Sonine polynomials

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

The Sonine polynomials are related to the associated Laguerre polynomials (see Margenau and Murphy, op. eft., p. 128) by... 

Sink (in graph theory), 258 "Slack variables, 294 Slightly-ionized gases, 46 "Slow time, 362 Small parameter methods, 350 S-matrix, 599,649,692 Smirnova, T. S., 726 Smoluchowski, R., 745 Sokolov, A. V., 768 Sommerfeld, C. M., 722 Sonine polynomials, 25 Source (in graph theory), 258 Space group... 

Enskog obtained a solution by expanding A, (Wj) and 6, (W,) in a finite series of Sonine polynomials [60,114,178]. This solution is usually found to converge in only one or two terms. More discussion of the solution itself can be found in Chapman and Cowling [60] and in Hirschfelder, Curtiss, and Bird [178]. We are concerned here with using the Enskog result to obtain transport coefficients. 

The conventionally quoted result of the exact kinetic theory [5], [9] is that obtained from a first-order expansion in Sonine polynomials, namely. 

Finally, solutions to the integral flux equations like (2.64) and (2.72) are then obtained by expressing the scalar functions A C,n,T) and B C,n,T) in terms of certain polynomials (i.e., Sonine polynomials). 

Chapman and Cowling [12] have shown that in kinetic theory the transport coefficients can be expressed in terms of the Sonine polynomial expansion coefficients which are complicated combinations of the bracket integrals. In the given solutions these integrals are written as linear combinations of a set of these collision integrals. See also Hirschfelder et al [39], sect 7-4. 

The solid curve represents the theoretical value for ri obtained using the Lennard-Jones potential and the first Sonine polynomial approximation,... 

The value of 2A/5t/c is only unity in the first Sonine polynomial approximation. A more complete calculation shows that this quantity differs from unity by a few percent and the deviations depend on temperature as well as on the interaction potential. In Fig. 12, the higher Sonine polynomial corrections have been taken into account for the 11-6-8 potential. 

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. 

The simplest way to determine the effective hard-sphere diameter d of the fluid molecules at temperature T and the corresponding value of x ts to fit exp( the experimental value for the coefficient of shear viscosity at low density, to the theoretical formula for hard-sphere molecules. That is, d is determined by using the first Sonine polynomial approximation to 17 for a gas of hard spheres [Eq. (115a)],... 

In more accurate comparisons, higher-order Sonine polynomials are also taken into account. 

The basic ingredients in the formulation of kinetic models are the matrix elements of the memory function calculated using an appropriate set of momentum basis functions. In the analysis of the Boltzmann collision operator it has been found convenient to use the Sonine polynomials. However, the k r dependence in (124) destroys the rotational symmetry present in the Boltzmann collision operator for a spherically symmetric potential therefore it is equally appropriate to choose the function... 

Finally, solutions to the integral flux equations like (2.189) and (2.197) are then obtained by expressing the scalar functions A C, n, T) and B(C, n, T) in terms of certain polynomials (i.e., Sonine polynomials). However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.267) is used to determine expressions for the viscous-stress tensor a and the heat flux vector q. 

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