Second-order fluctuation equation

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

Equations (12.30) and (12.14) show that the contributions to the entropy change due to fluctuations in the equilibrium state are of second order, expressed by i rS/2... 

Other Contributions to the Kerr Constant. In the case of naturally anisotropic molecules, the deviation tensor df in the expansion (195) does not vanish, causing a cross-effect as well as higher-order fluctuational effects to appear, in addition to the previous angular correlation effect (I77a). The cross-effect appears owing to co-operation between molecular angular correlations-the first term of equation (195) and the translational fluctuations described by the second term of (195). We shall consider this effect in the next subsection. 

The next category of turbulence closures, i.e., impl3ung to be more accurate than the very simple algebraic models, is a hierarchy of turbulent models based on the transport equation for the fluctuating momentum field. These are the first-order closure models, i.e., those that require parameterizations for the second moments and the second-order closure models, i.e., those that... 

Re3Tiolds decomposition and time averaging were then applied to the instantaneous variables in the volume average model equations. However, it was assumed that none of the densities fluctuate. The terms of fluctuating quantities with order higher than two were considered small compared to those of first and second order and thus neglected. 

The last four terms depend only on the reference density, po, and represent the repulsive energy contribution, Flcp, discussed above. Thus, we just have to deal with the second-order terms. The second-order term in the charge density fluctuations dp(r), that is, the second term in Equation 5.51, is approximated by writing Ap as a superposition of atomic contributions, Ap0 = Apv. This approach decays quickly with the increasing distance from the corresponding center. To simplify the second term further, Elstner applied a monopole approximation ... 

The second-order correlation of the fluctuations a b is not known and does not appear in the Navier-Stokes equations. Additional equations need to be provided, therefore giving rise to the closure problem. The closures are provided for an area called turbulence modeling for RANS (Reynolds-averaged Navier-Stokes) and LES (large eddy simulation) methodologies. 

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