Lagrangian form

When (7.10)-(7.12) are combined with the expressions for mass and momentum conservation, we are then able to compare assumptions regarding and v with macroscale observations such as wave profiles, for example. The conservation laws are (in Lagrangian form Pq dX = p dx )... 

Note that the spatial velocity (u) is arbitrary and may be the material velocity (u). If the spatial velocity is the material velocity (u = v), then the region of space moves with the material and the Lagrangian forms of the equations are generated. If the spatial velocity is zero, then the region of space is fixed and the equations take the Eulerian form. 

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... 

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as... 

There is not an analytical velocity function for the y-direction velocity at the flights, so the wide channel approximation is used for demonstration purposes with a pressure gradient of zero. Using the equation developed previously for screw rotation for a very wide shallow channel, the transformed Lagrangian form of is the same as the laboratory form for barrel rotation and is as follows ... 

This transformation will exist if there is a linear transformation expressing the Q s in terms of the q s, and equations (8) and (9) will result by suitably choosing the coefficients chi. By introducing equations (8) and (9) into the Lagrangian form of the equations of motion, viz.,... 

Equation (3.10) goes over into the lagrangian form... 

The field equations given in Lagrangian form are (a) the conservation of momentum (quasi-static, no body forces) and mass equations... 

In the third formulation, the so-called Hamiltonian formulation, the velocities Legendre transformation. The generalized momentum pi, conjugate to the coordinate qi, is defined as... 

By appropriate manipulation as before, this can be written in Lagrangian form as... 

The basis for any derivation of the momentum equation is the relation commonly known as Newton s second law of motion which in the material Lagrangian form (see Fig. I.IB) expresses a proportionality between the applied forces and the resulting acceleration of a fluid particle with momentum density, P, (e.g., [89]) ... 

For dispersed multiphase flows a Lagrangian description of the dispersed phase are advantageous in many practical situations. In this concept the individual particles are treated as rigid spheres (i.e., neglecting particle deformation and internal flows) being so small that they can be considered as point centers of mass in space. The translational motion of the particle is governed by the Lagrangian form of Newton s second law [103, 148, 120, 38] ... 

Following a single particle, the rates of change of the mesoscale variables can be written in a Lagrangian form ... 

Since the above derivation could be carried out for any value of j, there are 3n such equations, one for each coordinate <7,. They are called the equations of motion in the Lagrangian form and are of great importance. The method by which they were derived shows that they are independent of the coordinate system. 

Using the coordinates g<, we now set up the classical equations of motion in the Lagrangian form (Sec. lc). In this case the kinetic energy T is a function of the velocities g< only, and the potential energy V is a function of the coordinates q> only, and in consequence the Lagrangian equations have the form... 

In DNS of single-phase flows, a complete set of compressible Navier-Stokes, energy, and scalar transport equations are calculated together with the equation of state and some constitutive relations [3]. In DNS of particle-laden flows, in addition to carrier-gas equations, the Lagrangian form of particle (droplet) equations are solved via standard difference schemes [5]. 

The parameter /i, which has the dimensions of mass, determines the time scale of the fluctuations in box shape and size described by B. Particle momenta are given by mixi = m,Bs,. The equations of motion in Lagrangian form are... 

In the Lagrangian approach, the elemental control volume is considered to be moving with the fluid as a whole. In the Eulerian approach, in contrast, the control volume is assumed fixed in the space, the fluid is assumed to flow through and pass the control volume. The particle-phase equations are formulated in Lagrangian form, and the coupling between the two phases is introduced through particle sources in the Eulerian gas-phase equations. The standard k-e turbulence model, finite rate chemistry, and DTRM (discrete transfer radiation model) radiation model are used. 

Thirdly, and perhaps most practically, one would like to be able to solve the path integrals, say with canonical Lagrangian form (4.32), in more direct way than to consider all multiple integrals involved in the measure (4.28). 

With the Lagrangian form known one can unfold the relativistic dynamics according with the analytical mechanics principles actually, for the momentum one uses the form of the conjugated canonical momentum, widely checked in the classical framework with the analytical Lagrangian formulation... 

The recent, most significant achievements in studying the structure of polymer systems are based on applying the field theory formalism. This section contauns the definitions and brief characterization of the main quantities and terms of the field theory in the Lagrangian form. Hereinafter, we follow Amit (1978). 

In this section, we develop the basic optimum theorem in the Lagrangian form, to include the cases where the allowable variations of the control variables are limited. In a later section, we shall introduce Pontryagin s treatment of cases where the state variables are similarly limited to make the distinetion clear, we shall refer to constraints on the control variables but restraints on the state variables. In the next section, the Lagrange formulation will be converted to the Hamilton form. 

This form can be proven to be correct. Note that the interactions act over all times . Thus the analog of the Hamiltonian formulation of quantum mechanics is not as convenient as the Lagrangian form since there is no simple analog of the conserved energy. 

Another useful form of the equations of motion, in addition to newtonian and lagrangian forms, are Hamilton s equations of motion. To see these, we first define the so-called conjugate momentum pi as... 

The velocity v is written either in the Lagrangian form or in the Eulerian form therefore the acceieration a can be represented in either form ... 

For a static equilibrium problem, the system of partial differential equations in Lagrangian form together with the boundary conditions is given by... 

For nonlinear problems such as elasto-plastic materials it is necessary to use a formulation based on an incremental form of the equation of equilibrium. We can introduce either the total Lagrangian form or the updated Lagrangianform. In the former case the incremental form is expressed in Lagrangian terms, while in the latter case the incremental form is given in an Eulerian description. 

The total Lagrangian form of the equation of equilibrium is obtained by differentiating (2.116) directly. Thus the partial differential equation system together with the... 

It must be noted that, as understood from (2.131), the updated Lagrangian form is expressed in Eulerian terms . 

The divergence theorem is applied to the third term of the r.h.s. of this equation and substitution of Stokes power formula gives the following Lagrangian form of the First Law of Thermodynamics (local balance of energy) ... 

Thermodynamic functions with chemical processes are detailed here for a solid undergoing finite strain. Following the outlines of Sects. 3.4.3 and 3.5.2, the Lagrangian forms are presented first. 

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