Eulerian

For diffraction studies with monocliromatic radiation, the crystal is connnonly mounted on an Eulerian cradle, which can rotate the crystal so that the nonnal to any set of planes bisects the angle between the incident and reflected beams, which is set for reflection from planes with a particular value of the interplanar spacing, d. 

Each of the Hamiltonians Hi is easily integrated in terms of rotations about one or the other of the Eulerian axes. 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

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... 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

Governing flow equations, originally written in an Eulerian framework, should hence be modified to take into account the movement of the mesh. The time derivative of a variable / in a moving framework is found as... 

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. 

As described in Chapter 3, Section 5.1 the application of the VOF scheme in an Eulerian framework depends on the solution of the continuity equation for the free boundary (Equation (3.69)) with the model equations. The developed algorithm for the solution of the described model equations and updating of the free surface boundaries is as follows ... 

Figure 5.4 The finite element mesh configurations in the Arbitrary Lagrangian-Eulerian scheme...

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

Trajectory models require spatiaUy and temporaUy resolved wind fields, mixing-height fields, deposition parameters, and data on the spatial distribution of emissions. Lagrangian trajectory models assume that vertical wind shear and horizontal diffusion are negligible. Other limitations of trajectory and Eulerian models have been discussed (30). 

Eig. 4. Schematic diagram of a Lagrangian trajectory model where Lf(/) represents the air column height in both Eulerian and Lagrangian (, Tj, ... 

Euleria.n Models. Of the Eulerian models, the box model is the easiest to conceptualize. The atmosphere over the modeling region is envisioned as a well-mixed box, and the evolution of pollutants in the box is calculated following conservation-of-mass principles including emissions, deposition, chemical reactions, and atmospheric mixing. 

While the Eulerian system has intuitive appeal, it is the Lagrangian coordinate system that is more convenient mathematically and in many practical applications. In this system, the coordinate is fixed to the material and moves with it. It is sometimes called the material coordinate system. In Fig. 2.2, the boxcars can be numbered, so the position of a car in this system never changes. By convention, the Lagrangian coordinate (h) is chosen so that it is equal to the Eulerian coordinate (x) at some time t = 0. Figure 2.10(b) illustrates a Lagrangian h-t diagram of the same system as shown in Fig. 2.10(a) with the Eulerian system. Because the flow is independent of the coordinate system chosen to describe it, both systems must lead to the same results. 

Figure 2.10. (a) An Eulerian x-t diagram of a shock wave propagating into a material in motion. The fluid particle travels a distance ut, and the shock travels a distance Uti in time ti. (b) A Lagrangian h-t diagram of the same sequence. The shock travels a distance Cti in this system. 

The relative shock velocity t/ = (7 — Uj is the Eulerian shock velocity often used because it is a material property and is independent of the motion of the... 

Figure 2.11. The sequence described in the text for (a) Eulerian and (b) Lagrangian coordinates. Hashmarks indicate material particle positions.

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition... 

The term shock velocity usually refers to this Eulerian shock velocity relative to the particle velocity of the unshocked material unless otherwise stated, and is usually written as (/ (without the prime). 

Since the Lagrangian walls are impermeable, the mass of the Lagrangian element is constant. At time t, when the walls of the element are separated by an Eulerian distance dx, the density of the fluid within it must be... 

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