P: ISSN No. 2394-0344 RNI No.  UPBIL/2016/67980 VOL.- VII , ISSUE- VIII November  - 2022
E: ISSN No. 2455-0817 Remarking An Analisation
First Order Reactant in Mhd Turbulence Before the Final Period of Decay
Paper Id :  16520   Submission Date :  14/10/2022   Acceptance Date :  22/11/2022   Publication Date :  25/11/2022
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Ajay Kumar Sonkar
Assistant Professor
Mathematics
Govt. Degree College, Pawanikala
,Sonbhadra, U.P., India
Abstract Following Deissler’s approach, the decay for the concentration fluctuations of dilute contaminant undergoing a first order chemical reaction in MHD turbulence at times before the final period in presence of dust particle is studied and have considered correlations between fluctuating quantities at two and three point correlations equations are obtained and the set of equations is made. The correlations equations are converted to spectrum form by taking the Fourier transforms. Finally we obtained the decay law of magnetic energy for the concentration fluctuations before the final period inpresence of dust particle by integrating the energy spectrum over all wave numbers.
Keywords Mhd Turbulence, Decay Before the Final Period, Concentration, Fluctuations, Energy Spectrum, Magnetic Energy, First Order Reactant, Dust Particle, Correlations.
Introduction
In this paper we have discussed the decay of MHD turbulence before the final period of decay for the case of multi-point and single-time. In this topic, we have extended the case of multi-point and single time for the case of multi-point and multi-time in a rotating system in presence of dust particles. Here, it is assumed that back reaction of the magnetic field on the velocity field may be neglected .Thus we have decoupled the magnetic effect from the Navier-Stokes equations so that the velocity and velocity spectrum will be independent quantities in the whole analysis.
Aim of study Two-Point, Two-Time Correlation And Spectral Equations, Three-Point, Three-Time Correlation And Spectral Equations.
Review of Literature
Molla, M.H.U. et.al: (2020) Res. J1. of Maths and stat. 4(2), 45. 
The used of Decay law of magnetic energy fluctuations of a dilute cantaminant undergoing a first order chemical reaction before the final period.

Dixit, T. : (2019) AR JPS 12 (1,2) 29.
This study shows that due to the effect of rotation of fluid in the flow field with chemical reaction of the first order in the Concentration the magnetic field. 

Bkar, P.K. : (2020) Res. 31. of Appl. Sci. Eng. and Tech. 5(2), 585.
The Study of as a wall vector dk̂ =dk1, dk₂ dk3. The magnitude of has the dimension 1/length and Can be considered to be be the reciprocal of an eddy Size.
Main Text

1. Basic Equations

The equations of motion and continuity for viscous, incompressible dusty fluid MHD turbulent flow in a rotating system are given by:


2. Two-Point, Two-Time Correlation And Spectral Equations

Under the condition that (I) the turbulence and the concentration magnetic field are homogeneous (ii) the chemical reaction has no effect on the velocity field, and (iii) the reaction rate and the magnetic diffusivity are constant, the induction equation of a magnetic field fluctuation of concentration of a dilute contaminant undergoing a first order chemical reaction at the points p p' and separated by the vector  could be written as:


where, R is the constant reaction rate.

Multiplying Eq. (2.1) by  and Eq. (2.2) by hi and taking ensemble average, we get


and


Using the relations of (Chandrasekhar, 1951)


Now we write Eq. (2.8) and (2.9) in spectral form in order to reduce it to an ordinary differential equation by use of the following three-dimensional Fourier transforms:

Interchanging the subscripts i and j then interchanging the points p and p' gives

whereis known as a wave number vector and . The magnitude of    has the dimension l/length and can be considered to be the reciprocal of an eddy size. Substitution of Eq. (2.10) to (2.12) into Eq. (2.8) and (2.9) leads to the spectral equations


tensor Eq. (2.13) and (2.14) becomes a scalar equation by contraction of the indices i and j

The terms on the right side of Eq. (2.15) and (2.16) are collectively proportional to what is known as the magnetic energy transfer terms.

3.Three-Point, Three-Time Correlation And Spectral Equations

Similar procedure can be used to find the three-point correlation equations. For this purpose we take the momentum equation of dusty fluid MHD turbulence in a rotating system at the point P and the induction equations of magnetic field fluctuations, governing the concentration of a dilute contaminant undergoing a first order chemical reaction at p' and p" separated by the vector as:


In order to convert Eq. (3.7) – (3.9) to spectral form, we can define the following six dimensional Fourier transforms:



Interchanging the points P and P' along with the indices I and j results in the relations


If the derivative with respect to x1 is taken of the momentum Eq. (3.1) for the point P, the equation multiplied by  and time average taken, the resulting equation:


Taking the Fourier transforms of Eq. (3.8)

Equation (3.20) can be used to eliminatefrom Eq. (3.16).

The tensor Eq. (3.16) to (3.18) can be converted to scalar equation by contraction of the indices i and j and inner multiplication by k1.

4.  Solution For Times Before The Final Period

It is well known that the equation for final period of decay is obtained by considering the two-point correlations after neglecting third-order correlation terms. To study the decay for times before the final period, the three-point correlations are considered and the quadruple correlation terms are neglected because the quadruple correlation terms decays faster than the lower-order correlation terms. The term  associated with the pressure fluctuations should also be neglected. Thus neglecting all the terms on the right hand side of Eq. (3.4) to (3.6).





depends on the initial conditions of the turbulence.

In order to find the solution completely and following (Loeffler and Deissler, 1961), we assume that:

where, 0 is a constant determined by the initial conditions. The negative sign is placed in front of 0 in order to make the transfer of energy from small to large wave numbers for positive value of 0.

Substituting Eq. (4.10) into (4.9), we get:

The series of Eq. (4.14) contains only even power of k and start with k4 and the equation represents the transfer function arising owing to consideration of magnetic field at three-point and three-times.

If we integrate Eq. (4.14) for t = 0 over all wave numbers, we find that:

which indicates that the expression for F satisfies the condition of continuity and homogeneity. Physically it was to be expected as F is a measure of the energy transfer and the total energy transferred to all wave numbers must be zero.

The linear Eq. (4.12) can be solved to give:

where   is a constant of integration and can be obtained as by Corrsin (1951). Substituting the values of F from Eq. (4.14) into (4.16) gives the equation:





This is the decay law of magnetic energy fluctuations of concentration of a dilute contaminant undergoing a first order chemical reaction before the final period for the case of multi-point and multi-time in MHD turbulence in a rotating system in presence of dust particle.
Result and Discussion

In Eq. (4.20) we obtained the decay law of magnetic energy fluctuations of a dilute contaminant undergoing a first order chemical reaction before the final period considering three-point correlation terms for the case of multi-point and multi-time in MHD turbulence in presence of dust particle in a rotating system.

If the fluid is non-rotating and clean then, f = 0, the Eq. (4.20) becomes:


Which was obtained earlier by Islam and Sarker (2001).

Which is same as obtained earlier by Sarker and Kishore (1991).

This study shows that due to the effect of rotation of fluid in the flow field with chemical reaction of the first order in the concentration the magnetic field fluctuation in dusty fluid MHD turbulence in rotating system for the case of multi-point and multi-time, i.e., the turbulent energy decays more rapidly than the energy for non-rotating clean fluid and the faster rate is governed by . Here the chemical reaction in MHD turbulence for the case of multi-point and multi-time causes the concentration to decay more they would for non-rotating clean fluid and it is governed by. The first term of right hand side of Eq. (4.20) corresponds to the energy of magnetic field fluctuation of concentration for the two-point correlation and the second term represents magnetic energy for the three-point correlation. In Eq. (4.20), the term associated with the three-point correlation die out faster than the two-point correlation. If higher order correlations are considered in the analysis, it appears that more terms of higher power of time would be added to the Eq. (4.20). 
Conclusion For larger times the last term in the Eq. (4.20) becomes negligible, leaving the -3/2 power decay law for the final period.
References
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