghvuc

ghvucghvu

ghg

hhuuhx

( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()1()1()1()1()1()1()1()1( ghvucghvu −−−−−−− δδψδφθδδδ

( ) ( ) ( ) ( ) ( ) ( ) [ ]

xxxd

xh

xh

xu

xu

xhhuu

x

cghc

′′−′′′′′′

∂

∂

∂∂

−∂

∂

∂∂

∂∂

−−∂∂

−

−−−−−

∫ )3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3()3()3()3()3(

)3(

)3()3()3()3()3()3()2()2()2()2(

41

α

β

β

α

α

β

β

α

αβαβα

β π

ψδφθδδψδφθδ

( ) ( ))3()3()3(

)3()3()3()3(2 2 vuv

vufuu mm −∂∂

×−+Ω∈−∇+ δνα

ααααβα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(


δδψδφθδδδ

−−−−−

−−−−−−−+

cvuc

ghvucghvu

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

)3()3()3(

)3(2)3()3()3()3()3(

ψδδδψδφθδ

δδψδφθδδδ

δλα

αβαβαβ

−−−−−

−−−−−−−+

−∂∂

∇+−∂∂

−

cghvuc

ghvucghvu

ghg

hhuuhx


1

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lJo

urna

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ontie

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IYea

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16


29

( A)

( ))3()3()3(

)3(2)3(

)3()3( φθδ

φθγθ

ββ −

∂∂

×∇+∂∂

−×x

u

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ))3()3()3(

3)3(2)3(

)3()3(

)3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδψδφθδ

δδψδφθδδδ

ββ −

∂

∂−∇+

∂

∂−

−−−−−

−−−−−−−+

cRccDxcu

ghvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

uucghvu

cghvucgh

−∂∂

∂

∂×−−−−

−−−−−−−=

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

hhcghvu

cghvucgh

−∂∂

∂

∂×−−−−

−−−−−−−−+

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) [ ])1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

41

α

β

β

α

α

β

β

α

απψδφθδδδ


xh

xh

xu

xu

xcghvu

cghvucgh

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−−−−

−−−−−−−+

∫

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ))1()1()1(

)1(2)3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()1(

vuv

ucghvuc

ghvucghvuvxx

xd

−∂∂

∇×−−−−−−

−−−−−−+−∂∂

−′′′′′′

×

δνψδφθδδδψδφθδ

δδψδφθδδδ

αα

α

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

2 vuv

ucghvu

cghvucgh

mm −∂∂

Ω∈×−−−−

−−−−−−−+

δψδφθδδδ


αααβ

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))1()1(

)1()1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuv

vufcghvu

cghvucgh

−∂∂

−×−−−−

−−−−−−−−+

δψδφθδδδ


ααα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))1()1()1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

uhcghvu

cghvucvu

−∂∂

∂

∂×−−−−

−−−−−−−+

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))1()1()1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

hucghvu

cghvucvu

−∂∂

∂

∂×−−−−

−−−−−−−−+

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghg

hcghvu

cghvucvu

−∂

∂∇×−−−−

−−−−−−−−+

δλψδφθδδδ


αα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θψδφθδδδ


ββ −

∂

∂

∂

∂×−−−−

−−−−−−−+

xucghvu

cghvucghvu



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)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θγψδφθδδδ


−∂

∂∇×−−−−

−−−−−−−−+

cghvu

cghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


ββ −

∂

∂

∂

∂×−−−−

−−−−−−−+

cxcucghvu

cghvughvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


−∂

∂∇×−−−−

−−−−−−−−+

ccDcghvu

cghvughvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


−∂

∂×−−−−

−−−−−−−+

cRccghvu

cghvughvu

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

uucghvu

cghcghvu

−∂∂

∂

∂×−−−−

−−−−−−−+

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

hhcghvu

cghcghvu

−∂∂

∂

∂×−−−−

−−−−−−−−+

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) [ ])2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

41

α

β

β

α

α

β

β

α

απψδφθδδδ


xh

xh

xu

xu

xcghvu

cghcghvu

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−−−−

−−−−−−−+

∫

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)2()2()2(


δα

−−−−−−−−+

−∂∂

′−′′′′′′

×

cghcghvu

vuvxx

xd

( ) ( ) ( ) ( ) ( ))2()2()2(

)2(2)3()3()3()3()3()3()3()3( vuv

ucghvu −∂

∂∇×−−−− δνψδφθδδδ

αα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

2 vuv

ucghvu

cghcghvu

mm −∂∂

Ω∈×−−−−

−−−−−−−+

δψδφθδδδ


αααβ

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))2()2(

)2()2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuv

vufcghvu

cghcghvu

−∂∂

−×−−−−

−−−−−−−−+

δψδφθδδδ


ααα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

uhcghvu

cvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−+

δψδφθδδδ


αβ

βα


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31

( A)

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

hucghvu

cvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−−+

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2(

)2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hcghvu

cvucghvu

−∂∂

∇×−−−−

−−−−−−−−+

δλψδφθδδδ


αα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδφθδδδ


ββ −

∂∂

∂∂

×−−−−

−−−−−−−+

xucghvu

cghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2(

)2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θγψδφθδδδ


−∂∂

∇×−−−−

−−−−−−−−+

cghvu

cghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


ββ −

∂∂

∂∂

×−−−−

−−−−−−−+

cxcucghvu

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


−∂

∂∇×−−−−

−−−−−−−−+

ccDcghvu

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


−∂

∂×−−−−

−−−−−−−+

cRccghvu

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

uucghc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−+

δψδφθδδψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

hhcghc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−−+

δψδφθδδψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) [ ])3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

41

α

β

β

α

α

β

β

α

απψδφθδδψδ


xh

xh

xu

xu

xcghc

ghvucghvu

∂

∂

∂∂

−∂

∂

∂∂

∂∂

×−−−−

−−−−−−−+

∫



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)

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3(

)3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

)3()3()3(

vuv

ucghc

ghvucghvu

vuvxx

xd

−∂∂

∇×−−−−

−−−−−−−−+

−∂∂

′′−′′′′′′

×

δνψδφθδδψδ


δ

αα

α

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

2 vuv

ucghc

ghvucghvu

mm −∂∂

Ω∈×−−−−

−−−−−−−+

δψδφθδδψδ


αααβ

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuv

vufcghc

ghvucghvu

−∂∂

−×−−−−

−−−−−−−−+

δψδφθδδψδ


ααα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

uhcvuc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−+

δψδφθδδψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

hucvuc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−−+

δψδφθδδψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hcvuc

ghvucghvu

−∂∂

∇×−−−−

−−−−−−−−+

δλψδφθδδψδ


αα

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδδδψδ


ββ −

∂∂

∂∂

×−−−−

−−−−−−−+

xucghvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θγψδδδψδ


−∂∂

∇×−−−−

−−−−−−−−+

cghvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδψδ


ββ −

∂∂

∂∂

×−−−−

−−−−−−−+

cxcughvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδ


−∂

∂∇×−−−

−−−−−−−−+

ccDghvu

ghvucghvu


1

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33

( A)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδ


−∂∂

×−−−

−−−−−−−+

cRcghvu

ghvucghvu (73)

Various terms in the above equation can be simplified as that they may be expressed in terms of one-, two-, three- and four - point distribution functions. The 1st term in the above equation is simplified as follows

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

uucghvu

cghvucgh

−∂∂

∂

∂×−−−−

−−−−−−−

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1()1(

vuvx

ucghvu

cghvucghu

−∂∂

∂∂

×−−−−

−−−−−−−=

δψδφθδδδ


αβ

α

β

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) )1 (since; )1(

)1()1()1(

)1()1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1()1(

=∂∂

−∂∂

∂∂

×−−−−

−−−−−−−−=

α

α

βα

α

β

δψδφθδδδ


vuvu

xvucghvu

cghvucghu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vux

ucghvu

cghvucgh

−∂∂

×−−−−

−−−−−−−−=

δψδφθδδδ


ββ

(74)

Similarly, 7th, 10th and 12th terms of right hand-side of equation (73) can be simplified as follows; 7th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

uhcghvu

cghvucvu

−∂∂

∂

∂×−−−−

−−−−−−−

δψδφθδδδ


αβ

βα (75)

10th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θψδφθδδδ


ββ −

∂

∂

∂

∂×−−−−

−−−−−−−

xucghvu

cghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

φθδψδφθδδδ


ββ −

∂

∂×−−−−

−−−−−−−−=

xucghvu

cghvucghvu

(76)

and 12th term

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


ββ −

∂

∂

∂

∂×−−−−

−−−−−−−

cxcucghvu

cghvughvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψδφθδδδ


ββ −

∂

∂×−−−−

−−−−−−−−=

cx

ucghvu

cghvughvu

(77)



34

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I( A

)


( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vux

ucghvu

cghvucgh

−∂

∂×−−−−

−−−−−−−−

δψδφθδδδ


ββ

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghx

ucghvu

cghvucvu

−∂

∂×−−−−

−−−−−−−−+

δψδφθδδδ


ββ

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(



ββ −

∂∂

×−−−−

−−−−−−−−+

xucghvu

cghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψδφθδδδ


ββ −

∂

∂×−−−−

−−−−−−−−+

cx

ucghvu

cghvughvu

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()1()1(


δδψδφθδδδββ

−−−−−−

−−−−−−−∂∂

−=

cghvuc

ghvucghvuux

)3,2,1(3

)1()1(

Fvx ββ∂

∂−= [Applying the properties of distribution functions ] (78)

Similarly, 15th, 21st, 24th

and 26th


15th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

uucghvu

cghcghvu

−∂∂

∂

∂×−−−−

−−−−−−−

δψδφθδδδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))2()2(

)2()2()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

vux

ucghvuc

ghcghvu

−∂

∂×−−−−−

−−−−−−−=



ββ

(79)

21st

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

uhcghvu

cvucghvu

−∂

∂

∂

∂×−−−−

−−−−−−−

δψδφθδδδ


αβ

βα

(80)

24th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδφθδδδ


ββ −

∂

∂

∂

∂×−−−−

−−−−−−−

xucghvu

cghvucghvu


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35

( A)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(



ββ −

∂

∂×−−−−

−−−−−−−−=

xucghvu

cghvucghvu (81)

And 26th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


ββ −

∂

∂

∂

∂×−−−−

−−−−−−−

cxcucghvu

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψδφθδδδ


ββ −

∂

∂×−−−−

−−−−−−−−=

cx

ucghvu

ghvucghvu

(82)

Adding equations (79) to (82), we get

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()2()2(



−−−−−−

−−−−−−∂∂

−

cghvuc

ghvucghvuux

)2(

)3,2,1(3)2(

ββ x

Fv

∂

∂−= (83)

Similarly, 29th, 35th, 38th

and 40th


29th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()2()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

uucghc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−

δψδφθδδψδ


αβ

βα

= ( ) ( ) ( ) ( ) ( ) ( ))2()2()3()3()1()1()1()1()1()1()1()1( ghvucghvu −−−−−−− δδψδφθδδδ

( ) ( ) ( ) ( ) ( ) ( ))3()3()3(

)3()3()3()3()3()3()3()2()2()2()2( vux

ucghc −∂∂

×−−−−− δψδφθδδψδφθδβ

β (84)

35th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

uhcvuc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−

δψδφθδδψδ


αβ

βα

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghx

ucvuc

ghvucghvu

−∂

∂×−−−−

−−−−−−−−=

δψδφθδδψδ


ββ

(85)

38th

term,



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)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θψδδδψδ


ββ −

∂∂

∂∂

×−−−−

−−−−−−−

xucghvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδψδδδψδ


ββ −

∂∂

×−−−−

−−−−−−−−=

xucghvuc

ghvucghvu (86)

and 40th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδψδ


ββ −

∂∂

∂∂

×−−−−

−−−−−−−

cxcughvuc

ghvucghvu

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδφθδδδψδ


ββ −

∂∂

×−−−−

−−−−−−−−=

cx

ughvuc

ghvucghvu (87)

Adding equations (84) to (87), we get

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()3()3(



−−−−−−

−−−−−−−∂

∂−

cghvuc

ghvucghvuux

= - )3(βv

)3(

)3,2,1(3

βx

F

∂

∂ (88)

Similarly , 2 nd, 8th, 16th, 22nd, 30th and 36th terms of right hand-side of equation (73) can be simplified as follows; 2nd term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuvx

hhcghvu

cghvucgh

−∂∂

∂

∂×−−−−

−−−−−−−−

δψδφθδδδ


αβ

βα

= - )1(βg )1(

)1(

α

α

vg∂

∂)1(

)3,2,1(3

βx

F

∂

∂ (89)

8th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghgx

hucghvu

cghvucvu

−∂

∂

∂

∂×−−−−

−−−−−−−−

δψδφθδδδ


αβ

βα

= - )1(βg )1(

)1(

α

α

gv∂

∂)1(

)3,2,1(3

βxF∂

∂ (90)

16th term,


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37

( A)

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))2()2()2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

hhcghvu

cghcghvu

−∂∂

∂

∂×−−−−

−−−−−−−−

δψδφθδδδ


αβ

βα

= - )2(βg )2(

)2(

α

α

vg∂

∂)2(

)3,2,1(3

βxF∂

∂ (91)

22nd term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

hucghvu

cvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−−

δψδφθδδδ


αβ

βα

= - )2(

βg )2(

)2(

α

α

gv∂

∂)2(

)3,2,1(3

βx

F

∂

∂ (92)

30th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuvx

hhcghc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−−

δψδφθδδψδ


αβ

βα

= - )3(βg )3(

)3(

α

α

vg∂

∂)3(

)3,2,1(3

βx

F

∂

∂ (93)

and 36th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ))3()3()3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghgx

hucvuc

ghvucghvu

−∂∂

∂

∂×−−−−

−−−−−−−−

δψδφθδδψδ


αβ

βα

= - )3(βg )3(

)3(

α

α

gv∂

∂)3(

)3,2,1(3

βx

F

∂

∂

(94)

4th

term can be reduced as

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuv

ucghvu

cghvucgh

−∂∂

∇×−−−−

−−−−−−−−

δνψδφθδδδ


αα

[ ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ])3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()1(2)1(


δδψδφθδδδν αα

−−−−−−

−−−−−−∇∂∂

−=

cghvuc

ghvucghvuuv



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[ ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ])3()3()3()3()3()3()3()3()2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()1()1()1(

2

)1(


δψδφθδδδν αββα

−−−−−−−

−−−−−∂∂∂

∂∂

−=

cghvucgh

vucghvuuxxv

[ ( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()4()4()4(

2

)1(

lim


ββα−−−−

∂∂

∂

→∂

∂−= cghvuu

xxxx

v

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ])3()3()3()3()3()3()3()3()2()2()2()2()2()2()2()2( ψδφθδδδψδφθδδδ −−−−−−−− cghvucghvu

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()3()3()3()3()3()3()3()3(

)4()4()4()4()4()4()4()4()4()4()4(

2

)1(

lim

)1()4(


ψδφθδδδν αββα

−−−−−−−

−−−−∂∂∂

→∂∂

−= ∫

ghvucghvu

cghvuuxx

xxv

( ) ( ) ( ) ( ) ( ) )4()4()4()4()1()1()1()1()1()1()1()1()2()2( ψφψδφθδδδψδ dddgdvcghvuc −−−−−

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)1(

lim

)1()4(ψφν α

ββαdddgdvFv

xxxx

v ∫∂∂

∂

→∂

∂−= (95)

Similarly, 9th ,11th

,13th

,18th

,23rd

,25th

,27th

,32nd

,37th

,39th and 41st

terms of right hand-side of equation (73) can

be simplified as follows;

9th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ghg

hcghvu

cghvucvu

−∂∂

∇×−−−−

−−−−−−−−

δλψδφθδδδ


αα

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)1(

lim

)1()4(ψφλ α

ββαdddgdvFg

xxxx

g ∫∂∂

∂

→∂

∂−= (96)

11th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

φθδφ

θγψδφθδδδ


−∂∂

∇×−−−−

−−−−−−−−

cghvu

cghvucghvu

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)1(

lim

)1()4(ψφφ

φγ

ββ

dddgdvFxx

xx∫

∂∂

∂

→∂

∂−= (97)

13th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


−∂∂

∇×−−−−

−−−−−−−−+

ccDcghvu

cghvughvu


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39

( A)

)4()4()4()4()4,3,2,1(4

)4()3()4(

2

)1(

lim

)1()4(ψφψ

ψ ββ

dddgdvFxx

xxD ∫∂∂

∂

→∂∂

−= (98)

18th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuv

ucghvu

cghcghvu

−∂∂

∇×−−−−

−−−−−−−−

δνψδφθδδδ


αα

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)2(

lim

)2()4(ψφν α

ββαdddgdvFv

xxxx

v ∫∂∂

∂

→∂

∂−= (99)

23rd term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hcghvu

cvucghvu

−∂

∂∇×−−−−

−−−−−−−−

δλψδφθδδδ


αα

)4()4()4()4()4,3,2,1(

4)4(

)4()4(

2

)2(

lim

)2()4(ψφλ α

ββαdddgdvFg

xxxx

g ∫∂∂

∂

→∂

∂−= (100)

25th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θγψδφθδδδ


−∂∂

∇×−−−−

−−−−−−−−

cghvu

cghvucghvu

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)2(

lim

)2()4(ψφφ

φγ

ββ

dddgdvFxx

xx∫

∂∂

∂

→∂

∂−= (101)

27th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2(2)3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

ψδφθδδδ


−∂∂

∇×−−−−

−−−−−−−−

ccDcghvu

ghvucghvu

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)2(

lim

)2()4(ψφψ

ψ ββ

dddgdvFxx

xxD ∫

∂∂

∂

→∂

∂−= (102)

32nd term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuv

ucghc

ghvucghvu

−∂∂

∇×−−−−

−−−−−−−−

δνψδφθδδψδ


αα



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)

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3(

lim

)3()4(ψφν α

ββαdddgdvFv

xxxx

v ∫∂∂

∂

→∂

∂−= (103)

37th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ghg

hcvuc

ghvucghvu

−∂∂

∇×−−−−

−−−−−−−−

δλψδφθδδψδ


αα

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3(

lim

)3()4(ψφλ α

ββαdddgdvFg

xxxx

g ∫∂∂

∂

→∂

∂−= (104)

39th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

φθδφ

θγψδδδψδ


−∂∂

∇×−−−−

−−−−−−−−

cghvuc

ghvucghvu

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3(

lim

)3()4(ψφφ

φγ

ββ

dddgdvFxx

xx∫

∂∂

∂

→∂

∂−= (105)

41st term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3(2)3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδψ

φθδδδψδ


−∂∂

∇×−−−−

−−−−−−−−

ccDghvuc

ghvucghvu

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3(

lim

)3()4(ψφψ

ψ ββ

dddgdvFxx

xxD ∫

∂∂

∂

→∂

∂−= (106)

We reduce the third term of right hand side of equation (73),

[ ] ( )⟩−∂∂

−′′′′′′

×∂

∂

∂

∂−

∂

∂

∂

∂

∂∂

×

−−−−−−

−−−−−−⟨

∫ )1()1()1()1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)1(

)3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

41

)()()()()()(

)()()()()()(

vuvxx

xdx

h

xh

x

u

xu

x

cghvuc

ghvucghvu

δπ


δδψδφθδδδ

αα

β

β

α

α

β

β

α

α

(107)

17th term,

∫ ∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

= ]))(1(41[ )4()4()4()4()4()4,3,2,1(

4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)1()4()1()1( ψφπ α

β

β

α

α

β

β

α

αα

dddgdvdxFxg

xg

xv

xv

xxxv


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41

( A)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

ψδφθδδδ


−−−−

−−−−−−−

cghvu

cghcghvu

[ ] ( ))2()2()2()2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(

)2(41 vu

vxxxd

xh

xh

xu

xu

x−

∂∂

′−′′′′′′

×∂

∂

∂∂

−∂

∂

∂∂

∂∂

× ∫ δπ αα

β

β

α

α

β

β

α

α

∫ ∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

= ]))(1(41[ )4()4()4()4()4()4,3,2,1(

4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)2()4()2()2( ψφπ α

β

β

α

α

β

β

α

αα

dddgdvdxFxg

xg

xv

xv

xxxv

(108)

Similarly, 31st

term,

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

[ ] ( ))3()3()3()3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3(

)3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1(

41 vu

vxxxd

x

h

x

hx

u

x

ux

cghc

ghvucghvu

−∂

∂′′−′′′

′′′

∂

∂

∂

∂−

∂

∂

∂

∂

∂

∂×

−−−−−

−−−−−−

∫ δπ


δδψδφθδδδ

αα

β

β

α

α

β

β

α

α

[ ( )( ) ])4()4()4()4()4()4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

141 ψφπ α

β

β

α

α

β

β

α

αα

dddgdvdxFxg

xg

xv

xv

xxxv ∂

∂

∂∂

−∂

∂

∂∂

−∂∂

∂∂

= ∫ (109)

5th

and 6th

terms

of right hand side of equation (73),

5th

term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

2 vuv

ucghvu

cghvucgh

mm −∂

∂Ω∈×−−−−

−−−−−−−

δψδφθδδδ


αααβ

[ ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ])3()3()3()3()3()3()3()3()2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()1(

)1(2


δψδφθδδδα

ααβ

−−−−−−−

−−−−−∂

∂Ω∈=

cghvucgh

vucghvuv

umm

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()1()1(

2


δψδφθδδδαα

αβ

−−−−−−−

−−−−−∂

∂Ω∈=

cghvucgh

vucghvuuv

mm

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2(

)2()2()2()2()1()1()1()1()1()1()1()1()1(

)1(2


δδψδφθδδδα

ααβ

−−−−−−

−−−−−−∂∂

Ω∈=

cghvuc

ghvucghvuvu

mm

)3,2,1(32 Fmm Ω∈= αβ

(110)

and 6th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))1()1()1(

)1()1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

vuv

vufcghvu

cghvucgh

−∂

∂−×−−−−

−−−−−−−−

δψδφθδδδ


ααα



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)

( ) [ ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )])3()3()3()3()3()3()3()3()2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()1(

)1()1(


δψδφθδδδα

αα

−−−−−−−

−−−−−∂

∂−−=

cghvucgh

vucghvuv

vuf

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3()2()2()2()2()2()2(

)2()2()1()1()1()1()1()1()1()1()1(

)1()1(


δψδφθδδδα

αα

−−−−−−−

−−−−−∂

∂−−=

cghvucgh

vucghvuv

vuf

( ) )3,2,1(3)1(

)1()1( Fv

vufα

αα∂

∂−−= (111)

Similarly, 19th, 20th, 33rd and 34th

terms of right hand side of equation (73),

19th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))2()2(

)2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

2 vuv

ucghvu

cghcghvu

mm −∂

∂Ω∈×−−−−

−−−−−−−

δψδφθδδδ


αααβ

)3,2,1(

32 Fmm Ω∈= αβ (112)

20th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))2()2(

)2()2()2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuv

vufcghvu

cghcghvu

−∂

∂−×−−−−

−−−−−−−−

δψδφθδδδ


ααα

( ) )3,2,1(3)2(

)2()2( Fv

vufα

αα∂

∂−−= (113)

33rd term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

2 vuv

ucghc

ghvucghvu

mm −∂

∂Ω∈×−−−−

−−−−−−−

δψδφθδδψδ


αααβ

)3,2,1(32 Fmm Ω∈= αβ (114)

34th term,

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))3()3(

)3()3()3()3()3()3()3()3()3()2()2(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

vuv

vufcghc

ghvucghvu

−∂

∂−×−−−−

−−−−−−−−

δψδφθδδψδ


ααα

( ) )3,2,1(

3)3()3()3( F

vvuf

ααα

∂

∂−−= (115)

14th

term of Equation (73)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

)3,2,1(3)1(

)1(

)1()1()1(

)1()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()2()2()1()1()1()1()1()1(

FR

cRccghvu

cghvughvu

ψψ

ψδψ

ψδφθδδδ


∂

∂=

−∂

∂×−−−−

−−−−−−−

(116)


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43

( A)

28th term of Equation (73)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

)3,2,1(3)2(

)2(

)2()2()2(

)2()3()3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

FR

cRccghvu

ghvucghvu

ψψ

ψδψ

ψδφθδδδ


∂∂

=

−∂∂

×−−−−

−−−−−−−

(117)

42nd term of Equation (73)

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

)3,2,1(3)3(

)3(

)3()3()3(

)3()3()3()3()3()3()3(

)2()2()2()2()2()2()1()1()1()1()1()1()1()1(

FR

cRcghvu

ghvucghvu

ψψ

ψδψ

φθδδδ


∂∂

=

−∂∂

×−−−

−−−−−−−

(118)

III. Results

Substituting the results (74) – (118) in equation (73) we get the transport equation for three- point distribution

function ),,,()3,2,1(3 ψφgvF in MHD turbulent flow in a rotating system in presence of dust particles under going a

first order reaction as

( ) [ ( ))1()1(

)1(

)1(

)1()1()3,2,1(

3)3()3(

)2()2(

)1()1(

)3,2,1(3

βα

α

α

αβ

ββ

ββ

ββ xg

vvg

gFx

vx

vx

vt

F

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂

( ) ( ) ] )3,2,1(3)3()3(

)3(

)3(

)3()3(

)2()2(

)2(

)2(

)2()2( F

xgv

vg

gxg

vvg

gβα

α

α

αβ

βα

α

α

αβ

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

ν

αββ

ααα

dddgdvFvxx

xxv

xxv

xxv

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

λ

αββ

ααα

dddgdvFgxx

xxg

xxg

xxg

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφφ

φφφγ

ββ

dddgdvFxx

xxxxxx

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+



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I( A

)

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφψ

ψψψ

ββ

dddgdvFxx

xxxxxxD

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

[ ( ) ( )

( ) ( ) )4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

)2()4()2()2()1()4()1()1(

141

1411

41

Fx

g

xg

x

v

xv

xxxv

xxxvxxxv

α

β

β

α

α

β

β

α

αα

αααα

π

ππ

∂

∂

∂

∂−

∂

∂

∂

∂×

−∂

∂

∂

∂+

−∂

∂

∂

∂+

−∂

∂

∂

∂−

∫

∫∫

] 6 )3,2,1(3

)4()4()4()4()4( Fdddgdvdx mm Ω∈+× αβψφ

[ ( ) ( ) ( ) ]

0)( )3,2,1(3)3(

)3()2(

)2()1(

)1(

)3,2,1(3)3(

)3()3()2(

)2()2()1(

)1()1(

=∂∂

+∂∂

+∂∂

−

∂∂

−+∂∂

−+∂∂

−+

FR

Fv

vufv

vufv

vuf

ψψ

ψψ

ψψ

ααα

ααα

ααα

(119)

Continuing this way, we can derive the equations for evolution of )4,3,2,1(4F , )5,4,3,2,1(

5F 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. Certain approximations will be required thus obtained.

IV. Discussions

If R=0,i.e the reaction rate is absent, the transport equation for three- point distribution function in MHD turbulent flow (119) becomes

( ) [ ( ))1()1(

)1(

)1(

)1()1()3,2,1(

3)3()3(

)2()2(

)1()1(

)3,2,1(3

βα

α

α

αβ

ββ

ββ

ββ xg

vvg

gFx

vx

vx

vt

F

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂

( ) ( ) ] )3,2,1(3)3()3(

)3(

)3(

)3()3(

)2()2(

)2(

)2(

)2()2( F

xgv

vg

gxg

vvg

gβα

α

α

αβ

βα

α

α

αβ

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

ν

αββ

ααα

dddgdvFvxx

xxv

xxv

xxv

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

λ

αββ

ααα

dddgdvFgxx

xxg

xxg

xxg

∫∂∂∂

×

→∂∂

+

→∂∂

+

→∂∂

+


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45

( A)

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφφ

φφφγ

ββ

dddgdvFxx

xxxxxx

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφψ

ψψψ

ββ

dddgdvFxx

xxxxxxD

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

[ ( ) ( )

( ) ( ) )4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

)2()4()2()2()1()4()1()1(

141

1411

41

Fx

g

xg

x

v

xv

xxxv

xxxvxxxv

α

β

β

α

α

β

β

α

αα

αααα

π

ππ

∂

∂

∂

∂−

∂

∂

∂

∂×

−∂

∂

∂

∂+

−∂

∂

∂

∂+

−∂

∂

∂

∂−

∫

∫∫

] 6 )3,2,1(3

)4()4()4()4()4( Fdddgdvdx mm Ω∈+× αβψφ

[ ( ) ( ) ( ) ] 0)3,2,1(3)3(

)3()3()2(

)2()2()1(

)1()1( =∂∂

−+∂∂

−+∂∂

−+ Fv

vufv

vufv

vufα

ααα

ααα

αα (120)

which was obtained earlier by Azad et al (2015d)

If the fluid is clean then f=0, the transport equation for three-point distribution function in MHD turbulent flow (119) becomes

( ) [ ( ) )1()1(

)1(

)1(

)1()1()3,2,1(

3)3()3(

)2()2(

)1()1(

)3,2,1(3

βα

α

α

αβ

ββ

ββ

ββ xg

vvg

gFx

vx

vx

vt

F

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂∂

( ) ( ) ] )3,2,1(3)3()3(

)3(

)3(

)3()3(

)2()2(

)2(

)2(

)2()2( F

xgv

vg

gxg

vvg

gβα

α

α

αβ

βα

α

α

αβ

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

ν

αββ

ααα

dddgdvFvxx

xxv

xxv

xxv

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

λ

αββ

ααα

dddgdvFgxx

xxg

xxg

xxg

∫∂∂∂

×

→∂∂

+

→∂∂

+

→∂∂

+



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)

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφφ

φφφγ

ββ

dddgdvFxx

xxxxxx

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφψ

ψψψ

ββ

dddgdvFxx

xxxxxxD

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

[ ( ) ( )

( ) ( ) )4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

)2()4()2()2()1()4()1()1(

141

1411

41

Fx

g

xg

x

v

xv

xxxv

xxxvxxxv

α

β

β

α

α

β

β

α

αα

αααα

π

ππ

∂

∂

∂

∂−

∂

∂

∂

∂×

−∂

∂

∂

∂+

−∂

∂

∂

∂+

−∂

∂

∂

∂−

∫

∫∫

] 0 6 )3,2,1(

3)4()4()4()4()4( =Ω∈+× Fdddgdvdx mmαβψφ (121)

This was obtained earlier by Azad

et al (2014b)

In the absence of coriolis force, mΩ =0, the transport equation

for three- point distribution function in MHD turbulent flow (116) becomes

( ) [ ( ) )1()1(

)1(

)1(

)1()1()3,2,1(

3)3()3(

)2()2(

)1()1(

)3,2,1(3

βα

α

α

αβ

ββ

ββ

ββ xg

vvg

gFx

vx

vx

vt

F

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂∂

( ) ( ) ] )3,2,1(3)3()3(

)3(

)3(

)3()3(

)2()2(

)2(

)2(

)2()2( F

xgv

vg

gxg

vvg

gβα

α

α

αβ

βα

α

α

αβ

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

ν

αββ

ααα

dddgdvFvxx

xxv

xxv

xxv

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

λ

αββ

ααα

dddgdvFgxx

xxg

xxg

xxg

∫∂∂∂

×

→∂∂

+

→∂∂

+

→∂∂

+


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( A)

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφφ

φφφγ

ββ

dddgdvFxx

xxxxxx

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφψ

ψψψ

ββ

dddgdvFxx

xxxxxxD

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

[ ( ) ( )

( ) ( ) )4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

)2()4()2()2()1()4()1()1(

141

1411

41

Fx

g

xg

x

v

xv

xxxv

xxxvxxxv

α

β

β

α

α

β

β

α

αα

αααα

π

ππ

∂

∂

∂

∂−

∂

∂

∂

∂×

−∂

∂

∂

∂+

−∂

∂

∂

∂+

−∂

∂

∂

∂−

∫

∫∫

]

)4()4()4()4()4( ψφ dddgdvdx×

[ ( ) ( ) ( ) ] 0)3,2,1(3)3(

)3()3()2(

)2()2()1(

)1()1( =∂

∂−+

∂

∂−+

∂

∂−+ F

vvuf

vvuf

vvuf

ααα

ααα

ααα (122)

This was obtained earlier by N. M. Islam

et al (2014)

If the fluid is clean and the system is nonrotating then f=0, mΩ =0 and the reaction rate is absent, R=0, the transport equation for three- point distribution function in MHD turbulent flow (119) becomes

( ) [ ( ) )1()1(

)1(

)1(

)1()1()3,2,1(

3)3()3(

)2()2(

)1()1(

)3,2,1(3

βα

α

α

αβ

ββ

ββ

ββ xg

vvg

gFx

vx

vx

vt

F

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂∂

( ) ( ) ] )3,2,1(3)3()3(

)3(

)3(

)3()3(

)2()2(

)2(

)2(

)2()2( F

xgv

vg

gxg

vvg

gβα

α

α

αβ

βα

α

α

αβ

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

ν

αββ

ααα

dddgdvFvxx

xxv

xxv

xxv

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφ

λ

αββ

ααα

dddgdvFgxx

xxg

xxg

xxg

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+



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)

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφφ

φφφγ

ββ

dddgdvFxx

xxxxxx

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

( )

)4()4()4()4()4,3,2,1(4

)4()4()4(

2

)3()2()1(

lim

)3()4(

lim

)2()4(

lim

)1()4(

ψφψ

ψψψ

ββ

dddgdvFxx

xxxxxxD

∫∂∂

∂×

→∂

∂+

→∂

∂+

→∂

∂+

[ ( ) ( )

( ) ( )

] 0

141

1411

41

)4()4()4()4()4(

)4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

)2()4()2()2()1()4()1()1(

=×

∂

∂

∂

∂−

∂

∂

∂

∂×

−∂

∂

∂

∂+

−∂

∂

∂

∂+

−∂

∂

∂

∂−

∫

∫∫

ψφ

π

ππ

α

β

β

α

α

β

β

α

αα

αααα

dddgdvdx

Fx

g

xg

x

v

xv

xxxv

xxxvxxxv

(123)

It was obtained earlier by Azad et al (2014a) If we drop the viscous, magnetic and thermal diffusive and concentration terms from the three point

evolution equation (119), we have

( ) [ ( ) )1()1(

)1(

)1(

)1()1()3,2,1(

3)3()3(

)2()2(

)1()1(

)3,2,1(3

βα

α

α

αβ

ββ

ββ

ββ xg

vvg

gFx

vx

vx

vt

F

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂∂

( ) ( ) ] )3,2,1(3)3()3(

)3(

)3(

)3()3(

)2()2(

)2(

)2(

)2()2( F

xgv

vg

gxg

vvg

gβα

α

α

αβ

βα

α

α

αβ

∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂+

[ ( ) ( )

( ) ( )

] 0

141

1411

41

)4()4()4()4()4(

)4,3,2,1(4)4(

)4(

)4(

)4(

)4(

)4(

)4(

)4(

)3()4()3()3(

)2()4()2()2()1()4()1()1(

=×

∂

∂

∂

∂−

∂

∂

∂

∂×

−∂

∂

∂

∂+

−∂

∂

∂

∂+

−∂

∂

∂

∂−

∫

∫∫

ψφ

π

ππ

α

β

β

α

α

β

β

α

αα

αααα

dddgdvdx

Fx

g

xg

x

v

xv

xxxv

xxxvxxxv

(124)

The existence of the term

( )1(

)1(

)1(

)1(

α

α

α

α

gv

vg

∂

∂+

∂

∂), ( )

)2(

)2(

)2(

)2(

α

α

α

α

gv

vg

∂

∂+

∂

∂ and ( )

)3(

)3(

)3(

)3(

α

α

α

α

gv

vg

∂

∂+

∂

∂

can be explained on the basis that two characteristics of the flow field are related to each other and describe the interaction between the two modes (velocity and magnetic) at point x(1) , x(2) and x(3) .

We can exhibit an analogy of this equation with the 1st equation in BBGKY hierarchy in the kinetic theory of gases. The first equation of BBGKY hierarchy is given Lundgren (1969) as


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( A)

)2()2()1(

)2,1(2

)1(2,1)1(

1)1()1(

)1(1 1 vdxd

vF

xnF

xv

mtF

ααββ

ψ

∂

∂

∂

∂=

∂

∂+

∂∂

∫∫ (125)

where )1()2(2,1 ααψψ vv −=

is the inter molecular potential.

Some approximations are required, if we want to close the system of equations for the distribution functions.

In the case of collection of ionized particles, i.e. in plasma turbulence, it can be provided closure form easily by decomposing F2

(1,2)

as F1(1) F1

(2). But it will be possible if there is no interaction or correlation between two particles. If we decompose F2

(1, 2)

as F2

(1, 2) = (1+∈) F1

(1) F1(2)

and

F3(1,2,3)

= (1+∈)2 F1

(1) F1(2) F1

(3)

Also

F4(1,2,3,4)

= (1+∈)3 F1

(1) F1(2) F1

(3) F1(4)

where ∈ is the correlation coefficient between the particles. If there is no correlation between the particles, ∈ will be zero and distribution function can be decomposed in usual way. Here we are considering such type of approximation only to provide closed from of the equation.

V. Acknowledgement

Authors are grateful to the Department of Applied Mathematics, University of Rajshahi, Bangladesh for giving all facilities and support to carry out this work.

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( A)

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48. M. A. K. Azad, Mst. Mumtahinah, and M. A. Bkar Pk, 2015. Transport equation for the joint distribution functions of certain variables in convective dusty fluid turbulent flow in a rotating system under going a first order reaction, American Journal of Applied Mathematics, 3(1): 21-30.

49. M. A. Bkar Pk, A. Malek and M. A. K. Azad, 2015a. 4-point correlations of dusty fluid MHD turbulent flow in a 1st order reaction, Global J. of Science Frontier Research: F Mathematics & Decision Sciences 15(2), version 1.0: 53-69.

50. M. A. K. Azad, Abdul Malek & M. A. Bkar Pk, 2015a. 3-Point Distribution Functions in the Statistical theory in MHD Turbulent flow for Velocity, Magnetic Temperature and Concentration under going a First Order Reaction, Global J. of Science Frontier Research: A Physics and Space Science, 15(2) Version 1.0, 13-45.

51. M. A. K. Azad, Abdul Malek & M. Abu Bkar Pk, 2015b. Statistical Theory for Three-Point Distribution Functions of Certain Variables in MHD Turbulent Flow in Existence of Coriolis Force in a First Order Reaction, Global J. of Science Frontier Research: A Physics and Space Science, 15(2) Version 1.0, 55-90.

52. M. A. K. Azad, M. Abu Bkar Pk, & Abdul Malek, 2015c. Effect of Chemical Reaction on Statistical Theory of Dusty Fluid MHD Turbulent Flow for Certain Variables at Three- Point Distribution Functions, Science Journal of Applied Mathematics and Statistics, 3(3): 75-98.

53. M. A. K. Azad, Mst. Mumtahinah & M. Abu Bkar Pk, 2015d. Statistical Theory of Three- Point Distribution Functions in MHD Turbulent flow for Velocity, Magnetic Temperature and Concentration in a Rotating System in presence of Dust Particles, Int. J. Scholarly Res. Gate, 3(3): 91-142.

54. M. A. Bkar Pk & M.A.K.Azad, 2015b. Decay of Energy of MHD Turbulence before the Final Period for Four-point Correlation in a Rotating System Undergoing a First Order Chemical Reaction, Int. J. Scholarly Res. Gate, 3(4): 143-160.

55. M. H. U. Molla , M. A. K. Azad & M.Z. Rahman, 2015. Transport Equation for the Joint Distribution Functions of Velocity, Temperature and Concentration in Convective Turbulent Flow in a Rotating System in Presence of Dust Particles, Int. J. Scholarly Res. Gate, 3(4): 161-175.



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First Order Reactant in the Statistical Theory of …...First Order Reactant in the Statistical Theory of Three- Point Distribution Functions in Dusty Fluid MHD Turbulent Flow for