Classical fluid dynamics approach

In Equation 4.56, the real quantities p, v, and j are the charge density, velocity field, and current density, respectively. The above equations provide the basis for the fluid dynamical approach to quantum mechanics. In this approach, the time evolution of a quantum system in any state can be completely interpreted in terms of a continuous, flowing fluid of charge density p(r,t) and the current density j(r,t), subjected to forces arising from not only the classical potential V(r, t) but also from an additional potential VqU(r, t), called the quantum or Bohm potential the latter arises from the kinetic energy and depends on the density as well as its gradients. The current... 

The spontaneous emergence of avalanches, droplets and rivulets is very difficult to simulate with classical fluid dynamical models, due to the critical nature (self-organized criticality) and threshold character of these nonlinear phenomena. Therefore, the role of statistical fluctuations in thin-film dynamics cannot be underestimated, especially in the mesoscale. Unlike the classical approaches, we need not introduce any external and artificial perturbations. All phenomena occur spontaneously due to thermal noise inherent in the nonlinearly interacting particle dynamics. 

Chemical engineers, however, have to find practical ways for dealing with turbulent flows in flow devices of complex geometry. It is their job to exploit practical tools and find practical solutions, as spatial variations in turbulence properties usually are highly relevant to the operations carried out in their process equipment. Very often, the effects of turbulent fluctuations and their spatial variations on these operations are even crucial. The classical toolbox of chemical engineers falls short in dealing with these fluctuations and its effects. Computational Fluid Dynamics (CFD) techniques offer a promising alternative approach. 

The thermodynamics of irreversible processes should be set up from the scratch as a continuum theory, treating the state parameters of the theory as field variables [32]. This is also the way in which classical fluid mechanic theory is formulated. Therefore, in the computational fluid dynamics literature, the transport phenomena and the extensions of the classical thermodynamic relations are both interpreted as closures of the fluid dynamic theory. The validity of the thermodynamic relations for fluid dynamic systems has been approached from the viewpoint of the kinetic theory of gases [13]. However, any Arm distinction between irreversible thermodynamics and fluid mechanics... 

In essence, CPMD is a fairly typical example of the hierar-chy-of-models approach in which the electrons are treated quantum-mechanically as a massless fluid while nuclei are described as a set of classical particles. A wide scope for the application of hybrid simulations is provided by problems in polymer science. There are hybrid methods coupling a classical molecular-dynamics model with an elastic continuum modd, " RISM theory," " " and so on. The... 

A rigorous approach to modelling the channel flows is based on 2D or 3D classical equations of viscous fluid dynamics (Dutta et al., 2000 Um et al., 2000 Wang et al., 2001 Wang, 2004). Numerical calculations, however, do not give parametric dependencies in the problem. 

Recently, a new type of phase separation called viscoelastic phase separation was observed in polymer solutions or dynamically asymmetric fluid mixtures [1-3]. It is an interesting feature of this phenomenon that network-like domains of more viscous phase emerge in a transient regime. It has little been understood what ingredient of physics is crucial to this phenomenon. Various numerical approaches have been made for the phase separation phenomena in binary fluid systems in the last decade [4-6]. Most of these studies have been concerned with classical fluids and have not involved viscoelasticity. A new numerical model was recently proposed by the author [7] based upon the two-fluid model [8,9] using the method of smoothed-particle hydrodynamics (SPH) [10,11]. In this model the Lagrangian picture for fluid is adopted and the viscoelastic effect can easily be incorporated. In this paper we carry out a computer simulation for the viscoelastic phase separation in polymer solutions with this model. 

In the first problem class mentioned above (hereinafter called class A), a collection of particles (atoms and/or molecules) is taken to represent a small region of a macroscopic system. In the MD approach, the computer simulation of a laboratory experiment is performed in which the "exact" dynamics of the system is followed as the particles interact according to the laws of classical mechanics. Used extensively to study the bulk physical properties of classical fluids, such MD simulations can yield information about transport processes and the approach to equilibrium (See Ref. 9 for a review) in addition to the equation of state and other properties of the system at thermodynamic equilibrium (2., for example). Current activities in this class of microscopic simulations is well documented in the program of this Symposium. Indeed, the state-of-the-art in theoretical model-building, algorithm development, and computer hardware is reflected in applications to relatively complex systems of atomic, molecular, and even macromolecular constituents. From the practical point of view, simulations of this type are limited to small numbers of particles (hundreds or thousands) with not-too-complicated inter-particle force laws (spherical syrmetry and pairwise additivity are typically invoked) for short times (of order lO" to 10 second in liquids and dense gases). 

As one would expect, developments in the theory of such phenomena have employed chemical models chosen more for analytical simplicity than for any connection to actual chemical reactions. Due to the mechanistic complexity of even the simplest laboratory systems of interest in this study, moreover, application of even approximate methods to more realistic situations is a formidable task. At the same time a detailed microscopic approach to any of the simple chemical models, in terms of nonequilibrium statistical mechanics, for example, is also not feasible. As is well known, the method of molecular dynamics discussed in detail already had its origin in a similar situation in the study of classical fluids. Quite recently, the basic MD computer model has been modified to include inelastic or reactive scattering as well as the elastic processes of interest at equilibrium phase transitions (18), and several applications of this "reactive" molecular dynamicriRMD) method to simple chemical models involving chemical instabilities have been reported (L8j , 22J. A variation of the RMD method will be discussed here in an application to a first-order chemical phase transition with many features analogous to those of the vapor-liquid transition treated earlier. 

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