Eulerian frame of reference

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... 

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. 

The averaged Eulerian-Eulerian multi-fluid model denotes the averaged mass and momentum conservation equations as formulated in an Eulerian frame of reference for both the dispersed and continuous phases describing the time-dependent motion. For multiphase isothermal systems involving laminar flow, the averaged conservation equations for mass and momentum are given by ... 

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. 

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). 

The continuity equations of Sections 2.1 to 2.4 have been derived for a reference frame which is stationary in space. This is known as an Eulerian frame of reference, and in it at the steady state the time derivatives vanish, as in... 

In an Eulerian framework the spatial coordinates form a fixed frame of reference through which the fluid flows. The velocity vector is considered to be a continuous function of time and space, which are independent variables,... 

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. 

Recently, an Eulerian derivation of the Coriolis force has been reported by Kageyama and Hyodo [45]. They present a general procedure to derive the transformed equations in the rotating frame of reference based on the local Galilean transformation and rotational coordinate transformation of field quantities. 

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... 

To describe the flow in a stirred tank the governing equations are often written in cylindrical coordinates, as listed in Sect. 7.7.1. The remaining task is thus to transform these governing Eulerian equations formulated in an inertial Laboratory frame into a relative rotating frame of reference. We outline this procedure by examining the transformation of the momentum equation components (7.91)-(7.93). 

Sampling of a frame of reference in which motion and a condition of the studied medium is presented. For example, Eulerian method when the coordinate are motionless, and moves medium concerning it, or system Lagranzha when the coordinate are connected with moving a corpuscle, and motion is studied concerning this corpuscle. 

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. 

In order to calculate the dipole correlation function, let us consider an ensemble of N identical rigid molecules, each possessing a dipole moment m. To describe the orientation of a molecule in space, two coordinate systems are introduced. The laboratory frame of reference (XYZ), which we will call LF, is traditionally defined as having the Z-axis in the direction of the probing electric field. The molecular frame of reference fixed within the molecule (xyz), which we shall refer to as mF, usually has axes chosen along the principal axes of the moment of inertia tensor (or any other molecular tensor). The orientation of the molecule is then given by the orientation of mF with respect to LF, which is determined by a set of Eulerian angles Q = a, jS, (see Fig. 4.4). The molecular dynam-... 

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... 

A list of typical dependent and independent variables for a furnace simulation is shown in Table VI. Coal particles typically are treated in a procedure that couples the continuum (Eulerian or fixed reference fame) gas-phase grid and the discrete (Lagrangian or moving reference frame) particles. Numerical solutions are repeated until the gas flow field is converged for the computed particle source terms, radiative fluxes, and gaseous reactions. Lagrangian particle trajectories are then calculated. After solving all particle-class trajectories, the new source terms... 

Consider the system and control volume as illustrated in Fig. 2.2. The Eulerian control volume is fixed in an inertial reference frame, described by three independent, orthogonal, coordinates, say z,r, and 9. At some initial time to, the system is defined to contain all the mass in the control volume. A flow field, described by the velocity vector (t, z,r, 9), carries the system mass out of the control volume. As it flows, the shape of the system is distorted from the original shape of the control volume. In the limit of a vanishingly small At, the relationship between the system and the control volume is known as the Reynolds transport theorem. 

Transformation Group of the Dynamical Variables. The transformation groups r(NCf) X), r(3) X and A(3) X all refer to the frame system 1 . By means of the relation between the frame and laboratory system Eq. (2.1) they may be used to define the transformations of the eulerian angles as follows ... 

This Lagrangian should be thought of as dependent on 3Ne + 6 generalized coordinates, qt, and velocities, g, respectively. These are the 3Ng coordinates be, Ce which describe the relative positions of the Ne electrons with respect to the nuclear frame three coordinates Xo, Vo and Zo which describe the position of the molecular center of mass as referred to the laboratory coordinate system, and three Eulerian angles 6, and x which describe the instantaneous orientation of the molecular coordinate system with respect to the space fixed X-, Y- and Z-axes. There are numerous ways of specifying Eulerian angles. Because of later reference we will follow the choice used by Wilson et where and 6 are the ordinary polar coordinates of the molecular c-axis O d n 0 < < 2n) and x is the angle between the nodal line N and the positive b axis as is illustrated in Fig. IV.2. x is positive for clockwise rotation about the c axis. 

In the Eulerian reference frame, the material derivative of F, i.e., DF/Dt = (dFIdt) + u-VE, must be equal to zero, which then leads to the kinematic boundary condition [2] ... 

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