Eulerian formulae

The case when Aw = w j-w is close to zero, is of special interest. This is when the irradiation frequency w is close to the energy difference between states i and n. Then the second term in (4.12) can be neglected and, using the Eulerian formulae, the first term becomes... 

The simplicity of these transformation formulae is to be traced back to the general formulae (3.33), (3.34) and the fact that the operators PH act on both the eulerian angles and the internal coordinates simultaneously, as expressed by the representation F. The analogy of the Eqs. (3.37) to the representation r(NC1) Sffs should be noted. 

Extension of these formulae for volume averaging in three-dimensional Eulerian-Lagrangian simulations is straightforward. 

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as... 

We derive the conservation law of mechanical energy, referred to as Stokes power formula, starting with the following equation of motion in an Eulerian framework ... 

Applying the divergence theorem to the third term of the r.h.s. and using Stokes power formula (3.18), we obtain the following local Eulerian form of the dissipative energy equation ... 

The corresponding formula for the Eulerian velocity is contained in the expression for the current... 

These formulas enable us to translate the Lagrangian flow equations into Eulerian language. Differentiating Equation 4.21 with respect to t and using Equation 4.22, we easily deduce the continuity equation... 

To check that this transformation is a synunetry of the trajectory law (Equation 4.11), we only need to confirm consistency between the transformation rules of the Lagrangian velocity and density implied by the first two members of Equation 4.39 (with Equation 4.83 substituted) and the Eulerian rules (Equation 4.79). This follows straightforwardly using the following formula for conversion to the Lagrangian picture, which holds for any Eulerian function/(x, t. A) ... 

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