For verification of this paper, please visit on http://www.socialresearchfoundation.com/remarking.php#8

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.

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

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 h_i 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.

For larger times the last term in the Eq. (4.20) becomes negligible, leaving the -3/2 power decay law for the final period.

1.Biscamp, D. : (2003), Magnetohydrodynamic Turbulence, Cambridge Univ.Press, Cambridge. 2.Chandrasekhar, S. : (1951) Proc, Roy, Soc, London, A 204, 435. 3.Corrsin, S : (1951) J. Applied Physics, 22, 469. 4.Deissler, R.G. : (1960) Physics of Fluids, 3, 176. 5.Islam, M.A. and Sarker, M.S.A: (2001) Indian J. Pure and appl. Math., 32(8), 1173. 6.Sarker, M.S.A. and : (1991) Int. J. Engng. Sci., 29, 21 Kishore, N. 1479. 7.Loeffler, A. L. and Deissler, R.G.: (1961) Int. J. Heat Mass Transfer, 1, 312. 8.Dixit, T. and Sunil Kumar: (1994) Indian Academy of Mathematics, 16, 31. 9.Dixit, T. : (2006) Progress of Mathematics 40 (1,2). 10.Batchelor, G.K. : (1976) An Introduction to Fluid Dynamics, Cambridge Univ. Press. London. 11.Donghoon Lee et al.: (2001), J. Fluid Mech. 439, 367. 12.Edward, S. : (1964) J. Fluid Mech., 18, 239. 13.Ghosh, K.M. : (1972) Indian, J. of Pure and Appl. Math., 3, 157. 14.Hopf, E. : (1957) In Proc. Symp. Appl. Math. 7, American Mathematical Society, 41. 15.Kishore, N and Singh, S.R.: (1985) Prog. Of Maths, B.H.U. 19 (1 and 2), 13. 16.Kishore, N and Upadhyay, B.N.: (2000) Accepted for Publication, Indian, J. of Pure and Applied Physics. 17.Lighthill, M.J. : (1960) Fourier Analysis and Generalized Function, Cambridge University Press. 18.Leislie, D.C. : (1973) Development in the 22 Theory of Turbulence, Clarendon Press Oxford. 19.Pope, S.B. : (2000), Turbulent Flows, Cambridge Univ. Press, Cambridge. 20.Sultan, S. : (2008) Journal of Engineering and Applied Sci. 3 (2), 166. 21.Stanisic, M.M. : (1985) Mathematical Theory of Turbulence, Springer – Verlag, New York. 22.Molla, M.H.U. et al.: (2020) Res. Jl. of Maths and Stat. 4 (2), 45.Dixit, T. : (2019) ARJPS 12 (1,2) 29. Nichels, T.B. : (2018), J. Fluid Mech. 408, 323. 23. Bkar, P.K. : (2020) Res. Jl. of Appl. Sci.Eng. and Tech. 5 (2), 585.