The Classical Fluid

The fundamental property of a classical fluid, to cite Lamb s treatise on hydrodynamics [1], is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the cominon surface. In other words, the only stress that a surface of an element of fluid at rest can sustain is a normal pressure. Pressure oriented other than normally can be resolved into a component perpendicular to the surface and a tangential component, the latter of which will induce motion. One of the fundamental distinctions between the response of an elastic solid and a classical fluid to tangential stress is that there is a limited displacement within the solid which is proportional to the stress whereas the motion of a fluid continues as long as the stress is maintained. 

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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. 

The flow of liquids or semisolids is described by viscosity, or, more precisely, by shear viscosity (unit Pa sec). The viscosity defines the resistance of the material against flow. Viscosity is not a coefficient, because it is a function of the shear strain rate y [ti = /(y)]. In the classical fluid mechanics, the dynamic viscosity is obtained using a viscometer. (A viscometer is a rheometer, i.e., an instrument for the measurement of rheological properties, limited to... 

Further discussions on the classical fluid d3mamic stability theory can be found elsewhere (e.g., [112], chap 1 [38] [139], chap 15). 

In general, the reaction mechanism of the classical fluid-solid reaction has intra- and inter- particle diffusion problems. Liquid-solid-liquid triphase catalysis is more complicated... 

The kinetics of the triphase reaction is complicated and not yet completely described by that of the classical fluid-solid system. These experimental results revealed that it is preferable to use an SR rather than an FBR for a triphase reaction but the reactor volume of SR was much larger than that of FBR. The goal of designing an SR could be to reduce the reactor volume and stop the catalyst from flowing out of the reactor. If no premix was set before an FR, the length of reactor was too short to reduce llte performance of the FR. 

The hydrodynamic characteristics of a PFB are essentially the same as those of fluidized or spouted beds. However, there is no peak in the pressure-velocity curve the pressure drop increases gradually with gas velocity, even with fully developed fluidization (Figure 6.4). Although the pressure drop in the PFB is of the same order as that in the classical fluid bed for the same gas velocity and the same free cross-sectional area of the supporting grid, the PFB technology offers a lower pressure drop because of the combined effects from the following ... 

The classical fluid systems are characterized with simplified Hamiltonian in which the semiempirical pairwise-additive interaction between two particles are included. Those semiempirical interactions substantially arise from the Pauh exclusion of two electrons at the same quantum state and from the electrostatic interactions among electrons and nuclei. As a result, both repulsion and attraction appear in the two-body interaction. Specifically, when all involved particles are spherical ones like atoms, ions, or coarsegrained beads, the systems are called simple fluids. Obviously, the pair interactions in simple fluid systems are simply distance-dependent. Toward the investigation of simple fluid systems, atomic DFT is developed. A notable merit of atomic DFT is that the contributions to the free energy functional from different interaction parts can be treated separately. To demonstrate, below we present the DFT investigations for the simple systems of HS fluids, LJ fluids, and charged systems. 

The theoretical formulation of heat and mass transfer in porous media is usually obtained by a change in scale. We can pass from a microscopic view where the size of tlie representative volume is small with regards to pores, to a microscopic view where the size of the representative volume CO is large with regard to the pores. Moreover, the heat and mass transfer equations can be deduced from Whitaker s theory. The macroscopic equations can be obtained by averaging the classical fluid mechanics, dififiision and transfer equations over the averaging volume co[m ]. The average of a function/is ... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

Stell G 1964 Cluster expansions for classical systems In equilibrium The Equilibrium Theory of Classical Fluids ed H L Frisch and J L Lebowitz (New York Benjamin)... 

Ebeling W and Grigoro M 1980 Analytical calculation of the equation of state and the critical point in a dense classical fluid of charged hard spheres Phys. (Leipzig) 37 21... 

Smith W R 1972 Perturbation theory in the classical statistical mechanics of fluids Specialist Periodical Report vol 1 (London Chemical Society)... 

These are the two components of the Navier-Stokes equation including fluctuations s., which obey the fluctuation dissipation theorem, valid for incompressible, classical fluids ... 

The hydrodynamical analogy now follows by comparing Eq. (B.6) to the conservation law for a classical fluid... 

This proves that the pseudoparticles in the quantum fluid obey classical mechanics in the classical limit. 

Verlet, L. Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 165 (1967) 98-103. Ryckaert, J.-P., Ciccotti,G., Berendsen, H.J.C. Numerical integration of the cartesian equations of motion of a system with constraints Molecular dynamics of n-alkanes. Comput. Phys. 23 (1977) 327-341. 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

A new generation of antiinflammatory agents having immunosuppressive activity has been developed. The appearance of preclinical and clinical reports suggest that these are near entry to the pharmaceutical market. For example, tenidap (CP-66,248) (12) has been demonstrated to inhibit IL-1 production from human peripheral blood monocytes in culture (55). Clinically, IL-1 in synovial fluids of arthritic patients was reduced following treatment with tenidap. Patients with rheumatoid or osteoarthritis, when treated with tenidap, showed clinical improvement (57,58). In addition to its immunological effects, tenidap also has an antiinflammatory profile similar to the classical NSAIDs (59). Other synthetic inhibitors of IL-1 production are SKF 86002 (20) andE-5110 (21) (55). 

Heat Exchangers Since most cryogens, with the exception of helium 11 behave as classical fluids, weU-estabhshed principles of mechanics and thermodynamics at ambient temperature also apply for ctyogens. Thus, similar conventional heat transfer correlations have been formulated for simple low-temperature heat exchangers. These correlations are described in terms of well-known dimensionless quantities such as the Nusselt, Reynolds, Prandtl, and Grashof numbers. 

The classic signature of erosion-corrosion is the formation of horseshoeshaped depressions, comet tads, grooves, or sand dunelike surface contours oriented along the direction of fluid flow (Figs. 11.1,11.2,11.3,11.5, and 11.8). Occasionally, erosion-corrosion will produce smooth, almost featureless, surface contours (Fig. 11.15), although even in this case oriented metal loss often exists around the perimeter of the affected region. If erosion-corrosion has been recently active, affected surfaces will be free of accumulated deposits and corrosion products. 

M. Schoen. Taylor-expansion Monte Carlo simulations of classical fluids in the canonical and grand canonical ensembles. J Comput Phys 775 159-171, 1995. 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [120], Gouy and Chapman [121,122], and Stern [123], and in most analytic studies, the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g. as a simple dipolar fluid. Computer simulations... 

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

Viewing things from the perspective of his physical theory of contact electricity, Volta was intrigued by the apparently endless power of the battery to keep the electric fluid in motion without the mechanical actions needed to operate the classical, friction, electrostatic machine, and the electrophorus. He called his batteiy alternately the artificial electric organ, in homage to the torpedo fish that had supplied the idea, and the electromotive apparatus, alluding to the perpetual motion (his words) of the electric fluid achieved by the machine. To explain that motion Volta relied, rather than on the concepts of energy available around 1800, on his own notion of electric tension. He occasionally defined tension as the effort each point of an electrified body makes to get rid of its electricity but above all he confidently and consistently measured it with the electrometer. 

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