Euler-Lagrange approach

In the Euler-Lagrange approach, nanoparticles are individually tracked as they are transported through the vascular system. Here, nanoparticle transport can be modeled via Newton s second law of motion, typically for NP diameters greater than 100 nm. It states that the particle s time rate of change of momentum is balanced by the forces acting on the particle ... 

The example of the variational procedure considered in this section was very simple, because we operated only with one independent variable (angle (p). Sometime one needs to minimize the energy with respect to two variables in fact, we met this case in Section 6.3.3 for an infinite medium. For two variables, the system of two Euler equations can be constructed using the same procedure as earlier. However, very often one deals with some constraints as, for example, in the case of the director that has three projections satisfying the constraint + tiy- - n =. In such cases the Lagrange multipliers are introduced to solve the variational problem, however this, more general Euler - Lagrange approach will not be used in this book. 

Mathematically the molecular field vector h can be found using the Euler-Lagrange approach by a variation of the elastic and magnetic (or electric) parts of the free energy with respect to the director variable n(r) (with a constraint of = 1). For the elastic torque, in the absence of the external field, the splay, twist and bend terms of h are obtained [9] fi-om the Frank energy (8.16) ... 

In addition to partition and elongation, the gas flow rate in the different compartments of horizontal fluidized bed equipment can be used to manipulate the residence time distribution of the particles and, thus, the quality of the outlet product. The use of CFD can support the design of compartments, partitions and weirs. Not only the here briefly presented two-fluid (Euler-Euler) approach, but also Euler-Lagrange approaches (so-called discrete particle models, DPM) can serve this goal, as will be discussed in Section 7.5.4. 

M.L., Delvigne, E, Crine, M., and Toye, D. (2015) Euler-Lagrange approach... 

Blei, S. (2005). On the interaction of non-uniform particles during the spray drying process Experiments and modelling with the Euler-Lagrange approach. PhD Dissertation, Martin-Luther-Universitat Halle-Wittenberg. 

The airflow throughout the computational domain was turbulent based on the calculation of Reynolds number. The re-normalization group (RNG) turbulence model [75], having more imiversality and improved predictions for turbulent flows was used as high streamline curvature flows and mass transfer were involved in the study. The Euler-Lagrange approach [75], suited for flows where particle streams are injected into a continuous phase flow with a well-defined entrance and exit condition was followed. A fundamental assumption made in this model was that the dispersed second phase occupies a low volume fiaction (usually less than 10-12%) for the fiber air-flows in both the web forming systems . For both the web forming systems, a three-dimensional 10-node tetrahedron volume element was used to discretize the computational domains. 

The effect of treating the particles as point particles (see also Balachandar, 2009) is that the detailed flow between the particles in response to the presence and motion of the particles remains unresolved (see, e.g., Derksen, 2003). As a result, a correlation is required in order to take the fluid—particle interaction into account. The Euler—Lagrange approach is most often used for simulating dilute gas—solid and liquid—sohd flows where the particle size is smaller than the smallest turbulent length scale (eddy) considered in the flow simulation of the carrier phase (Balachandar and Eaton, 2010). 

In the Euler—Lagrangian approach of two-phase flow (see, e.g., Crowe et al, 1996), the particles are treated as point particles the finite volume of the particles is not considered and the flow around the particles is not resolved. The motion of the particle is simulated by means of Newton s second law and that is why the fluid—particle interaction force is needed and the empirical correlations enter. Although the flow around the particles is not resolved, any empirical correlation does reflect the hydrodynamics of the canonical case involved. The use of the Euler—Lagrange approach, or point-particle method, is usually restricted to the more dilute gas—solid and liquid—soHd systems. Ignoring the mutual interaction of particles is therefore not too serious a simplification. The fluid-particle interaction can be treated in the simpler one-way mode or according to the more complicated two-way coupling mode in which the particles also affect the carrier phase flow field (Decker and Sommerfeld, 2000 Derksen, 2003 Derksen et al, 2008). 

On the basis of all information gathered, it is fair to conclude that one of the major drivers behind the occurrence, the shape, and the dynamics of these mesoscale structures is the fluid—particle interaction force that plays a dominant role, both in stability analyses and in CFD simulations of any type. This role is related to the difference in inertia of the two phases and, as a result, to the temporally and spatially varying difference in velocities of dispersed phase (particles) and carrier (or continuous) phase. Cluster and strand formation seem to be closely related to the continuous chaotic accelerations in a turbulent carrier fluid (in the Euler—Lagrange approach) or a turbulent continuous phase (in the Euler-Euler or two-fluid approach). An interesting explanation for cluster formation is the sweep-stick mechanism proposed by Goto and Vasillicos (2008). 

Decker S, Sommerfeld M Numerical calculations of two-phase flows in agitated vessek using the Euler/Lagrange approach. In Proceedings of ASME 2000fluid engineering division summer meeting, Boston, MA, FEDSM OO 11154, 2000, pp 1—8. 

Lain S, Broder D, Sommerfeld M Numerical modelling of the hydrodynamics in a bubble column using the Euler-Lagrange approach. In Proceedings of the MFTP-2000—intematiorud symposium on multiphase flow and transport phenomena, Antalya (Turkey), 2000. 

Currently there are two approaches for the numerical calculation of multiphase flows the Euler-Lagrange approach and the Euler-Euler approach. The Euler-Lagrange approach solves the time-averaged Navier-Stokes... 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Moreau, M., B. Bedat, and O. Simonin, From Euler-Lagrange to Euler-Euler large eddy simulation approaches for gas-particle turbulent flows, in ASME Fluids Engineering Summer Conference, Houston. 2005, ASME FED. 

Euler-Lagrange equations, electron nuclear dynamics (END), time-dependent variational principle (TDVP) basic ansatz, 330-333 free electrons, 333-334 Evans-Dewar-Zimmerman approach, phase-change rule, 435... 

On the analogy of simulating the process of adding blobs of a miscible liquid, two-phase flow in stirred tanks in a RANS context may be treated in two ways Euler-Lagrange or Euler-Euler, with the second, dispersed phase treated according to a Lagrangian approach and from a Eulerian point of view, respectively. 

In Section 4.5, we have introduced the Euler-Lagrange optimization approach to decoherence control that is optimally tailored to the bath spectrum in question. This approach has then been applied in Section 4.6 to optimized state transfer in hybrid system, from noisy to quiet qubits or through noisy spin channels. 

Attempts to extend RANS formulation to LES of two-phase combustion may be found in [318 354 317 255 292]. They are all based on a Euler-Lagrange (EL) description of the dispersed phase in which the flow is solved using an Eulerian method and the particles are tracked with a Lagrangian approach. An alternative is the Euler-Euler (EE) description, also called two-fluid approach, in which both the gas and the dispersed phases are... 

In the limit of nonvanishing temperatures, simple analytic forms for the w s, such as the ones compiled in Table 4.1, do not exist. Hence we need to resort to a numerical scheme to solve the Euler-Lagrange equations [see Eq. (4.86)]. This can be accomplished by an approach detailed in Appendix D.2.1 that starts from the set of (exact) morphologies M compiled in Table 4.1 at T = 0 as starting solutions for a temperature T > 0. Once convergence has been attained, the algorithm yields new morphologies for this sufficiently... 

An approach similar to the one adopted in the previous subsection can be used to derive the Euler-Lagrange equations for a functional that possesses -dependent variables yi.yi - fn to give the following system of equations ... 

The Euler-Lagrange equation is used in our variational approach. By variation of the energy functional with respect to S, we arrive at an elliptic equation... 

Inhomogeneous or multiphase reaction systems are characterised by the presence of macroscopic (in relation to the molecular level) inhomogeneities. Numerical calculations of the hydrodynamics of such flows are extremely complicated. There are two opposite approaches to their characterisation [63, 64] the Euler approach, with consideration of the interfacial interaction (interpenetrating continuums model) and the Lagrange approach, of integration by discrete particle trajectories (droplets, bubbles, and so on). The presence of a substantial amount of discrete particles in real systems makes the Lagrange approach inapplicable to study motion in multicomponent systems. Under the Euler approach, a two-phase flow is described... 

Nanoparticle transport and targeting in the cardiovascular system can be modeled by either the Euler-Euler or the Euler-Lagrange modeling approach. For both methods, the conservation of mass and momentum are solved for the blood transport through the vascular system ... 

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