From hydrodynamic equations equation

Equations (15-25) and (15-26) can be obtained from hydrodynamic considerations. Equations (15-25) have a simple physical interpretation. The quantity... 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

From various studies" " it is becoming clear that in spite of a heat flux, the overriding parameter is the temperature at the interface between the metal electrode and the solution, which has an effect on diffusion coefficients and viscosity. If the variations of these parameters with temperature are known, then / l (and ) can be calculated from the hydrodynamic equations. 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

In gridpoint models, transport processes such as speed and direction of wind and ocean currents, and turbulent diffusivities (see Section 4.8.1) normally have to be prescribed. Information on these physical quantities may come from observations or from other (dynamic) models, which calculate the flow patterns from basic hydrodynamic equations. Tracer transport models, in which the transport processes are prescribed in this way, are often referred to as off-line models. An on-line model, on the other hand, is one where the tracers have been incorporated directly into a d3mamic model such that the tracer concentrations and the motions are calculated simultaneously. A major advantage of an on-line model is that feedbacks of the tracer on the energy balance can be described... 

The results presented here are quite remarkable. The theory underlying derivation of the hydrodynamic equations assumes that all gradients and forces acting on the fluid are small. The MD fluids are under the influence of extremely large gradients and forces. Yet, we find results which are in both qualitative and quantitative agreement with macroscopic predictions. The appearance of spatial structure on such a small scale (10 cm) provides strong indications that fluid dynamics can be understood from a microscopic viewpoint. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

The hydrodynamic equations can be derived from the MPC Markov chain dynamics using projection operator methods analogous to those used to obtain... 

Here 0 is the Heaviside function. The projection operator formalism must be carried out in matrix from and in this connection it is useful to define the orthogonal set of variables, k,uk,5k > where the entropy density is sk = ek — CvTrik with Cv the specific heat. In terms of these variables the linearized hydrodynamic equations take the form... 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

With the surface-velocity expression known from the hydrodynamics, Equation 2 can be rewritten as... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

Although trajectories are not computed in QFD-DFT, it is clear that there is a strong connection between this approach and the trajectory or hydrodynamical picture of quantum mechanics [20], independently developed by Madelung [21], de Broglie [22], and Bohm [23], which is also known as Bohmian mechanics. From the same hydrodynamical equations, information not only about the system... 

Of course, as was shown in Section V-A, this latter expression may also be derived starting from the hydrodynamical equations for the pair distribution and the Poisson equation it is also the final result of the theories developed independently by Falken-hagen and Ebeling,9 and by Friedman 12-13 in these two approaches, the starting point is a Liouville equation for the system of ions with an ad hoc stochastic term describing the interactions with the solvent. 

Cook equation of state, using covolume approximation) 63-4 (Other coyolume equations of state) 65 (Jones, Jones-Miller and Lennard-Jones equations of state) 66 (Cot-trell-Paterson equation of state) 12aj) W. Fickett W.W. Wood, Physics of Fluids 1, 528(1958 (A Detonation-Product Equation of State Observed from Hydrodynamic Data)... 

H) W. Fickett W.W. Wood, The Physics of Fluids 1 (6), 528-34 (Nov-Dec 1958) (Detonation-product equations of state, known as "constant-/ and "constant-)/ , obtained from hydrodynamic data) I) J.J. Erpenbeck D.G. Miller, IEC 51, 329-31 (March 1959) (Semiempirical vapor pressure relation based on Dieterici s equation of state J) K.A. Kobe P.S. Murti, IEC 51, 332 (March 1959) (Ideal critical volumes for generalized correlations) (Application to the Macleod equation of state) Kj) S. Katz et al, jApplPhys 10, 568-76(April 1959) (Hugoniot equation of state of aluminum and steel) K2) S.J. Jacobs, jAmRocketSoc 30, 151(1960) (Review of semi-empirical equations of state)... 

Detonation - Product Equation of State Obtained from Hydrodynamic Data is discussed by W. Fickett W.W. Wood in the Physics of Fluids 1, 528-34(1958)... 

Accdg to Dunkle (Ref 28), Brode (Ref 14), in order to solve detonation problems without recourse to empirical values derived from explosion measurements, integrated the hydrodynamical equations of motion (which constitute a set of nonlinear partial... 

A new relationship in the hydrodynamic theory of expl waves) 95) W.W. Wood J.G. Kirkwood, JChemPhys 29, 956(1958) (Present status of deton theory) 95a) W. Fickett 8t W.W. Wood, Phys of Fluids 1 (6), 528-34 (Nov-Dec. 1958) (A detonation-product equation of state obtained from hydrodynamic data) 96) Cook... 

In suspensions of particles with an aspect ratio (length to diameter) greater than 1, particle rotation during flow results in a large effective hydrodynamic volume, and Kh > 2.5 (see Figure 4.7). At particle volume fractions above about 5-10%, interaction between particles during flow causes the viscosity relationship to deviate from the Einstein equation. In such instances, the reduced viscosity is better described by the following relationship ... 

The hydrodynamic shape factor and axial ratio are related (see Eigure 4.18), but are not generally used interchangeably in the literature. The axial ratio is used almost exclusively to characterize the shape of biological particles, so this is what we will utilize here. As the ellipsoidal particle becomes less and less spherical, the viscosity deviates further and further from the Einstein equation (see Eigure 4.19). Note that in the limit of a = b, both the prolate and oblate ellipsoid give an intrinsic viscosity of 2.5, as predicted for spheres by the Einstein equation. 

The Maxwell theory is well known to be a material fluid flow theory [6],4 since the equations are hydrodynamic equations. In principle, anything that can be done with fluid theory can be done with electrodynamics, since the fundamental equations are the same mathematics and must describe consistent analogous functional behavior and phenomena [5]. This means that EM systems with electromagnetic energy winds from their active external atmosphere ... 

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