Lagrangian Descriptions

Casey, J. and Naghdi, P.M., On the Relationship between the Eulerian and Lagrangian Descriptions of Finite Rigid Plasticity, Arch. Rational Mech. Math. 102, 351-375 (1988). 

In the two-fluid formulation, the motion or velocity field of each of the two continuous phases is described by its own momentum balances or NS equations (see, e.g., Rietema and Van den Akker, 1983 or Van den Akker, 1986). In both momentum balances, a phase interaction force between the two continuous phases occurs predominantly, of course with opposite sign. Two-fluid models therefore belong to the class of two-way coupling approaches. The continuum formulation of the phase interaction force should reflect the same effects as experienced by the individual particles and discussed above in the context of the Lagrangian description of dispersed two-phase flow. 

A Lagrangian description of the velocity field can be used to find the location X(f) of the fluid element at time 0 < t that started at X(0). In the Lagrangian description, (3.3) implies that the scalar field associated with the fluid element will remain unchanged, i.e., (X(f), 0 = (X(0), 0). 

Bakker, R. A. and H. E. A. van den Akker (1996). A Lagrangian description of micromixing in a stirred tank reactor using 1 D-micromixing models in a CFD flow field. Chemical Engineering Science 51, 2643-2648. 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

For dispersed multiphase flows a Lagrangian description of the dispersed phase are advantageous in many practical situations. In this concept the individual particles are treated as rigid spheres (i.e., neglecting particle deformation and internal flows) being so small that they can be considered as point centers of mass in space. The translational motion of the particle is governed by the Lagrangian form of Newton s second law [103, 148, 120, 38] ... 

An important intermediate level of description of mixing can be given in terms of the trajectories of fluid elements in the flow. This is the so-called Lagrangian description. Various characteristics of the ensemble of trajectories, like absolute and relative dispersion, contain useful information for predicting the evolution of the spatial distribution of quantities of interest. 

Equation (4.13) is a particular case. Ci is a set of concentrations interacting through the reaction terms Ri and vx = (t)x is the, possibly time-dependent, transverse velocity field pointing towards the center of the filament located at x = 0. As in Sect. 2.7.1 this is to be understood as a local Lagrangian description. 

A free surface or a deformable interface between two fluid phases requires that fluid particles move only tangentially along the interface. As such, if the position of the interface E in a Cartesian coordinate system (x,y,z) is geometrically defined by z = h(x,y,t), then, given that the interface itself is a streamline, a Lagrangian description of a fluid particle as it follows the streamline can be geometrically described by the implicit function ... 

We can generalize this procedure If a function

Eulerian description. The choice of the form is arbitrary but will be influenced by any advantage of a problem formulation in either description. For example, in solid mechanics, the Lagrangian description is commonly used, while in fluid mechanics the Eulerian description is popular. This is because in solid mechanics we can attach labels (e.g., visualize strain gauges at various points) on the surface of a solid body, and each material point can be easily traced from the reference state to the current state. On the other hand for a fluid we measure the velocity V or pressure p at the current position jc, therefore the Eulerian description better represents the fluid (note that for a fluid it is difficult to know the exact reference point X corresponding to all the current points jr). 

We distinguish between the coordinate system Ei of the Lagrangian description and the coordinate system of the Eulerian description in order to understand the... 

It is understood that the density function of the second Piola-Kirchhoff stress = TjjEI 0 is the energy-conjugate or dual stress to E under the definition of the increment of internal energy dua = [Pg.87]

Stokes Power Formula in a Continuum Lagrangian Description ... 

From (3.47), the internal energy in Eulerian and Lagrangian descriptions is... 

The external power in the Lagrangian description is given by (3.38). Then the First Law of Thermodynamics in the Lagrangian description is written as... 

As we did for the case of the First Law of Thermodynamics we can obtain the Second Law of Thermodynamics (Clausius-Duhem inequality) in a Lagrangian description. The entropy density in a Lagrangian description can be written as... 

Problems in solid mechanics mostly employ a Lagrangian description. Here we rewrite the thermodynamic functions treated in the previous Subsection in a Lagrangian description. 

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