Eulerian coordinates

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. 

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. 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

Consider the impact of a semi-infinite space on a plate of thickness dp, separated from an identical plate by a gap of width d. If the impactor and plates are all composed of the same materials, what is the subsequent behavior Plot in both Lagrangian and Eulerian coordinates. 

Eulerian coordinates The coordinate system in which the spatial position (x) and time (t) are the independent variables. The dependent variables... 

In Eulerian coordinates x and t, the mass and momentum conservation laws and material constitutive equation are given by (u = = particle velocity,, = longitudinal stress, and p = material density)... 

Sunderland and Grosh (Sll) use an explicit numerical scheme for solving the Landau problem. An Eulerian coordinate system is used with the origin at the melt interface. As noted by Landau, the numerical integration is simplified by appropriate choice of the ratio of the space and time intervals. Extension to time-dependent heat flux by either a numerical or a graphical technique is indicated. 

Three-dimensional, time-dependent methods (25, 26) have been recently proposed, but results for reactive atmospheres have not been reported at this writing. Simplified chemistry must be used in each of these approaches because of the emphasis on details of advection and diffusion. The body of data for most aff basins falls short of the input requirements for any transport formulation of this complexity. In some cases it may be difficult to avoid the problem of allowing too many unspecified parameters to obscure the physically based portions of the calculation. One new method uses an Eulerian coordinate frame (La-grangian coordinates refer to a fiuid mass which is followed in time and space in contrast with an Eulerian frame which has fluid moving relative... 

Now we see that the slope of the jump condition on the P-u plane is also a function of shock velocity. The interesting part to note is that U in this equation is the shock velocity in laboratory or Eulerian coordinates. The quantity (U -Uq) is the shock velocity in Lagrangian coordinates, or relative to the material. So we see that for the jump condition on the P-u plane... 

Ansumali et al.196 propose that the grand potential, written in the Eulerian coordinate system as... 

To complete the computation of the concentration field of the puff in Eulerian coordinates, the position of the puff centroid must be updated based on the wind field velocity at the puff centroid the entire (Lagrangian) puff is then assumed to translate affinely with the centroid. A puff-splitting algorithm may be used to overcome the inaccuracies that arise as the puff dimensions become sufficiently large that the approximation inherent in assuming constant wind velocities throughout the puff becomes invalid (Sykes and Henn 1995). 

Eulerian coordinate system A coordinate system that is fixed in space. 

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. 

In Eulerian coordinates, shrinking causes an advective mass flux, which is difficult to handle. By changing the coordinate system to Lagrangian, i.e., the one connected with dry mass basis, it is possible to eliminate this flux. This is the principle of a method proposed by Kechaou and Roques (1990). In Lagrangian coordinates Equation 3.91 for one-dimensional shrinkage of an infinite plate becomes ... 

If the gradient is used with respect to Eulerian coordinates with the basis e,, it is denoted as (2.14). If we explicitly explain the gradient with respect to the Eulerian system, it is denoted as... 

Spatial (Eulerian) Coordinate in X Direction Material (Lagrangian) Coordinate in X Direction Yield Strength... 

For the description of granular flow, a formulation in EuLERian coordinates finme is used. There, the coordinate frame is not moving, and hence the relative velocity c becomes equal to the material velocity v. Then, the following differential equations hold for the CAUCHY stress... 

---