Higher-order hydrodynamic equations

The boundary conditions to be used with the Burnett equations have also been determined for a BGK model by Sone, and for more general models by de Wit using variational methods, but in fact this set of boundary conditions is not complete. The Burnett and higher-order hydrodynamic equations have nonphysical solutions showing spatial variations on the length scale of the mean free path. One would like to have boundary conditions that could be used to reject these unphysical solutions. However, the available set of boundary conditions is not sufficient for that. Instead one must postulate that the rapidly varying solutions are absent and then use the available boundary conditions to determine the remaining hydrodynamic solution. 

In most cases of physical interest the higher-order hydrodynamic equations give only a small improvement, if any, over the Navier-Stokes equations. However in Section 2.3.3 we will discuss one case, sound propagation, where the Burnett and higher-order equations do successfully improve the description of experimental results. ... 

Integral methods include all those which attempt solution of the moment equation (Maxwell s equation of transfer) of the Boltzmann equation. The well known integral methods which have been applied include Mott-Smith s bimodal distribution [2.100], Grad s 13 moment equations [2.101], Lees two-stream Maxwellian [2.102], and Waldmann s higher-order hydrodynamics and boundary conditions [2.103]. 

It can be seen from Fig. 1 that gas flows in micron size channels are typically relevant to the slip flow regime, at any rate for usual pressure and temperature conditions. For lower sizes, i.e., for Knudsen numbers higher than 10 , the slip flow regime could remain valid, provided that classical velocity slip and temperature jump boundary conditions are modified (taking into account higher-order terms as explained below) and/or that Navier—Stokes equations are extended to more general sets of conservation equatiOTis, such as the quasi-gasodynamic (QGD), the quasi-hydrodynamic (QHD), or the Burnett equatiOTis [3]. 

It is clear that the general procedure used to derive the Navier-Stokes equations can be used to obtain the corrections to the hydrodynamic equations to higher order in the uniformity parameter. The order equations— the... 

These are called slip boundary conditions, since the gas slips along the walls. tAdditional boundary conditions are needed for these higher-order equations, since they involve third- and higher-order spatial gradients of the hydrodynamic variables. 

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,... 

What is the structure of the linear and of the nonlinear hydrodynamic equations for three-dimensional systems Do the apparent divergences of the Burnett and higher-order transport coefficients in the linearized equations mean simply that nonlocal— but linear—effects must be taken into account, or must nonlinear effects be taken into account before one gets a completely well-behaved theory Moreover, almost nothing is known about the theory of nonlinear hydrodynamic equations for dense gases. This is an area that certainly needs to be explored both from a theoretical viewpoint, and also with an eye toward suggesting experiments by means of which one can test the theories. 

The above treatment using Stokes law applied only to very dilute suspensions (volume fraction < 0.01). For more concentrated suspensions, the particles no longer sediment independent of each other and one has to take into account both the hydrodynamic interaction between the particles (which applies for moderately concentrated suspensions) and other higher order interactions at relatively high volume fractions. A theoretical relationship between the sedimentation velocity v of non-flocculated suspensions and particle volume fraction has been derived by Maude and Whitmore [87] and by Batchelor [88]. Such theories apply to relatively low volume fractions (< 0.1) and they show that the sedimentation velocity i at a volume fraction rj) is related to that at infinite dilution (the Stokes velocity) by an equation of the form... 

The higher order dependence on for spheres, in eq. 10.2.15, is shown in Figure 10.2.3. This departure from the Einstein equation is due to hydrodynamic interactions between spheres and to other interparticle forces. We will examine these effects in Section 10.4, but first we look at the influence of particle shape on the rheology of dilute suspensions. 

Equation 10.5.2 fits available data (see Figure 10.2.3 and de Kruif et al., 1985) within measurement accuracy. Higher order expansions do not seem to be usefiil because they are applicable over increasingly small concentration regions. Various approaches are being used to compute viscosities at higher concentrations. The hydrodynamics for multiple particle interactions become very involved. They have been studied mainly by simulation (e.g., Brady and Bossis, 1988 Phillips et al., 1988). Other workers have used an approach based on nonequilibrium thermodynamics (Russel and Cast, 1986). Finally, Woodcock (e.g., 1984) uses molecular dynamics simulations, ignoring the medium viscosity, to calculate the flow-induced structure and then the viscosity. 

Dynamically raised processes in the dispersion, such as Brownian molecular motion, cause variations in the intensities of the scattered light with time, which is measured by PCS. Smaller the particle, higher the fluctuations by Brownian motion. Thus, a correlation between the different intensities measured is only possible for short time intervals. In a monodisperse system following first-order kinetics, the autocorrelation function decreases rather fast. In a half logarithmic plot of the auto correlation function, the slope of the graph enables the calculation of the hydrodynamic radius by the Stokes-Einstein equation. With the commercial PCS devices the z-average is determined, which corresponds to the hydrodynamic radius. 

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