Lagrangian framework

As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow... 

The geometrical flexibility of the VOF scheme can be significantly improved if in its formulation, instead of using a fixed framework, a combination of a Lagrangian-Eulerian approach is adopted. The most common approach to develop such a combined framework is the application of the Arbitrary... 

FINITE ELEMENT MODELLENG OF POLYMERIC FLOW PROCESSES 3.5.3 VOF method in Lagrangian frameworks... 

Therefore, in the Lagrangian framework the population balanee ean be written as... 

Abstract. The —> um° decay is studied using the method of phenomenological chiral Lagrangians. Obtained in the framework of this method the expression of weak hadronic currents between vector and pseudoscalar mesons has been checked and it is shown that this decay channel proceeds only due to the — p - mixing diagram. 

Note, that we studied (Nasriddinov, 1998) the t —> ir riuT decay of the r lepton in the framework of this method with taking into account the isotopic spin violation of chiral symmetry in the Oakes scheme.In this case the 7r° — rj - mixing Lagrangian has the form... 

For canonical turbulent flows (Pope 2000), the flow parameters required to complete the CRE models are readily available. However, for the complex flow fields present in most chemical reactors, the flow parameters must be found either empirically or by solving a CFD turbulence model. If the latter course is taken, the next logical step would be to attempt to reformulate the CRE model in terms of a set of transport equations that can be added to the CFD model. The principal complication encountered when following this path is the fact that the CRE models are expressed in a Lagrangian framework, whilst the CFD models are expressed in an Eulerian framework. One of the main goals of this book... 

The PFR model ignores mixing between fluid elements at different axial locations. It can thus be rewritten in a Lagrangian framework by substituting a = Tpfrz, where a denotes the elapsed time (or age) that the fluid element has spent in the reactor. At the end of the PFR, all fluid elements have the same age, i.e., a = rpfr. Moreover, at every point in the PFR, the species concentrations are uniquely determined by the age of the fluid particles at that point through the solution to (1.2). 

In the CRE literature, the residence time distribution (RTD) has been shown to be a powerful tool for handling isothermal first-order reactions in arbitrary reactor geometries. (See Nauman and Buffham (1983) for a detailed introduction to RTD theory.) The basic ideas behind RTD theory can be most easily understood in a Lagrangian framework. The residence time of a fluid element is defined to be its age a as it leaves the reactor. Thus, in a PFR, the RTD function E(a) has the simple form of a delta function ... 

Figure 24.2 Calculation of the time-dependent concentration along the river axis x of a reactive compound. The effect of the reaction is calculated in the Lagrangian framework the effect of advection is accounted for by the relation between flow time t and distance x, x(t).

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

Parison inflation models use a Lagrangian framework with most of them employing the thin-shell formulation and various solidlike or liquid constitutive equations, generally assuming no-slip upon the parison contacting the mold. The first attempts to simulate polymeric parison inflation were made by Denson (83), who analyzed the implications of elongational flow in various fabrication methods, as discussed in the following example. 

Despite the simple form of Equation (1.83), the detailed formulation of an extended Lagrangian for CPCM is not a straightforward matter and its implementation remains challenging from the technical point of view. Nevertheless, is has been attempted with some success by Senn and co-workers [31] for the COSMO-ASC model in the framework of the Car-Parrinello ab initio MD method. They were able to ensure the continuity of the cavity discretization with respect to the atomic positions, but they stopped short of providing a truly continuous description of the polarization surface charge as suggested,... 

The strategy to obtain a Lagrangian formulation of PCM is to consider the PCM apparent charges as a set of dynamic variables, exactly as the solute nuclear coordinates. The algorithm proposed in the present chapter is applied within the MM framework, since it allows a simplified notation and faster calculations. However, we point out that it can be straightforwardly extended to QM calculations. 

In order to overcome the limitations of currently available empirical force field param-eterizations, we performed Car-Parrinello (CP) Molecular Dynamic simulations [36]. In the framework of DFT, the Car-Parrinello method is well recognized as a powerful tool to investigate the dynamical behaviour of chemical systems. This method is based on an extended Lagrangian MD scheme, where the potential energy surface is evaluated at the DFT level and both the electronic and nuclear degrees of freedom are propagated as dynamical variables. Moreover, the implementation of such MD scheme with localized basis sets for expanding the electronic wavefunctions has provided the chance to perform effective and reliable simulations of liquid systems with more accurate hybrid density functionals and nonperiodic boundary conditions [37]. Here we present the results of the CPMD/QM/PCM approach for the three nitroxide derivatives sketched above details on computational parameters can be found in specific papers [13]. 

A. Lagrangian Framework. An ideal subgrid model should be constructed on a Lagrangian hydrodynamics framework moving with the macroscopic flow. This requirement reduces purely numerical diffusion to zero so that realistic turbulence and molecular mixing phenomena will not be masked by non-physical numerical smoothing. This requirement also removes the possibility of masking purely local fluctuations by truncation errors from the numerical representation of macroscopic convective derivatives. 

The time-dependent (hyperbolic) Lagrangian framework should also generalize to three dimensions as well as resolve reactive interfaces dynamically. 

The study of mixing effects on chemical reactions has been an active area of research since the pioneering papers of Danckwerts (1958) and Zwietering (1959). The topic has become a part of classical Chemical Reaction Engineering and has been discussed in textbooks (Froment and Bischoff, 1990 Levenspiel, 1999 Westerterp et al., 1984) and review articles (Villermaux, 1991). Historically, this study has progressed in two parallel branches, based on the Lagrangian and Eulerian frameworks of description, respectively. 

Sub-models for the particle motion, heat transfer, and combustion are used in the framework of a Lagrangian approach. During its flight the particle interacts with the gas phase flow field producing source terms which are implemented using the particle source in cell method (PSIC), in the momentum, enthalpy and mass equations [10, 11]. The particle-particle interactions are neglected since the particle phase is dilute in the furnace studied here. 

Lagrangian incorporating the equations of motion and varying the quasienergy is used to allow variational calculations to be carried out on the interacting system within the coupled-cluster framework (which, when applied straightforwardly, is non-variational).56 58... 

Other interesting questions concern the modification of the front geometry as a consequence of advection. In particular, one may ask if the presence of Lagrangian chaos has a role in the front dynamics. We shall briefly discuss this problem in the framework of the geometrical optics limit. 

In this framework an interesting example is the Lagrangian motion in velocity field given by a simple model for Rayleigh-Benard convection [31], which is given by the stream function ... 

The main advantage of using a Lagrangian framework for dispersed phase particles is that particle-level phenomena can be modeled rigorously. Species and enthalpy conservation equations for individual particles can be written ... 

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