Lagrangian frame of reference

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... 

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... 

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... 

Analysis of the motion of the gas bubbles can be conducted either in the Eulerian or Lagrangian frame of reference. In the Eulerian frame of reference, the problem is formulated in terms of partial differential equations which describe the balances of mass and momentum, while in the latter approach, the trajectories of individual bubbles are tracked by solving ordinary differential equations in time. The Lagrangian method has distinct advantages over the Eulerian method in terms of simplicity of formulation, ability to accommodate complicated exchange process, computer memory requirements, and computational efforts. 

The particle motion is calculated in a Lagrangian frame of reference, which considers a discrete resolved particle travelling in a continuous fluid medium. The changes of the position and angular displacement of the particle, xp and p, as well as the translational and angular components of the particle velocity, up and cop, are calculated by solving a set of ordinary differential equations (ODEs) along the particle trajectory ... 

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

The approach in this section is lagrangian i.e., the model is for a drying object (particle, drop, sheet, etc.) as it moves through the drying process in time. More complicated models can use a eulerian frame of reference by simulating the dryer with material moving into and out of the dryer. 

To overcome some of the limitations just mentioned that are associated with purely empirical models, simulations that include various aspects of the inhaled aerosol dynamics have been developed. The simplest of these belong to a class of models we refer to as Lagrangian dynamical models (LDMs), meaning that the model simulates some of the dynamical behavior of the aerosol in a frame of reference that travels with the aerosol (i.e., a Lagrangian viewpoint ). 

For mathematical convenience, boundary conditions and initial conditions must be prescribed. For the simple marine propeller problem, a Lagrangian viewpoint was adopted. The frame of reference was attached to the propeller so that the propeller was fixed but the vessel was rotating. The boundary condition was then a zero velocity on the impeller, while the vessel wall rotated at -Qimpdier- The free surface was considered to be fiat, therefore the normal velocity was zero and a shear-free condition was assumed. It should be noted that in the Lagrangian viewpoint, the frame of reference is in rotation. The fluid is therefore subjected to a constant acceleration and the momentum conservation equation [Eq. (6)] must be modified to account for centrifugal forces and Coriolis forces.An advantage is, however, that the flow can be solved numerically at steady state provided the flow is fully periodic, which limits the computational efforts significantly. 

In the case of the coaxial mixer, the rotation kinematics is much more complex since the two sets of agitators counter-rotate at different speeds. For the sake of simplicity, we decided to simulate the flow using the frame of reference of the anchor. In this Lagrangian viewpoint, the anchor is fixed but the vessel wall rotates at —Qanchor and the turbine rotates at anchor + turbine- such a situation. Contrary to the simple propeller problem, the resolution of the flow equations is time-dependent as the position of the central agitator changes with time. 

The adjective Lagrangian is used to indicate that the correlation relates to moving fluid particles (e.g., [167], p. 46 [113], p. 539). The adjective Eulerian is used whenever correlations between two fixed points in a fixed frame of reference are considered. 

This is messy problem analytically. It is fairly easy if you take the viewpoint of someone riding on the jump (the lagrangian viewpoint) and solve by trial and error for the jump velocity that satisfies the hydraulic jump equation in the moving frame of reference. 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

D Alembert s principle in the Lagrangian version has been obtained in Section 3.4.5 in terms of virtual displacements and actual accelerations. Since it needs to be accounted for a superimposed guided motion, the position p x,s,t) in the inertial frame of reference, as described by Eq. (7.65), has to be taken into consideration. With the density p s, n) in accordance with Remark 7.1, the virtual work of inertia forces originating from Eq. (3.59) then reads... 

Eulerian [82] and the Lagrangian methods [83-85]. The Eulerian approach uses a coordinate system fixed in the frame of reference of the laboratory, and it takes account of the velocity of the body relative to that frame as the volume of the body changes. The Lagrangian method uses a coordinate system fixed in the gel, such that a fixed volume of solid phase is contained in any volume element. The solution is obtained in terms of the material coordinate, m, where... 

In a Eulerian frame of reference time derivatives are stated in a stationary frame. Therefore, in order to write the time derivative of a property for a flnid element, we have to include both the local derivative (the rate of change of at the stationary point) and the convective derivative (the rate of change of in the direction in which the fluid element is moving). The Eulerian frame is in contrast to a Lagrangian frame where we state time derivatives following a fluid element (or a particle). 

Generally speaking, Eulerian indicates that the frame of reference for the description of the flow field is stationary, while Lagrangian indicates that the frame of reference is a material particle, i.e. following the flow. 

The conservation equations are more commonly written in the initial reference frame (Lagrangian forms). The time derivative normally used is d /dt. Equation (9.5) is used to derive (9.2) from the Lagrangian form of the conservation of mass... 

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, ...

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 ... 

Of course, the static condition refers to a certain distinguished reference frame (due to this static field of nuclei at rest there is no Lorentz invariance and hence also no natural preference any more of a Lagrangian formalism against a Hamiltonian one). We further assume spatial periodicity in a large periodic spatial volume V with respect to that reference frame, and refer all integrated quantities to that volume V (toroidal three-space). 

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