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)
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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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)
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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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
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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( 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
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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( ) ( ) ( ) ( ) ( ))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,
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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( 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)
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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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
)1()2(ψδφθδδδν α
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
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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( 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∂
∂
∂
∂−
∂
∂
∂
∂×
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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)
)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
φθδδδψδφθδδ
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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(39)
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( A)
( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(
)1(
)1()2()2(
)2()2()2()2()2()2()1()1()1()1()1()1(
ghgt
hc
ghvucvu
−∂∂
∂∂
−
−−−−−−−+
δψδ
φθδδδψδφθδδ
( ) ( ) ( ) ( ) ( ) ( )( ) ( ))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
φθδδψδφθδδδ
δψδ
φθδδψδφθδδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
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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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
φθδδψδφθδδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
αααβ
( ) ( ) ( ))2()2()2(
2)2(2)2(
)2()2()2()2( ψδ
ψφθδ
ββ −
∂
∂×−∇+
∂
∂−− cRccD
xcu
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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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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδφθδδδ
ββ −
∂∂
∂∂
×−
−−−−−−+
cxcuc
ghvughvu
( ) ( ) ( ) ( ) ( ) ( )( ) ( ))1()1(
)1()1(2)2()2(
)2()2()2()2()2()2()1()1()1()1()1()1(
ψδψ
ψδ
φθδδδφθδδδ
−∂
∂∇×−
−−−−−−−+
ccDc
ghvughvu.
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ))1()1()1(
)1()2()2(
)2()2()2()2()2()2()1()1()1()1()1()1(
ψδψ
ψδ
φθδδδφθδδδ
−∂
∂×−
−−−−−−+
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
φθδδψδφθδδδ
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
φθδδψδφθδδδ
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
φθδδψδφθδδδ
αα
β
β
α
α
β
β
α
α
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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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
φθδδψδφθδδδ
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
φθδδψδφθδδδ
αααβ
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))2()2(
)2()2()2()2()2(
)2()2()2()2()1()1()1()1()1()1()1()1(
vuv
vufc
ghcghvu
−∂∂
−×−
−−−−−−−+
δψδ
φθδδψδφθδδδ
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
φθδδψδφθδδδ
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
φθδδψδφθδδδ
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
φθδδψδφθδδδ
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)
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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© 2016 Global Journals Inc. (US)
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
αβ
βα
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ))1()1()1(
)1()2()2(
)2()2()2()2()2()2()1()1()1()1()1()1(
ghx
uc
ghvucvu
−∂∂
×−
−−−−−−−=
δψδ
φθδδδψδφθδδ
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
φθδδδφθδδδ
ββ −
∂∂
∂∂
×−
−−−−−−
cxcuc
ghvughvu
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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( ) ( ) ( ) ( ) ( )( ) ( ) ( ) .)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)
Adding these equations from (42) to (45), we get
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ))1()1()1(
)1()2()2(
)2()2()2()2()2()2()1()1()1()1()1()1(
vux
uc
ghvucgh
−∂∂
×−
−−−−−−−
δψδ
φθδδδψδφθδδ
ββ
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ))1()1()1(
)1()2()2(
)2()2()2()2()2()2()1()1()1()1()1()1(
ghx
uc
ghvucvu
−∂∂
×−
−−−−−−−+
δψδ
φθδδδψδφθδδ
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
φθδδδφθδδδ
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
φθδδψδφθδδδ
αβ
βα
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ))2()2()2(
)2()2()2(
)2()2()2()2()1()1()1()1()1()1()1()1(
vux
uc
ghcghvu
−∂∂
×−
−−−−−−−=
δψδ
φθδδψδφθδδδ
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
φθδδψδφθδδδ
αβ
βα
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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( A)
( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(
)2()2()2()2(
)2()2()2()2()1()1()1()1()1()1()1()1(
ghx
uc
vucghvu
−∂∂
×−
−−−−−−−=
δψδ
φθδδψδφθδδδ
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)
Adding these equations from (46) to (49), we get
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ))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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
αβ
βα
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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I( A
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= - )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
φθδδψδφθδδδ
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
φθδδψδφθδδδ
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
φθδδδψδφθδδ
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
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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( 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
φθδδδψδφθδδ
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
φθδδδφθδδδ
−∂
∂∇×−
−−−−−−−
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
φθδδψδφθδδδ
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,
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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)
( ) ( ) ( ) ( ) ( ) ( )( ) ( ))2()2(
)2()2(2)2()2(
)2()2()2()2()1()1()1()1()1()1()1()1(
ghg
hc
vucghvu
−∂
∂∇×−
−−−−−−−
δλψδ
φθδδψδφθδδδ
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
φθδδδψδφθδδ
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
φθδδψδφθδδδ
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,
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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( 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
φθδδδψδφθδδ
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
φθδδδψδφθδδ
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
φθδδψδφθδδδ
αααβ
= )2,1(22 Fmm Ω∈ αβ
(68)
and 20th term,
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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( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ))2()2(
)2()2()2()2()2(
)2()2()2()2()1()1()1()1()1()1()1()1(
vuv
vufc
ghcghvu
−∂
∂−×−
−−−−−−−
δψδ
φθδδψδφθδδδ
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
φθδδδφθδδδ
−∂
∂×−
−−−−−−
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 ∫∂∂
∂
→∂∂
+
→∂∂
+
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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( 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
−∂∂
−−−−−
−−−−−−+
−∂∂
−−−−−
−−−−−−=
δψδφθδδδψδ
φθδδδψδφθδδ
δψδφθδδδψδ
φθδδδψδφθδδ
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ))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(
φθδδψδφθδδδ
ψδψδφθδδδψδ
φθδδδφθδδδ
−−−−−−+
−∂∂
−−−−−
−−−−−−+
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
δψδφθδδδψδ
φθδδψδφθδδδ
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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( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ))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(
ψδφθδδδψδφθδ
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
−∂∂
∂∂
−−−−−
−−−−−−−=
δψδφθδδδψδ
φθδδδψδφθδδ
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))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
ψδφθδδψδφθδδδ
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
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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( 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
φθδδδψδφθδδδ
δψδφθδδψδ
φθδδδψδφθδδδ
δψδφθδδψδ
φθδδδψδφθδδδ
−∂
∂∂
∂−−−−
−−−−−−−−+
−∂
∂∂
∂−−−−
−−−−−−−−+
−∂
∂∂
∂−−−−
−−−−−−−−+
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
φθδδδψδφθδδδ
−∂∂
∂∂
−−−−
−−−−−−−−+
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
φθδδδψδφθδδ
−−−−−
−−−−−−−=∂
∂
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
φθδδδψδφθδδ
−−−−−
−−−−−−−+
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
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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( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ))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
φθδδδφθδδδ
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
φθδδψδφθδδδ
−−−−−
−−−−−−−+
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(
ψδφθδδψδφθδ
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
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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© 2016 Global Journals Inc. (US)
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( 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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδδδ
ββ −
∂
∂
∂
∂×−−−−
−−−−−−−+
xucghvu
cghvucghvu
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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I( A
)
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))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
ψδφθδδδψδδδ
−∂
∂∇×−−−−
−−−−−−−−+
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
ψδφθδδδφθδδδ
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
ψδφθδδδφθδδδ
−∂
∂∇×−−−−
−−−−−−−−+
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
ψδφθδδδφθδδδ
−∂
∂×−−−−
−−−−−−−+
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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
xh
xh
xu
xu
xcghvu
cghcghvu
∂
∂
∂∂
−∂
∂
∂∂
∂∂
×−−−−
−−−−−−−+
∫
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ))2()2()2()2()2()2()1()1()1()1()1()1()1()1(
)2()2()2(
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
αβ
βα
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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© 2016 Global Journals Inc. (US)
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( 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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
ψδδδψδφθδδδ
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
ψδδδψδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−+
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
−∂
∂∇×−−−−
−−−−−−−−+
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
φθδδδψδφθδδδ
−∂
∂×−−−−
−−−−−−−+
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
xh
xh
xu
xu
xcghc
ghvucghvu
∂
∂
∂∂
−∂
∂
∂∂
∂∂
×−−−−
−−−−−−−+
∫
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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I( A
)
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ))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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−+
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
−∂
∂∇×−−−
−−−−−−−−+
ccDghvu
ghvucghvu
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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( 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
φθδδδψδφθδδδ
−∂∂
×−−−
−−−−−−−+
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδδδ
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
ψδφθδδδψδδδ
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
ψδφθδδδφθδδδ
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
ψδφθδδδφθδδδ
ββ −
∂
∂×−−−−
−−−−−−−−=
cx
ucghvu
cghvughvu
(77)
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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Adding these equations from (74) to (77), we get
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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(
φθδψδφθδδδ
ψδφθδδδψδδδ
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
ψδφθδδδφθδδδ
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(
ψδφθδδδψδφθδ
δδψδφθδδδββ
−−−−−−
−−−−−−−∂∂
−=
cghvuc
ghvucghvuux
)3,2,1(3
)1()1(
Fvx ββ∂
∂−= [Applying the properties of distribution functions ] (78)
Similarly, 15th, 21st, 24th
and 26th
terms of right hand-side of equation (73) can be simplified as follows;
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
ψδφθδδψδφθδδδ
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
−∂
∂×−−−−−
−−−−−−−=
δψδφθδδδψδ
φθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
ψδδδψδφθδδδ
ββ −
∂
∂
∂
∂×−−−−
−−−−−−−
xucghvu
cghvucghvu
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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( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))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(
φθδψδφθδδδ
ψδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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(
ψδφθδδδψδφθδ
δδψδφθδδδββ
−−−−−−
−−−−−−∂∂
−
cghvuc
ghvucghvuux
)2(
)3,2,1(3)2(
ββ x
Fv
∂
∂−= (83)
Similarly, 29th, 35th, 38th
and 40th
terms of right hand-side of equation (73) can be simplified as follows;
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
ββ
(85)
38th
term,
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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( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ))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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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(
ψδφθδδδψδφθδ
δδψδφθδδδββ
−−−−−−
−−−−−−−∂
∂−
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
αβ
βα
= - )1(βg )1(
)1(
α
α
gv∂
∂)1(
)3,2,1(3
βxF∂
∂ (90)
16th term,
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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( 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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
ψδφθδδδψδφθδδ
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(
ψδφθδδδψδφθδ
δδψδφθδδδν αα
−−−−−−
−−−−−−∇∂∂
−=
cghvuc
ghvucghvuuv
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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)
[ ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ])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(
ψδφθδδδψδφθδδ
δψδφθδδδν αββα
−−−−−−−
−−−−−∂∂∂
∂∂
−=
cghvucgh
vucghvuuxxv
[ ( ) ( ) ( ) ( ))1()1()1()1()1()1()1()1()4()4()4(
2
)1(
lim
)1()4(ψδφθδδδν α
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(
φθδδδψδφθδδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδδδ
−∂∂
∇×−−−−
−−−−−−−−
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
ψδφθδδδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−+
ccDcghvu
cghvughvu
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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( 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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
ψδδδψδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−
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
φθδδδψδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−
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
φθδδδψδφθδδδ
αα
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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)
)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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−
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
φθδδδψδφθδδδ
−∂∂
∇×−−−−
−−−−−−−−
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
ψδφθδδδψδφθδ
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
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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( A)
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ))3()3()3()3()3()3()3()3(
)2()2()2()2()2()2()1()1()1()1()1()1()1()1(
ψδφθδδδ
ψδφθδδψδφθδδδ
−−−−
−−−−−−−
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
ψδφθδδψδφθδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδδ
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
ψδφθδδδψδφθδ
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
ψδφθδδδψδφθδδ
ααα
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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I( A
)
( ) [ ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )])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(
ψδφθδδδψδφθδδ
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(
ψδφθδδδψδφθδδ
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
ψδφθδδψδφθδδδ
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
ψδφθδδψδφθδδδ
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
φθδδδψδφθδδδ
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
φθδδδψδφθδδδ
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
ψδφθδδδφθδδδ
∂
∂=
−∂
∂×−−−−
−−−−−−−
(116)
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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( 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
φθδδδψδφθδδδ
∂∂
=
−∂∂
×−−−−
−−−−−−−
(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
φθδδδψδφθδδδ
∂∂
=
−∂∂
×−−−
−−−−−−−
(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
∫∂∂
∂×
→∂
∂+
→∂
∂+
→∂
∂+
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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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
∫∂∂∂
×
→∂∂
+
→∂∂
+
→∂∂
+
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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( 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
∫∂∂∂
×
→∂∂
+
→∂∂
+
→∂∂
+
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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IV
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(
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
∫∂∂∂
×
→∂∂
+
→∂∂
+
→∂∂
+
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
1
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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
∫∂∂
∂×
→∂
∂+
→∂
∂+
→∂
∂+
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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( )
)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
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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( 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.
References
1. Hopf E. (1952). Statistical hydrodynamics and functional calculus, J. Rotational Mech. Anal., 1, 87-123.
2. Kraichanan R H. (1959), Distribution functions in the statistical theory of covective MHD turbulent flow, J. Fluid Mech., 5, 497.
3. Edward S. (1964). The statistical dynamics of homogeneous turbulence J. Fluid Mech. 18(2), 239-273.
4. Herring J. R (1965).Self-consistent field approach to turbulence theory, Phys. Fluid, 8, 2219-2225.
5. Ta-You W., 1966. Kinetic Theory of Gases and Plasma. Addision Wesley Phlelishing.
6. Lundgren T.S., 1967. Hierarchy of coupled equations for multi-point turbulence velocity distribution functions. Phys. Fluid., 10: 967.
7. Lundgren T.S., 1969. Hierarchy of coupled equations for multi-point turbulence velocity distribution functions. Phys. Fluid., 12: 485.
8. Kishore N., 1978. Distribution functions in the statistical theory of MHD turbulence of an incompressible fluid. J. Sci. Res., BHU, 28(2): 163.
9. Pope S. B., 1979.The statistical theory of turbulence flames. Phil. Trans Roy. Soc. London A, 291: 529.
10. Pope S.B., 1981.The transport equation for the joint probability density function of velocity and scalars in turbulent flow. Phys. Fluid., 24: 588.
11. Kollman W. and J. Janica, 1982.The transport equation for the probability density function of a scalar in turbulent shear flow. Phys. Fluid., 25: 1955.
12. Kishore N., Singh S. R., 1984. Transport equation for the bivariate joint distribution function of velocity and temperature in turbulent flow. Bull. Tech. Univ. Istambul, 37: 91-100
13. Kishore N., Singh S. R., 1985. Transport equation for the joint distribution function of velocity, temperature and concentration in convective turbulent flow. Prog. of Maths. 19(1&2):13-22.
14. Dixit T., Upadhyay B.N., 1989. Distribution functions in the statistical theory of MHD turbulence of an incompressible fluid in the presence of the Coriolis force, Astrophysics and Space Sci., 153: 297-309.
15. Sarker M.S.A. and Kishore N., 1991. Distribution functions in the statistical theory of convective MHD turbulence of an incompressible fluid. Astrophys. Space Sci., 181: 29.
16. Sarker M.S.A. and N. Kishore, 1999. Distribution functions in the statistical theory of convective MHD turbulence of mixture of a miscible incompressible fluid. Prog. Math., BHU India, 33(1-2): 83.
17. Sarker M.S.A. and M.A. Islam, 2002. Distribution functions in the statistical theory of convective MHD turbulence of an incompressible fluid in a rotating system. Rajshahi University Studies. Part-B. J. Sci., Vol: 30.
18. Azad, M. A. K., Sarker M. S. A., 2003. Decay of MHD turbulence before the final period for the case of multi-point and multi-time in presence of dust particle, Bangladesh J. Sci. Ind. Res. 38(3-4):151-164.
19. Azad, M. A. K., Sarker M. S. A., 2004a. Statistical theory of certain distribution functions in MHD
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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16XVI
Iss u
e er
sion
IV
I( A
)
turbulence in a rotating system in presence of dust particles. Rajshahi university studies. Part -B. J. Sci., 32: 193-210.
20. Sarker M. S. A., Azad M. A. K., 2004b. Decay of MHD turbulence before the final period for the case of multi-point and multi-time in a rotating system. Rajshahi. University Studies. Part-B, J. Sci., 32: 177-192.
21. Sarker M. S. A., Azad M. A. K., 2006. Decay of temperature fluctuations in homogeneous turbulence before the final period for the case of multi-point and multi-time in a rotating system. Bangladesh. J. Sci. & Ind. Res., 41 (3-4): 147-158.
22. Islam M.A. and M.S.A. Sarker, 2007. Distribution functions in the statistical theory of MHD turbulence for velocity and concentration undergoing a first order reaction. Rajshahi university studies. Part-B. J. Sci., Vol: 35.
23. Azad, M. A. K., Sarker M. S. A., 2008. Decay of temperature fluctuations in homogeneous turbulence before the final period for the case of Multi- point and Multi- time in a rotating system in presence of dust particle. J. Appl. Sci. Res., 4(7): 793- 802.
24. Azad, M. A. K., Sarker M. S. A., 2009a. Decay of temperature fluctuations in MHD turbulence before the final period in a rotating system,. Bangladesh J. Sci & Ind. Res, 44(4): 407-414.
25. Azad M. A. K., Aziz M. A., Sarker M. S. A., 2009b.
First order reactant in Magneto-hydrodynamic turbulence before the final Period of decay in presence of dust particles in a rotating System. J. Phy. Sci., 13: 175-190.
26. Azad M. A. K., Aziz M. A., Sarker M. S. A., 2009c. First order reactant in MHD turbulence before the final period of decay in a rotating system. J. Mech. Contin. & Math. Sci., 4(1): 410-417.
27. Aziz M. A., Azad
M. A. K., Sarker M. S. A., 2009d.
First order reactant in Magneto- hydrodynamic turbulence before the final period of decay for the case of multi-point and multi-time in a rotating system. Res. J. Math. & Stat., 1(2): 35-46.
28. Aziz M. A., Azad
M. A. K., Sarker M. S. A., 2010a.
Statistical Theory of Certain Distribution Functions in MHD Turbulent Flow Undergoing a First Order Reaction in Presence of Dust Particles, Journal of Modern Mathematics and Statistics, 4(1):11-21.
29. Aziz M. A., Azad M. A. K., Sarker M. S. A., 2010b. Statistical Theory of Distribution Functions in Magneto-hydrodynamic Turbulence in a Rotating System Undergoing a
First Order Reaction in
Presence of Dust Particles, Res. J. Mathematics and Statistics, 2(2):37-55.
30. Aziz M. A., Azad
M. A. K., Sarker M. S. A., 2010c.
First order reactant in MHD turbulence before the final period of decay for the case of multi-point and
multi-time in a rotating system in presence of dust particle. Res. J. Math. & Stat., 2(2): 56-68, 2010c.
31. Azad, M. A. K., Aziz M. A., Sarker M. S. A., 2011. Statistical Theory of certain Distribution Functions in MHD Turbulent flow for Velocity and Concentration Undergoing a First Order Reaction in a Rotating System, Bangladesh Journal of Scientific & Industrial Research, 46(1): 59-68.
32. Azad, M. A. K., Molla M. H. U., Rahman M. Z., 2012. Transport Equatoin for the Joint Distribution Function of Velocity, Temperature and Concentration in Convective Tubulent Flow in Presence of Dust Particles, Research Journal of Applied Sciences, Engineering and Technology( RJASET) 4(20): 4150-4159.
33. Molla M. H. U., Azad M. A. K., Rahman M. Z., 2012. Decay of temperature fluctuations in homogeneous turbulenc before the finaln period in a rotating system. Res. J. Math. & Stat., 4(2): 45-51.
34. Bkar Pk M. A., Azad M. A. K. Sarker M.S.A., 2012. First-order reactant in homogeneou dusty fluid turbulence prior to the ultimate phase of decay for four-point correlation in a rotating system. Res. J. Math. & Stat., 4(2): 30-38.
35. Bkar Pk M. A., Azad M. A. K. Sarker M.S.A., 2013a. First-order reactant in homogeneous turbulence prior to the ultimate phase of decay for four-point correlation in presence of dust particle. Res. J. Appl. Sci. Engng. & Tech., 5(2): 585-595.
36. Bkar Pk. M. A., Sarker M. S. A., Azad M. A. K., 2013b. Homogeneous turbulence in a first-order reactant for the case of multi-point and multi-time prior to the final period of decay in a rotating system. Res. J. Appl. Sci., Engng. & Tech. 6(10): 1749-1756.
37. Bkar Pk. M. A., Sarker M. S. A., Azad M. A. K., 2013c. Decay of MHD turbulence before the final period for four- point correlation in a rotating system. Res. J. Appl. Sci., Engng. & Tech. 6(15): 2789-2798.
38. Bkar Pk M. A., Azad M. A. K. Sarker M.S.A., 2013d. Decay of dusty fluid MHD turbulence for four- point correlation in a rotating System. J. Sci. Res. 5 (1):77-90.
39. Azad, M. A. K., Molla M. H. U., Rahman M. Z., 2014. Transport Equatoin for the Joint Distribution Functions in Convective Tubulent Flow in Presence of Dust Particles undergoing a first order reaction, Rajshahi University Journal of Science and Engineering( accepted for publication), volume 41:
40. Azad M. A. K., Mumtahinah Mst., 2013. Decay of Temperature fluctuations in dusty fluid homogeneous turbulence prior to final period, Res. J. Appl. Sci., Engng and Tech., 6(8): 1490-1496.
41. Bkar PK. M. A., Sarker M. S. A., Azad M. A. K., 2013. Decay of MHD Turbulence prior to the ultimate phase in presence of dust particle for Four-
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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( A)
point Correlation, Int. J. of Appl. Math and Mech. 9 (10): 34-57.
42. Molla M. H. U., Azad M. A. K., Rahman M. Z., 2013. Transport equation for the joint distribution functions of velocity, temperature and concentration in convective turbulent flow in presence of coriolis force, Int. J. Scholarly Res. Gate, 1(4): 42-56.
43. Azad M. A. K., Mumtahinah Mst., 2014. Decay of Temperature fluctuations in dusty fluid homogeneous turbulence prior to the ultimate period in presence of coriolis force, Res. J. Appl. Sci. Engng. and Tech. 7(10):1932-1939.
44. Azad M.A.K., Islam M. N., Mumtahinah Mst., 2014a.Transport Equations of Three- point Distribution Functions in MHD Turbulent Flow for Velocity, Magnetic Temperature and Concentration, Res. J. Appl. Sci. Engng. and Tech. 7(24): 5184-5220.
45. Azad M. A. K. Islam M. N., 2014b. Transport Equations of Three Point Distribution Functions in MHD Turbulent Flow for Velocity, Magnetic Temperature and Concentration in a Rotating System, Int. J. Scholarly Res. Gate, 2(3): 75-120.
46. Islam M. N., Azad M.A.K., 2014. Transport equations of three-point distribution functions in MHD turbulent flow for velocity, magnetic temperature and concentration in presence of dust particles, Int. J. Scholarly Res. Gate, 2(4): 195-240.
47. M. Abu Bkar Pk, Abdul Malek, M. Abul Kalam Azad, 2015. Effects of first-order reactant on MHD turbulence at four-point correlation, Applied and Computational Mathematics, 4(1): 11-19.
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.
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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