Boltzmann equations linearized

Fukui, S., and Kaneko, R., Analysis of Uitra-Thin Gas Fiim Lubrication Based on Linearized Boltzmann Equation First Report—Derivation of a Generalized Lubrication Equation In-ciuding Thermai Creep Fiow," ASME J. Tribal., Voi. 110, 1988,pp.253-262. 

Fukui, S. and Kaneko, R., Analysis of Ultrathin Gas Film Lubrication Based on Linearized Boltzmann Equation First... 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

Figure 2.2 Non-dimensionalized velocity distribution across a channel for Kn = 0.1 (left) and Kn = 2.0 (right), taken from [2]. The results were obtained by DSCM using two different collision cross-sections and by solution of the linearized Boltzmann equation.

Eq. (437) may be transformed into a true transport equation for f this transport equation is the generalized linearized Boltzmann equation for f, as it also appears in the theory of thermal transport coefficients. More precisely, we get ... 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

Before proceeding further, however, it is appropriate to stress a logical inconsistency in working with the unlinearized P-B equation (3.131). The unlinearized Boltzmann equation (3.10) implies a non/mear relationship between charge density and potential. In contrast, the linearized Boltzmann equation (3.16) implies a linear relationship of g,. to y/. ... 

Q.l5.7 What assumption of the Debye-Hiickel model permits the use of the linearized Boltzmann equation Is this assumption valid if ZiCo r/kT = 0.5 0.1 0.01 ... 

A.15.7 The linearized Boltzmann equation can be used if ZiCo r/kT is much smaller than 1. Tliis is because the third term in the Taylor series expansion squares Zieo -kr/kT, making the term negligible if it is very small. If Zieoifr/kT = 0.1, Pr = 1 — 0.1 — 0.005. The third term isn t negligible so tlie assumption wouldn t be valid. If Zieo rr/kT = 0.01, Pr = 1 — 0.01 — 0.00005. The third term is small relative to the first two so the assumption might be reasonable. If ZiSo r/kT = 0.01, p, = 1 - 0.01 - 0.00005. The third term is very small when compared to the first two so the assumption would be reasonable. 

From the DSMC results and solutions of the linearized Boltzmann equation, it is evident that the velocity profiles in pipes, channels and ducts remain approximately parabolic for a large range of Knudsen number. This is also consistent with the analysis of the Navier-Stokes and Burnett equations in long channels, as documented in Ref [1]. Based on this observation, we model the velocity profile as parabolic in the entire Knudsen regime, with a consistent slip condition. We write the dimensional form for velocity distribution in a channel of height h. 

Ohwada T, Sone Y, Aoki K (1989) Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall an the basis of the linearized Boltzmann equation for hard-sphca e molecules. Phys Fluid A 1(9) 1588-1599... 

Substituting Eq. 22 into Eq. 8 the linearized Boltzmann equation is obtained... 

Lorenzani S (2011) Higher order slip accmding to the linearized Boltzmann equation with general boundary conditions. Philos Trans R Soc Lond Ser A 369(1944) 2228-2236... 

It is known that Maxwell s and Knudsen s boundary conditions are not correct in the slip regime. Kuscer s analysis [2.88] of the velocity slip problem using the linearized Boltzmann equation yields ... 

By use of an inner-and-outer expansion procedure for solution of the linearized Boltzmann equation. PAO and WILLIS [2.109] have shown for three dimensional problems that the expansion for any transfer function, such as F, in Knj about KnT = 0 is nonanalytic and has the form... 

Having obtained and solved each equation for 4>, we found that we could always add an arbitrary solution of the homogeneous linearized Boltzmann equation to the particular solution of the inhomogeneous equation. By requir-... 

Another method that has been applied to a number of problems of gas flows is based on a variational principle that can be derived from the linearized Boltzmann equation/ One particularly useful feature of the variational method is that quantities of physical interest such as the drag on an object or the flow rate can be directly related to the stationary point in the variational procedure. One particularly striking application of this method is the computa-tion of the force on a sphere in a gas stream at low Mach numbers for all values of the ratio of the mean free path to the diameter of the sphere. The results are in very good agreement with the experimental results from the Millikan oil drop experiment. ... 

Since Eq. (138a) is a linear equation for it is much easier to solve than the nonlinear equation. In spite of this simplification, it is still difficult to produce explicit solutions to this equation unless the operator L has a simple form and the geometry of the boundaries is simple enough. The operator L can be characterized by its sp>ectrum, and only for Maxwell molecules is this spectrum known explicitly. For this case the spectrum is discrete and the corresponding eigenfunctions can be expressed in terms of Sonine polynomials and spherical harmonics in y/ 8-52.6i-64) however, even for this special potential, it is still difficult to solve the linearized Boltzmann equation. 

If one replaces L in Eq. (138b) by Lbgk> the linearized Boltzmann equation can be solved for a number of interesting cases. The simplest case where Eqs. (138b) and (138c) have been solved completely is the so-called Kramers problem. Here one considers the flow of a gas in a semi-infinite space bounded by a plane wall with which the molecules make diffusive collisions. For this problem one can show that there is a kinetic boundary layer near the wall and that the Chapman-Enskog normal solution is correct for points that... 

In this connection it is worth pointing out that for the Kramers problem, an exact solution to the linearized Boltzmann equation for a BGK model gas has been constructed by Cercignani. Even for this case the answers to questions (a) and (b) are not known. ... 

A. Velocity distribution following a fast neutron pulse. The physical processes occurring in neutron thermalization are best illustrated by considering the velocity distribution of neutrons as a function of time following a burst of fast neutrons at = 0. We consider an infinite homogeneous medium at a uniform temperature T with an absorption cross section varying inversely as the neutron velocity. The linearized Boltzmann equation describing the neutron distribution in velocity and time is... 

Solution of the linearized Boltzmann equation for the slab geometryy Duke... 

Collecting now first-order terms on both sides of equation (9.29) one obtains the linearized Boltzmann equation on which all subsequent discussions will be based, ... 

To proceed further with the solution of the linearized Boltzmann equation one assumes the magnetic field to be sufficiently weak so that the electrons undergo collisions long before they can complete a cyclotron orbit. Then one can regard the solution for the case B = 0 as slightly perturbed by the addition of the magnetic field and hence write down a power-series expansion of (P — M) in the form... 

The linearized Boltzmann equation can be obtained assuming that the unknown distorted distribution function f y,r) deviates only slightly from the local Fermi distribution function /o(e, r), i.e. if we neglect higher-order terms in f(y, r) — /o( , r). 

The linearized Boltzmann equation for electrons in a metal under a uniform electric field E and a uniform thermal gradient VT is given by ... 

Several methods have been developed to solve the linearized Boltzmann equation. The simplest is the relaxation-time approximation. This procedure is based on the assumption that the collision term can be replaced by ... 

The previous section concerned the application of the Boltzmann formalism. An exact solution of the linearized Boltzmann equation (eq. (11)) is very difficult. However, there are many examples of calculations which are in good agreement with experiment. 

For the analysis of these data Gratz and Nowotny (1985) used the Nordheim-Gorter rule, which is, in the case of a non-magnetic compound like YAI2, given by S T) = (pJp)Sq + (p Jp)S. The temperature variation of p and,.and the value of po re taken from the resistivity measurements. A calculation ofSo andSp within the scope of the linearized Boltzmann equation revealed that Sq is linear in... 

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