Decay of Temperature Fluctuations in Dusty Fluid Homogeneous Turbulence Prior to the Ultimate Period in Presence of Coriolis Force

Using deissler’s method we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence before the final period in presence of coriolis force and have considered correlations between fluctuating quantities at twoand three-point. The equations for twoand three-point correlation is obtained and the set of equations is made to determinate by neglecting the forth-order correlation in comparison to the secondand third-order correlations. For solving the correlation equations are converted to spectral form by taking their Fourier transform. Finally, integrating the energy spectrum over all wave numbers, the energy decay law of temperature fluctuations in homogeneous dusty fluid turbulence before the final period in presence of coriolis force is obtained.


INTRODUCTION
Interest in motion of dusty viscous fluid has developed rapidly in recent years.Such situations occur in movement of dust-laden air, in problems of fluidization, in the use of dust in gas cooling system and in sedimentation problem in tidal rivers.
In geophysical flows, the system is usually rotating with a constant angular velocity.Such large-scale flows are generally turbulent.When the motion is referred to axes, which rotate steadily with the bulk of the fluid, the coriolis and centrifugal force must be supposed to act on the fluid.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 force plays an important role in a rotating system of turbulent flow, while centrifugal force with the potential is incorporated in to the pressure.Taylor (1935) has been pointed out that the equation of motion of turbulence relates the pressure gradient and the acceleration of the fluid particles and the mean-square acceleration can be determined from the observation of the diffusion of marked fluid particles.The behavior of dust particles in a turbulent flow depends on the concentration of the particles and the size of the particles with respect to the scale of turbulent fluid.Saffman (1962) derived and equation that describe the motion of a fluid containing small dust particle, which is applicable to laminar flows as well as turbulent flow.Kishore and Sarker (1990) studied the rate of change of vorticity covariance in MHD turbulent flow of dusty incompressible fluid.Also Rahman (2010) studied the Rate of change of vorticity covariance in MHD turbulent flow of dusty fluid in a rotating system.Kishore and Sinha (1988) also studied the rate of change of vorticity covariance of dusty fluid turbulence.Corrsin (1951b) had made an analytical attack on the problem of turbulent temperature fluctuations using the approaches employed in the statistical theory of turbulence.His results pertain to the final period of decay and for the case of appreciable convective effects, to the "energy" spectral from in specific wave-number ranges.Deissler (1958Deissler ( , 1960) ) developed a theory for homogeneous turbulence, which was valid for times before the final period.Following Deissler's theory Loeffler and Deissler (1961) studied the decay of temperature fluctuations in homogeneous turbulence before the final period.Sarker and Azad (2006), Azad and Sarker (2006), Azad and Sarker (2008), Azad et al. (2006), Azad and Sarker (2009) and Azad et al. (2007Azad et al. ( -2008) ) also studied the decay of temperature fluctuations in homogeneous and MHD dusty fluid turbulence.Azad et al. (2012) studied 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) also studied decay of temperature fluctuations in homogeneous turbulenc before the final period in a rotating system.Bkar Pk et al. (2012) studied first-order reactant in homogeneou dusty fluid turbulence prior to the ultimate phase of decay for four-point correlation in a rotating system.They considered dust particles and Coriolis force on their won works.In their study, they considered twoand three-point correlations and neglecting fourth-and higher-order correlation terms compared to the secondand third-order correlation terms.Sinha (1988) had considered the effect of dust particles on the acceleration of ordinary turbulence.Kishore and Singh (1984) had studied the statistical theory of decay process of homogeneous hydro-magnetic turbulence.Dixit and Upadhyay (1989a) also had deliberated the effect of coriolis force on acceleration covariance in MHD turbulent dusty flow with rotational symmetry.Kishore and Golsefied (1988) considered the effect of coriolis force on acceleration covariance in MHD turbulent flow of a dusty incompressible fluid.Shimomura and Yoshizawa (1986) discussed the statistical analysis of an isotropic turbulent viscosity in a rotating system.
In the present study, by analyzing the above theories we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence prior to the final period in presence of coriolis force considering the correlations between fluctuating quantities at two-and -three point and single time.In this study, Deissler's method is used to solving the problem.Throughout the study we have obtained the energy decay law of temperature fluctuations in homogeneous dusty fluid turbulence prior to the final period due to corilis force.In result, it is shown that the energy decays more rapidly than non rotating clean fluid.

CORRELATION AND SPECTRAL EQUATIONS
For an incompressible fluid with constant properties and for negligible frictional heating, the energy equation may be written at the point P': (1) may be written: where, From the case of homogeneity it follows that = 0 and in addition the usual assumption is made that ˠ is independent of time and that ˯ = 0 Thus Eq. ( 2) simplifies to: where, ˜ = , Prandtl number v = Kinematic Viscosity Equation (3) holds at the arbitrary point P.For the point P' the corresponding equation can be written: Multiplying Eq. ( 3) by T', Eq. ( 4) by T and taking time average and adding the two equations gives: The continuity equation is: Substitution of Eq. ( 6) into (5) yields: By use of a new independent variable: This equation is converted into spectral form by use of the following three dimensional Fourier transforms: And by interchanging P and Pˈ: Substitution of Eq. ( 9)-( 13) into Eq.( 8) leads to the spectral equation: Equation ( 14) is analogous to the two-point spectral equation governing the decay of velocity fluctuations and therefore the quantity may be interpreted as a temperature fluctuation "energy" contribution of thermal eddies of size 1/k.Equation ( 14) expresses the time derivative of this "energy" as a function of the convective transfer to other wave numbers and the "dissipation" due to the action of thermal conductivity.The second term on the left hand side of Eq. ( 14) is the so called transfer to term while the term on the right hand side is "dissipation" term.

THREE POINTS CORRELATION AND SPECTRAL EQUATIONS
In order to obtain single time and three-point correlation and spectral equation we consider three points P, P' and P" with position vectors J and J ′ are considered: For the two points P' and P" we can write a relation according to Eq. ( 7): Equation ( 15) multiplied through by u j the j-th velocity fluctuation component at point P. Then the equation can be written in a rotating system at the point P: The momentum equation at point P in presence of dust particles: ( )  17) into ( 16) the result on taking time averages is: Making use of the relations allows Eq. ( 18) can be written as: Six-dimensional Fourier transforms for quantities this equation may be defined as: Interchanging the points P' and P'' shows that: Using Eq. ( 20)-( 24) into Eq.( 19) then the transformed equation can be written as: If the derivative with respect to x j is taken of the momentum Eq. ( 18) for point P and taking time average the resulting equation is: In terms of the displacement vectors J and J ′ this becomes: Taking the Fourier transform of Eq. ( 27) and then solving for Equation ( 28) can be used to eliminate θ θ α ′ ′ ′ from Eq. ( 25):

SOLUTION FOR TIMES PRIOR TO THE ULTIMATE PERIOD
To obtain the equation for final period of decay the third-order fluctuation terms are neglected compared to the second-order terms.Analogously, it would be anticipated that for times before but sufficiently near to the final period the fourth-order fluctuation terms should be negligible in comparison with the third-order terms.If this assumption is made then Eq. ( 28) shows that the term associated with the pressure fluctuations, should also be neglected.Thus Eq. ( 25) simplifies to: where, ˞ ′ ′′ = ′ ′′ and 1-R = S, R and S are arbitrary constant.
Inner multiplication of Eq. ( 29) by k j and integrating between t 0 and t gives: Now, letting r' = 0 in Eq. ( 20) and comparing the result with the Eq. ( 10) shows that: Substituting of Eq. ( 30) and (31) into Eq.( 14), we obtain: can be expressed in terms of k' and ξ as: (33) Deissler (1958) Substituting Eq. ( 33) into (32) yields: In order to find the solutions completely and following Loeffler and Deissler (1961), we assume that: where, δ 0 constant depending on the initial condition.The negative sign is placed in front of δ 0 in order to make the transfer of energy from small to large wave no. for positive value of δ 0 .Substituting Eq. ( 35) into (34): Multiplying both sides of Eq. ( 36) by k 2 and defining the spectral energy function: And the resulting equation is: where, ( ) Integrating Eq. ( 39) w.r.t.ξ, we have: )  The Eq. ( 38) indicates that w must begin as k 4 for small k.The condition of w is fulfilled by the Eq. ( 41).It can be shown, using Eq. ( 41) that: It was to be expected physically since w is a measure of the transfer of "energy" and the total energy transferred to all wave numbers must be zero.
The necessity for Eq. ( 41) to hold can be shown as follows if Eq. ( 10) is written for both ˫ IJˤ − ˫ and resulting equations differentiated with respect to r i and added, the result is, for: Since according to the Eq. ( 36), ( 38) and ( 14): So the Eq. ( 43) can be written as: as dH = 4πk 2 dk for w = w (k, t) Then the Eq. ( 42) becomes: The linear Eq. ( 38) can be solved for w as: where, J (k) is an arbitrary function of k.
For large times, Corrsin (1951b) has shown the correct form of the expression for E to be: where, N 0 is an constant which depends on the initial conditions.Using Eq. ( 45) to evaluate J (k) in Eq. ( 44) yields:   Putting J = 0 in Eq. ( 9) and we use the definition of E given by the Eq. ( 47), the result is: Substituting Eq. ( 47) into (50) gives: Equation ( 51) is the decay law of temperature fluctuation in homogeneous dusty fluid turbulence prior to the ultimate period in presence of coriolis force.The first term of the right side of Eq. ( 51) corresponds to the temperature energy for two-point correlation and the second terms represents the energy for the three-point correlation.This second term becomes negligible at large times leaving the final period decay law previously found by Corrsin (1951b).ˠ $ Is the total "energy" (the mean square of the temperature fluctuations).

RESULTS AND DISCUSSION
Equation ( 51) is the decay law of temperature fluctuation in homogenous dusty fluid turbulence before the final period in presence of coriolis force.In the absence of the dust particle and coriolis force, i.e., f = 0 then the Eq. ( 51) becomes: which was obtained earlier by Molla et al. (2012).In this study, they had studied the decay of temperature fluctuation in homogeneous turbulenc before the final period in a Rotating System.They considered two-and three-point correlations and neglecting fourth-and higher-order correlation terms compared to the secondand third-order correlation terms and derived the above equation.If which was obtained earlier by Azad and Mumtahinah (2013).
In the absence of the dust particle and the coriolis force i.e., f = 0 and 0 = Ω m , the Eq. ( 51) becomes: Due to the effect of coriolis force in homogeneous dusty fluid turbulence, the temperature energy fluctuations decays more rapidly than the energy for non rotating clean fluid prior to the ultimate period.For large times, the second term in the Eq. ( 51) becomes negligible leaving the -3/2 power decay law for the ultimate period.
In their study, they considered two-and three-point correlations and neglecting fourth-and higher-order correlation terms compared to the second-and thirdorder correlation terms.
In the present study, I have studied the decay of temperature fluctuations in homogeneous turbulence prior to the final period taking dust particle and coriolis force considering the correlations between fluctuating quantities at two-and three-point and single time.In this study, we have used Deissler (1958) method to solving the problem.Through the study we have obtained the Eq. ( 51) for energy decay law of temperature fluctuations in homogeneous dusty fluid turbulence prior to the final period in a rotating system.In this result, it is shown that the energy decays more rapidly than clean fluid and non rotating system.
of a uniform rotation N : Constant number density of dust particle Substituted Eq. ( . (40) w.r.t.k' we have:( function of Prandtl no.