Hydrodynamics framework

A. Lagrangian Framework. An ideal subgrid model should be constructed on a Lagrangian hydrodynamics framework moving with the macroscopic flow. This requirement reduces purely numerical diffusion to zero so that realistic turbulence and molecular mixing phenomena will not be masked by non-physical numerical smoothing. This requirement also removes the possibility of masking purely local fluctuations by truncation errors from the numerical representation of macroscopic convective derivatives. 

Geochimica et Cosmochimica Acta, Vol. 8, pp. 22-48, 137-170, 198-212 Chiarelli, A., 1978. Hydrodynamic framework of Eastern Algerian Sahara. Influence on hydrocarbon occurrence. The American Association of Petroleum Geologists Bulletin, Vol. 62, no. 4, pp. 667-685... 

In large measure the paradigm within which work is carried out is strongly influenced by the objectives of the work, the background of the investigator, and the particular materials model under study. From a strictly fluid mechanics, hydrodynamic, or continuum framework, defect issues are not overtly at issue. From a strictly mechanical framework, the defective solid... 

The intent of this chapter is to establish a comprehensive framework in which the physicochemical properties of permeant molecules, hydrodynamic factors, and mass transport barrier properties of the transcellular and paracellular routes comprising the cell monolayer and the microporous filter support are quantitatively and mechanistically interrelated. We specifically define and quantify the biophysical properties of the paracellular route with the aid of selective hydrophilic permeants that vary in molecular size and charge (neutral, cationic, anionic, and zwitterionic). Further, the quantitative interrelationships of pH, pKa, partition... 

In what follows, we present in this short review, the basic formalism of TDDFT of many-electron systems (1) for periodic TD scalar potentials, and also (2) for arbitrary TD electric and magnetic fields in a generalized manner. Practical schemes within the framework of quantum hydrodynamical approach as well as the orbital-based TD single-particle Schrodinger-like equations are presented. Also discussed is the linear response formalism within the framework of TDDFT along with a few miscellaneous aspects. 

The purpose of this chapter is to show and discuss the connection between TD-DFT and Bohmian mechanics, as well as the sources of lack of accuracy in DFT, in general, regarding the problem of correlations within the Bohmian framework or, in other words, of entanglement. In order to be self-contained, a brief account of how DFT tackles the many-body problem with spin is given in Section 8.2. A short and simple introduction to TD-DFT and its quantum hydrodynamical version (QFD-DFT) is presented in Section 8.3. The problem of the many-body wave function in Bohmian mechanics, as well as the fundamental grounds of this theory, are described and discussed in Section 8.4. This chapter is concluded with a short final discussion in Section 8.5. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The following analysis holds for Type B fluidization and for Type A bubbling fluidization, when the region of particulate fluidization is so small that it can be ignored. In the framework of the two-phase model (see the subsection Hydrodynamic modeling of bubbling fluidization), the bed expansion in terms of the fraction of the bed occupied by bubbles is... 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Let us briefly review the essential ingredients to this procedure (for more details of the method see [30] and for our model [42]). For a given system the hydrodynamic variables can be split up into two categories variables reflecting conserved quantities (e.g., the linear momentum density, the mass density, etc.) and variables due to spontaneously broken continuous symmetries (e.g., the nematic director or the layer displacements of the smectic layers). Additionally, in some cases non-hydrodynamic variables (e.g., the strength of the order parameter [48]) can show slow dynamics which can be described within this framework (see, e.g., [30,47]). 

Following the standard procedure within the framework of irreversible thermodynamics we find the following set of macroscopic hydrodynamic equations [30, 31,42, 47] ... 

Multidisciplinary analytical and numerical models require development. These models should involve considerations of equilibrium and irreversible thermodynamics and kinetics of carbonate mineral-organic matter-water interactions within a sound hydrodynamic and basin evolution framework. 

Complex fluids are the fluids for which the classical fluid mechanics discussed in Section 3.1.4 is found to be inadequate. This is because the internal structure in them evolves on the same time scale as the hydro-dynamic fields (85). The role of state variables in the extended fluid mechanics that is suitable for complex fluids play the hydrodynamic fields supplemented with additional fields or distribution functions that are chosen to characterize the internal structure. In general, a different internal structure requires a different choice of the additional fields. The necessity to deal with the time evolution of complex fluids was the main motivation for developing the framework of dynamics and thermodynamics discussed in this review. There is now a large amount of papers in which the framework is used to investigate complex fluids. In this review we shall list only a few among them. The list below is limited to recent papers and to the papers in which I was involved. 

In the development above, it has been convenient to consider collisions, via particle transport, and reactions, the probability of particle attachment, as separate steps. There are a number of considerations indicating that this conceptual framework may have outlived its usefulness, as advancements in particle science, analytical capabilities, and supercomputers obviate the necessity of this artificial separation. Adler [4], Han and Lawler [3], and others have demonstrated using trajectory modeling the significant influence of hydrodynamics on particle collisions, and show how the lumped collision efficiency (inclusive of hydrodynamics) is a function of the type of collision mechanism. Thus, for a given particle pair, the collision efficiency will be different depending on whether the collision is a result of Brownian motion, fluid shear, or differential sedimentation. 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

In conclusion, much remains to be done in the field of many-body hydrodynamic interactions. Existing results need to be embedded into a unified and systematic framework and extended from quiescent to sheared suspensions. The methods of Mazur and co-workers can be used to derive far-... 

Nevertheless, it is useful to point out that zeolites with higher framework Si content generally prefer larger, less hydrated cations (e.g. Cs with its smaller hydrodynamic radii), while those with Si/Al 1 take up hydrated polyvalent cations selectively. The role of water is apparent here and also with the growing realization that water in immediate contact with an aluminous zeolite surface is hydrolyzed with two foreseeable consequences ... 

The simplest type of flow of a medium that yields itself to an analytical description within the framework of the precise hydrodynamical equations of viscous liquids (Navier-Stocks equations) is the Couette flow. This flow occurs under the impact of tangential stresses generated in a viscous liquid by a solid surface moving in it. The magnitude of the force that has to be applied to this surface to securse its movement in the viscous medium characterizes the tangential stresses and the velocity of its movement — the shear velocity. 

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