LaGrangian flow

These formulas enable us to translate the Lagrangian flow equations into Eulerian language. Differentiating Equation 4.21 with respect to t and using Equation 4.22, we easily deduce the continuity equation... 

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... 

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... 

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

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). 

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 introduction of Lagrangian coordinates in the previous section allows a more natural treatment of a continuous flow in one dimension. The derivation of the jump conditions in Section 2.2 made use of a mathematical discontinuity as a simplifying assumption. While this simplification is very useful for many applications, shock waves in reality are not idealized mathematical... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

Equations (2.45), (2.46), and (2.49) express the conservation of mass, momentum, and energy in Lagrangian coordinates for continuous flow. 

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. 

When we translate these observations into Lagrangian wave speed, the data would look like that shown in the lower diagram of Fig. 7.11. The points e and q represent volume strains at whieh elastie-perfeetly-plastie release (e) and quasi-elastie release (q) would undergo transition to large-seale, reverse plastie flow (reverse yield point). The question is the following What is responsible for quasi-elastie release from the shoeked state, and what do release-wave data tell us about the mieromeehanieal response in the shoeked state ... 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

This flow field can be maintained in a steady state, at least in the Eulerian sense, either by use of a four-roll mill [18] as in Figure 2.8.4(a) or by means of opposed jet flow as in Figure 2.8.4(b). However, it is important to note that the flow is still transient in the Lagrangian sense. That is, pure planar extension is confined to the central stagnation... 

Chaiken, J., Chevray, R., Tabor, M., and Tan, Q. M., Experimental study of Lagrangian turbulence in Stokes flow, Proc. Roy. Soc. Lond. A408, 165-174 (1986). 

Binder, J. L., and T. J. Hanratty, 1992, Use of Lagrangian Methods to Describe Drop Deposition and Distribution in Horizontal Gas-Liquid Annular Flows, Int. J. Multiphase Flow 7S(6) 803 821. (3)... 

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