Lagrangian frame

In practice, particle tracking is usually performed in a Lagrangian frame of reference, and the motion of a particle is governed by... 

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. 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

Figure 7.10 Transformed (Lagrangian) frame for the analysis of extruder fluid flow. Here the reference frame is positioned on the bottom of the screw channel. The observer on the frame would see the barrel move with the component velocities of and V, ...

Cross-channel velocity in the transformed (Lagrangian) frame ... 

Down-channei veiocity KizI in the transformed (Lagrangian) frame is provided by Eq. 7.23. This equation provides the veiocity in the z direction due to the rotation of the screw. 

It follows that the dissipation rate should be different for screw rotation In the laboratory frame from the expected dissipation In the Lagrangian frame for screw rotation and the laboratory frame for the barrel rotation for the same pumping rate of a particular screw. 

Evaluating the process aty = 0.5//, one obtains for the Lagrangian frame or barrel rotation the dissipation rate ... 

X component of velocity of the screw flight at the barrel wall z component of velocity of the screw flight at the barrel wall velocity of barrel as observed in the Lagrangian frame X component of velocity of the barrel as observed in the Lagrangian frame z component of velocity of the barrel as observed in the Lagrangian frame velocity component in the x direction... 

Equation A7.13 is the cross-channel flow in the transformed (Lagrangian) frame and concludes the derivation of Eq. 7.18. Equation A7.13 also applies to a physical device where the barrel is actually rotated. Transforming Eq. A7.13 to the laboratory (Eule-rian) reference frame as follows for a physical device where the screw is rotated ... 

Equation (2) is expressed in the Eulerian frame of reference, in which the volume element under consideration is fixed in space, and material is allowed to flow in and out of the element. An equivalent representation of very different appearance is the Lagrangian frame of reference, in which the volume element under consideration moves with the fluid and encapsulates a fixed mass of material so that no flow of mass in or out is permitted. In this frame of reference, Eq. (2) becomes... 

By using the vector identities relating Eulerian and Lagrangian frames together with the equation of continuity, one can convert Eq. (10) to an equivalent form ... 

The conservation of mass equation can be expressed in a frame of reference which moves with the fluid, the Lagrangian frame. 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

We now turn to the prediction of the suspension mechanism of the ballotini versus the speed ratio in the coaxial mixer. The average volume concentration is 1%, and the solids are initially at rest in the tank bottom. The first case investigated corresponds to the motion of the sole anchor arm at a speed of 40 rpm. Simulations are carried out in the Lagrangian frame of reference (fixed anchor, rotating vessel). Fig. 12 shows the predicted and experimental solid volume fraction at equilibrium. The computation of the solid-liquid interface at the bottom is fairly well... 

We have now derived the four basic (time-independent) equations of stellar structure. These are mass continuity (Eq. (14)), hydrostatic equilibrium (Eq. (17)), conservation of energy (Eq. (28)), and energy transport (Eq. (33)). These form a set of coupled first order ordinary differential equations relating one independent variable, e.g. r, to four dependent variables i.e., m, /, / //, which uniquely describe the structure of the star, note that any variable could be used as the independent variable. In an Eulerian frame, the spatial coordinate r is the independent variable. For most problems in stellar structure and evolution it is usually more convenient to work in a Lagrangian frame, with mass as the independent variable. Transforming, we obtain ... 

To solve such a system, boundary conditions are required. In the Lagrangian frame, the boundaries are at m = 0 (centre) and m = M (surface). In the centre, the enclosed mass and luminosity are defined ... 

The deterministic approach of direct numerical simulation (DNS) and the probabilistic approach of probability density function (PDF) modeling are implemented for prediction of droplet dispersion and polydis-persity in liquid-fuel combustors. For DNS, a multidomain spectral element method was used for the carrier phase while tracking the droplets individually in a Lagrangian frame. The geometry considered here is a backward-facing step flow with and without a countercurrent shear. In PDF modeling, the extension of previous work to the case of evaporating droplets is discussed. 

Kinematics of Mixing Spencer and Wiley [1957] have found that the deformation of an interface, subject to large unidirectional shear, is proportional to the imposed shear, and that the proportionality factor depends on the orientation of the surface prior to deformation. Erwin [1978] developed an expression, which described the stretch of area under deformation. The stretch ratio (i.e., deformed area to initial area) is a function of the principal values of the strain tensor and the orientation of the fluid. Deformation of a plane in a fluid is a transient phenomenon. So, the Eulerian frame of deformation that is traditionally used in fluid mechanical analysis is not suitable for the general analysis of deformation of a plane, and a local Lagrangian frame is more convenient [Chella, 1994]. 

Lagrangian Frame I. Equations of Motion and Conservation Laws. 

Lagrangian Frame II. Geometric Formulation of Time-Dependent Density Functional Theory. 

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