Euler. Eulerian

On the analogy of simulating the process of adding blobs of a miscible liquid, two-phase flow in stirred tanks in a RANS context may be treated in two ways Euler-Lagrange or Euler-Euler, with the second, dispersed phase treated according to a Lagrangian approach and from a Eulerian point of view, respectively. 

As a matter of fact, in comparison with the Euler-Lagrangian approach, the complete Eulerian (or Euler-Euler) approach may better comply with denser two-phase flows, i.e., with higher volume fractions of the dispersed phase, when tracking individual particles is no longer doable in view of the computational times involved and the computer memory required, and when the physical interactions become too dominating to be ignored. Under these circumstances, the motion of individual particles may be overlooked and it is wiser to opt for a more superficial strategy that, however, still has to take the proper physics into account. 

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Four-circle with Euler geometry. The sample is placed at the center of a Eulerian cradle and the detector rotates in the equatorial plane. Data can be recorded from full spheres in the reciprocal lattice except if blind regions are introduced on the Euler angles (o>, x, <1>) by, for example, the sample environment assembly. 

Attempts to extend RANS formulation to LES of two-phase combustion may be found in [318 354 317 255 292]. They are all based on a Euler-Lagrange (EL) description of the dispersed phase in which the flow is solved using an Eulerian method and the particles are tracked with a Lagrangian approach. An alternative is the Euler-Euler (EE) description, also called two-fluid approach, in which both the gas and the dispersed phases are... 

Note that according to Truesdell and Toupin [170] the material coordinates were introduced by Euler in 1762, although they are now widely referred to as the Lagrangian coordinates, while the spatial coordinates, often called Eulerian coordinates, where introduced by Jean le Rond d Alembert (1717-1783) in 1752. 

The main advantage of the Eulerian-Lagrangian approach (i.e., compared to the alternative Euler-Euler model described in the next subsection) is its flexibility with respect to the incorporation of the microscopic transport phenomena. Particle dynamics can in principle be described in detail, a particle size distribution can easily be incorporated, direct particle-particle interactions can be accounted for as well as the hydrodynamic interaction between neighboring particles. 

Eulerian fluid dynamics A formulation of fluid mechanics In which the velocity of the fluid at fixed postions Is analysed, i.e. a formulation that describes changes at fixed positions in the fluid, in contrast to La-gran an fluid dynamics. It is named after the Swiss mathematician and physicist Leonhard Euler (1707-83). 

The GYS has an asymmetric matrix corresponding to the Euler equations. Its structure can be represented by three one-dimensional MGYs [5, 6] called an EJS (Eulerian junction structure), as Eig. 9.10 shows. 

As described in Sect. 3.2.2 the procedure becomes complicated for the case of a generalized Eulerian strain measure e (n). Only an isotropic material body can form the rate of internal energy dua e, s) together with the rate c " of the generalized Eulerian strain measure and its energy-conjugatecorotational Euler... 

Inside each fluid, p and p are constants. Equations (95) and (96) were solved by using a finite difference method on a fixed two- or three-dimensional grid. The spatial terms were discretized by second-order finite differences on a staggered Eulerian grid. The discretization of time was achieved by an expKdt Euler method or a second order Adams Bashforth method. The boundary conditions used in their study were either periodic or full sKp in the horizontal directions and rigid, stress-free on the top and bottom. 

We concluded this study with an examination of Newton s third law, and to do so we consider the two-particle system illustrated in Fig. 1-3. We begin by appl3dng Euler s first law to the isolated two-particle system identified by Eulerian Cut I. Since there is no external force acting on the two-particle system, we can use Euler s first law to obtain the relation... 

At this point, we have shown that Euler s first law contains all the results available in Newton s three laws. In addition, we have found that Euler s second law provides no new information concerning the processes Illustrated in Figs. 1-1 and 1-2. It might seem that Euler s second law is devoid of content for mass-point mechanics however, application of Eq. 1-3 to the Eulerian Cut I illustrated in Fig. 1-3 will indeed provide some new information. Since there are no external torques acting on the two-particle system, Euler s second law yields... 

In the first coarse-grained approach, the discrete phase is treated as an Eulerian continuum, interpenetrating with the real continuous phase. The particle-particle interactions are then captured by an effective particle phase rheology obtained from kinetic theory of granular flows. These so-called two-fluid (Euler-Euler) models have been very successful at predicting the dynamic properties of, e.g., gas-solid fluidized beds (see Van derHoefet al, 2008 Verma et al, 2013). Despite their success, two-fluid models also have their limitations they are usually limited to idealized cases ofmonodisperse hard sphere particles, while extensions to polydisperse mixtures (e.g., in size or in contact properties) are difficult to make. Also, because no particles are explicitly tracked, it is difficult to include particle properties which may vary from particle to particle, such as particle temperature, surface moisture concentration, or chemical surface species concentrations. 

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