Navier-Stokes transport coefficients

In view of the very slow time decay of the time correlation functions, it is not clear to what extent the Navier-Stokes transport coefficients can be used even in three dimensions to describe phenomena that vary on a time scale of 50tc, for on this time scale there is not yet a clear separation of microscopic and macroscopic effects. However, usually the Navier-Stokes equations are applied to phenomena that vary on a much longer time scale, and then the slow decay of the correlation functions does not interfere with the hydrodynamic processes. Nevertheless, the divergences of the Burnett and higher-order transport coefficients do appear to have experimental consequences even for three-dimensional systems. In particular, it appears that the dispersion relation for the sound wave frequency wave number k can no longer be expressed as a power series in k as was done in Eq. (133) but instead that fractional powers of the form for /i = l,2,... 

In view of the apparent divergence of Navier-Stokes transport coefficients for two-dimensional systems, can one find the correct form of the hydrodynamic equations in two dimensions ... 

If we design the coupling of the external field to the system in such a way that the dissipative flux is equal to one of the Navier-Stokes fluxes (such as the shear stress in planar Couette flow or the heat flux in thermal conductivity), it can be shown - provided the system satisfies a number of fairly simple conditions (Evans Morriss 1990) - that the response is proportional to the Green-Kubo time integral for the corresponding Navier-Stokes transport coefficient. This means that the linear response of the system to the fictitious external field is exactly related to linear response of a real system to a real Navier-Stokes force, thereby enabling the calculation of the relevant transport coefficient. 

The primary targets of these calculations are the thermal (or Navier-Stokes) transport coefficients (TTCs) which are characteristic properties of the fluid at a given thermodynamic state and inhomogeneity. The latter is quantified by the gradient of an intensive variable and, formally, can be described as an external field acting on the system. It is an essential simplification both in theoretical and in numerical studies to fix these external fields in time and consider only time-independent (i.e., steady) states. 

Table 1 Examples of Green-Kubo Expressions for Navier-Stokes Transport Coefficients...

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. 

In the macroscopic limit, this model running on a square lattice tends to the Navier-Stokes equation, but a hexagonal lattice rather than a square lattice gives a simulation that is more scientifically justifiable and permits the determination of a range of parameters, such as transport coefficients. 

The physics of the problem under study is assumed to be governed by the compressible form of the Favre-filtered Navier-Stokes energy and species equations for an ideal gas mixture with constant specific heats, temperature-dependent transport properties, and equal diffusion coefficients. The molecular Schmidt, Prandtl, and Lewis numbers are set equal to 1.0, 0.7, and 1.43, respectively [17]. 

These equations are called the Navier-Stokes equations, and when supplemented by the state equation for fluid pressure and species transport equations, they form the basis for any computational model describing the flows in fires. For simplicity, several approximations are inherent (see Equation 20.3) (no Soret/Dufour effects, no viscous dissipation, Fickian diffusion, equal diffusion coefficients of all species, unit Lewis number). 

A major limitation of the present work is that it deals only with well-defined (and mostly unidirectional) flow fields and simple homogeneous and catalytic reactor models. In addition, it ignores the coupling between the flow field and the species and energy balances which may be due to physical property variations or dependence of transport coefficients on state variables. Thus, a major and useful extension of the present work is to consider two- or three-dimensional flow fields (through simplified Navier-Stokes or Reynolds averaged equations), include physical property variations and derive lowdimensional models for various types of multi-phase reactors such as gas-liquid, fluid-solid (with diffusion and reaction in the solid phase) and gas-liquid-solid reactors. 

Our theoretical calculations indicated viscous flow to be dominant at high densities, as depicted the agreement of the theory in the inset in Figure 1. A viscous flow model may be used over length scales larger than the mean free path, which is largely satisfied for mesopores. To obtain the theoretical transport coefficient we solved die Navier Stokes equation... 

Transport coefficients of molecular model systems can be calculated by two methods [8] Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. 

The propeller setup was used for this purpose. From a computational standpoint, a mesh of the vessel-propeller set was created containing 8746 elements yielding 54,333 velocity equations and 8746 concentration equations. The surface mesh of the propeller (Fig. 5) comprised 964 control points. A maximum of three control points per element was used to avoid locking. Unsteady state flow simulations were performed with a 1-s time step and three coupling iterations between the Navier-Stokes equations and the solid transport equation were required per time step. Steady state was deemed obtained when the solids concentration coefficient of variation did not change. 

Brenner (1980) has explored the subject of solute dispersion in spatially periodic porous media in considerable detail. Brenner s analysis makes use of the method of moments developed by Aris (1956) and later extended by Horn (1971). Carbonell and Whitaker (1983) and Koch et al. (1989) have addressed the same problem using the method of volume averaging, whereby mesoscopic transport coefficients are derived by averaging the basic conservation equations over a single unit cell. Numerical simulations of solute dispersion, based on lattice scale calculations of the Navier-Stokes velocity fields in spatially periodic structures, have also been performed (Eidsath et al., 1983 Edwards et al., 1991 Salles et al., 1993). These simulations are discussed in detail in the Emerging Areas section. 

The ordinary kinetics theory of neuter gas, the Boltzmann equation is considered with collision term for binary collisions and is despised the body s force F . This simplified Boltzmann equations is an integro - differential non lineal equation, and its solution is very complicated for solve practical problems of fluids. However, Boltzmann equation is used in two important aspects of dynamic fluids. First the fundamental mechanic fluids equation of point of view microscopic can be derivate of Boltzmann equation. By a first approximation could obtain the Navier-Stokes equations starting from Boltzmann equation. The second the Boltzmann equation can bring information about transport coefficient, like viscosity, diffusion and thermal conductivity coefficients (Pai, 1981 Maxwell, 1997). 

The kinetic theory of gases attempts to explain the macroscopic nonequilibrium properties of gases in terms of the microscopic properties of the individual gas molecules and the forces between them. A central aim of this theory is to provide a microscopic explanation for the fact that a wide variety of gas flows can be described by the Navier-Stokes hydrodynamic equations and to provide expressions for the transport coefficients appearing in these equations, such as the coefficients of shear viscosity and thermal conductivity, in terms of the microscopic prop>erties of the molecules. We devote most of our attention in this article to this problem. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

The normal solution method just outlined leads to two principal results, both of which can be tested experimentally. These are (i) explicit expressions for the coefficients of viscosity 17 and thermal conductivity A for dilute monatomic gases, in terms of the intermolecular potential < (r), and (ii) an explicit form for the Burnett and higher-order corrections to the Navier-Stokes equation, together with expressions for the associated (higher-order) transport coefficients in terms of the intermolecular potential. 

Here Vq is the adiabatic sound velocity for an ideal gas. The coefficient A is completely determined by the transport coefficients rj and A appearing in the Navier-Stokes equations B depends on these as well as on the additional transport coefficients that appear in the Burnett hydrodynamic equations C depends on all of the transport coefficients in A and B, as well as the super-Burnett transport coefficients, and so on.t... 

Here we will not go through the detailed calculations that lead to the Enskog theory values for the transport coeflicients of shear viscosity, bulk viscosity, and thermal conductivity appearing in the Navier-Stokes hydro-dynamic equations. Instead we shall merely cite the results obtained and refer the reader to the literature for more details. One finds that the coefficient of shear viscosity 17 is given by ... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

Other macroscopic transport equations can be used to derive rela--tions for other transport coefficients as functionals of time-correlation functions. For example, the Navier-Stokes equation can be used to derive a time-correlation function expression for the coefficient of shear viscosity. 

While the gas concentration distribution in the gas charmel is obtained by the solution of the Navier-Stokes equation along with the governing equation for mass species transport, the overall resistance for convective mass transport and convection mass transfer rate from bulk gas stream to the adjacent electrode surface is often given by the convection mass transfer coefficient. For such a case, the convective mass transfer rate equation over a surface of area A can be written as... 

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