Classical fluids

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)... 

Chandler D and Andersen H C 1972 Optimized cluster expansions for classical fluids II. Theory of molecular liquids J. Chem. Phys. 57 1930... 

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... 

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

Verlet L 1967 Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules Phys. Rev. f59 98-103... 

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

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... 

L 1967. Computer Experiments on Classical Fluids. II. Equilibrium Correlation Functions. tysical Review 165 201-204. 

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. 

M. Schoen, D. J. Diestler, J. H. Cushman. Fluids in micropores. I. Structure of a simple classical fluid in a slit-pore. J Chem Phys 27 5464-5476, 1987. 

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

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

As mentioned earlier, in principle, one can model the dynamics of a simple classical fluid by means of MD simulations. This technique, although straightforward, is relatively time-consuming, and therefore not suitable for observation of large-scale macroscopic phenomena in the fluid. However, one often does not need such a detailed description of the microdynamics as provided by MD. In such cases, it would be more efficient to strip the MD model down to its barest essentials, where the only requirement is that the model behaves like a fluid macroscopically, but is still atomistic in character—i.e., the mechanism underlying the fluid motion is the movement of particles. From the derivation of... 

Here we focus on yet another implementation, the single-particle hydrodynamic approach or QFD-DFT, which provides a natural link between DFT and Bohmian trajectories. The corresponding derivation is based on the realization that the density, p(r, t), and the current density, j(r, t) satisfy a coupled set of classical fluid, Navier-Stokes equations ... 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

Thermodynamic Equation of State for Classical Fluid Mixtures of Molecules Interacting with Alpha-exponential-six Pair Potentials up to High Densities. 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

MSN.92. I. Prigogine, M. Theodosopoulou, and A. Grecos, On the derivation of linear irreversible thermodynamics for classical fluids, Proc. Natl. Acad. Sci. USA, 75, 1632-1636 (1978). 

Andersen, H. C. Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. III. Cluster Analysis of the Renormalized Interactions and a Second Diagrammatic Representation of the Correlation Functions. J. Phys. Chem. B 2003, 107, 10234-10242. 

Note that the velocity at the wall (y = 0 and y = b) is zero, meaning nothing moves at the wall, according to classical fluid mechanics. 

To obtain a realistic and detailed picture of how individual molecules rotate and translate in these classical fluids. 

Sanchez I.C., R.H. Lacombe, "An Elementary Molecular Theory of Classical Fluids. Pure Fluids", J. Phvs. Chem.. 1976, 80(21),2352-2362. 

Evans R. Nature of the liquid-vapor interface and other topics in the statistical-mechanics of nonuniform, classical fluids. Adv. Phys., 1979 28(2) 143-200. 

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