Lagrangian trace

Figure 15.10 Lagrangian trace of the tracer particle during one stay in a riser (From [14]).

After breakup, droplets continue to interact with the surrounding environment before reaching thein final destination. In theory (24), each droplet group produced during primary breakup can be traced by using a Lagrangian calculation procedure. Droplet size and velocity can be deterrnined as a function of spatial locations. 

The origin of this problem traces back to the fact that, because the Lagrangian density is degree zero in the temporal ordering parameter, it is then invariant with respect to any transformation of this parameter that preserves the ordering. 

The Lagrangian particle equations are solved for a representative number of particle trajectories. Typically, about 600,000 trajectories have been traced, with nearly 1000 starting locations. Particle size of SO fm radius have been chosen and injected at each location. The (PSIC) procedure, briefly described previously, is repeated until convergence is achieved on the particle source terms. 

The Lagrangian density whose vanishing variation leads to the field equations in qk is chosen to be the trace of the scalar curvature ... 

The techniques for describing the statistical properties of the concentrations of marked particles, such as trace gases, in a turbulent fluid can be divided into two categories Eulerian and Lagrangian. The Eulerian methods attempt to formulate the concentration statistics in terms of the statistical properties of the Eulerian fluid velocities, that is, the velocities measured at fixed points in the fluid. A formulation of this type is very useful not only because the Eulerian statistics are readily measurable (as determined from continuous-time recordings of the wind velocities by a fixed network of instruments) but also because the mathematical expressions are directly applicable to situations in which chemical reactions are taking place. Unfortunately, the Eulerian approaches lead to a serious mathematical obstacle known as the closure problem, for which no generally valid solution has yet been found. 

The non-chiral model contains the scalar-isoscalar field s and its strange counterpart z, the vector-isoscalar fields ty and 0, and the p-meson p as well as the photon Af fields. For more details see [36]. In contrast to the non-chiral model, the SU 3)l X SU 3)r Lagrangian contains the dilaton field x introduced to mimic the trace anomaly of QCD in an effective Lagrangian at tree level (for an explanation of the chiral model see [36,13]). 

Atmospheric mesoscale models are based on a set of conservation equations for velocity, heat, density, water, and other trace atmospheric gases and aerosols. The equation of state used in these equations is the ideal law. The conservation-of-velocity equation is derived from Newton s second law of motion (F = ma) as applied to the rotating earth. The conservation-of-heat equation is derived from the first law of thermodynamics. The remaining conservation equations are written as a change in an atmospheric variable (e.g., water) in a Lagrangian framework where sources and sinks are identified. 

The results of these experimental measurements of the liner trajectory were compared with a numerical model for the implosion obtained from a 1-D, Lagrangian, hydrodynamic code containing a detailed treatment of the equation-of-state of the liner material. Figure 17 shows such a comparison for a 1.4 MJ shot. The point labeled as "abrupt current change" marks the discontinuity in the slope of the current trace caused by the sudden ending of the IL contribution to the voltage when the liner hits the axis. The... 

We can generalize this procedure If a function

Lagrangian description, and if

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). 

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