Eulerian Descriptions

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

In simulating physical operations carried out in stirred vessels, generally one has the choice between a Lagrangian approach and a Eulerian description. While the former approach is based on tracking the paths of many individual fluid elements or dispersed-phase particles, the latter exploits the continuum concept. The two approaches offer different vistas on the operations and require different computational capabilities. Which of the two approaches is most... 

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

In consideration of the principle of material frame indifference the fundamental laws for the multi-component mixture in the Eulerian description are formulated in the following local forms. 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

Usually, the intrinsic average reflects the real physical property or quantity such as density and velocity, while the phase average gives a pseudoproperty or quantity based on the selection of control volume. Phase averages are used to construct the continuum of each phase to which Eulerian description can be applied. 

The Lagrangian map gives a full description of the flow that is equivalent to the so called Eulerian description based on specifying... 

Although we have derived the desired equations, we have at the same time generated new dependent variables u iu kc )J,k= 1,2,3. If we generate additional equations for these variables, we find that still more dependent variables appear. The closure problem becomes even worse if a nonlinear chemical reaction is occurring. If the single species decays by a second-order reaction, then the term (R) in (18.4) becomes -k((c)2 + (c 2)), where (c 2) is a new dependent variable. If we were to derive an equation for (c 2), we would find the emergence of new dependent variables ( jc 2), (c 3), and (0c /0x -0c /0xy). It is because of the closure problem that an Eulerian description of turbulent diffusion will not permit exact solution even for the mean concentration (c). 

As we have seen, the Eulerian description of turbulent diffusion leads to the so-called closure problem, as illustrated in (18.4) by the new dependent variables (njcj),/ = 1, 2, 3, as well as any that might arise in (/ , ) if nonlinear chemical reactions are occurring. Let us first consider only the case of chemically inert species, that is, R, = 0. The problem is to deal with the variables ( c ) if we wish not to introduce additional differential equations. 

Initial reinforcement layup, followed by compaction and resin injection, are phases common to RTM. The volume of literature devoted to RTM can thus be applied directly to these phases (e.g., references 14,15). Once the resin has been injected and the gates have been closed, wet compression is initiated. The first theoretical analyses on the compression of saturated fibrous materials were carried out by Dave et and Gutowski et in the context of autoclave processing and bleeder ply moulding. Therein it was noted the similarity between these processes and the classic geomechanics consolidation problem, that is, the compression of a water-saturated soil or clay. i The theory can be formulated in terms of Lagrangian coordinates, but is more conveniently presented for the present purposes, as in what follows, in terms of an Eulerian description. ... 

The elastic theory of SmC has beep considered by the Orsay groupand by Rapini. Both approaches use the Oseen description of smectic A, neglecting all changes in internal parameters such as density, interlayer distance, and tilt angle. The Orsay group used the Lagrai an description for the elastic strains with a vector IKF) describing the local rotation of the director. In order to be consistent with the elastic theory for NLC s, we shall use the Eulerian description developed by Rapini based on the director. 

We can generalize this procedure If a function

Lagrangian description, and if

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). 

We distinguish between the coordinate system Ei of the Lagrangian description and the coordinate system of the Eulerian description in order to understand the... 

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. 

For simplicity, we describe the equations without mass density p. Since the speciflc stress a" (x,t) is given in terms of an Eulerian description, we treat here the simplest response for that description. To satisfy the principle of determinism and the principle of local action mentioned in the previous section, the stress ai(x,t) can be written in terms of v and Vv ... 

Substituting Stokes law (2.234) into the equation of motion (2.104) under the Eulerian description yields the following equation of motion for the unknown velocity v ... 

Stokes Power Formulation as Mechanical Conservation of Energy in a Continuum Eulerian Description... 

For the deformed body (i.e., Eulerian description) we can introduce a Legendre transformation it should be noted that material isotropy is imposed. In the analysis of finite strain problems the updated Lagrangian equilibrium equation, which is a form of an Eulerian description (cf. Sect. 2.6.2) is usually used, meaning that the body is required to be isotropic. 

The usage of the term micro/macro is justified by the two-level description of the fluid that lies at its core a microscopic approach permits evaluation of the stress tensor to be used in closing the macroscopic conservation equations. The macro part can be thought of as corresponding to the Eulerian description of the fluid, while the micro part has Lagrangian character. 

Secondly, the authors give an Eulerian description of the motion of N independent and identical Brownian particles in terms of the phase space density /(r,u f) obeying the Fokker-Planck equation (FPE)... 

The past studies of turbulence effects on (conventional) helicopter flight mechanics, loads and vibrations and also stability have all neglected the effects of rotating frame turbulence (RFT) which requires rotationally-sampled (non-Eulerian) description. As a case in point, we also mention the earlier studies of RFT effects on the compound helicopters in high-speed cruising with rotor off-loading by a fixed wingi2-i4 and on the horizontal-axis and vertical-... 

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