Sonin

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 polynomials13 of order n, index l +, and variable g1 are defined as ... 

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

Polarization wave function of photon, 557 Polynomials expansion, 25 Sonine, 25... 

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... 

Blinova SL, Ivanova ES. Culturing the nematode-bacterial complex of Neoaplectana carpocapsae in insects. In Sonin MD, editor. Helminths of insects. New Delhi Amerind Publishing 1987. 

T. G., Pavlova V.E., and Shmakov S.G. Valuable comments were given by Kostenko N.P., Plyashkevich L.N., Yakymenko E.L., and others. Sonin, l.l., who was chief of the team at Aerogeologia and Solovov A.P. gave invaluable help. Thanks to all who assisted. The second author participated in works in 1981-1983. 

Baveja R, Kresge N, Bian X, Yokoyama Y, Sonin NV, Clemens MG, Zhang JX, Hepatology 1999, 30,232A. 

Sonin, A. A., Osmosis and Ion Transport in Charged Porous Membranes A Macroscopic Mechanistic Model. In Charged Gels and Membranes I (E. Selegny, Ed.) D. Reidel, Dordrecht, Holland, 1976. 

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. 

Belaykov, V. A., Sonin, A. S. Optica kholestericheskikh zidkih kristallov (Optics of cholesteric liquid Crystals), Moscow, Nauka, 1982... 

As shown in Table III, several authors (Fidaleo and Moresi, 2005a Kraaijeveld et al., 1995 Kuroda et al., 1983 Sonin and Isaacson, 1974) established power function relationships between the Sherwood number (Sh) and the Reynolds (Re) and Schmidt (Sc) numbers in ED cells equipped with different eddy promoters, even if different definitions of the equivalent diameter were used to calculate the Reynolds number. 

Sonin, A.A. and Isaacson, M.S. 1974. Optimization of flow design in forced flow electrochemical systems with special application to electrodialysis. Ind. Eng. Chem. Process Des. Develop. 13, 241-248. 

Huber P.W., and Sonin, A.A., A model for the electric charging process in fuel filtration, AIChE Loss Prevention, 9,127-129,1975. [Note this paper is a condensed version of 124.]... 

Huber P.W., and Sonin, AA, Comments on Gavis and Wagner s correlation for the charging current generated during liquid hydrocarbon filtration, Chem. Eng. Sci. 32, No. 3, 331-333, 1977. 

The phenomenon has been studied by Gao and Sonin [16] and it has been shown that the cross-sectional area and apparent contact angle of a printed bead is a function of the substrate temperature and the print-scan velocity. A quantitative analysis of jet-printed phase change materials was reported by Schiaffino and Sonin [17] for isothermal conditions. The results showed that the contact angles for the printed drops could be varied as a function of the substrate temperature. By adjusting the substrate temperature and the scanning speed, the feature size and print quality can be adjusted without having to modify the substrate surface energy. 

M.D. Arney, G. Zaccai, E.K. Achter, E.J. Burke, G. Miskolczy and A.A. Sonin. Airsampling Apparatus with Easy Walk-in Access, US patent no. 4896547. 1990. Availabele at http /www.fr eepatentsonline.com/4896547.html. 

The authors are greatly indebted to M. Ballard, J. Bird, N. Kajitani, and R. Sonin for the samples used in this study. This project was sponsored by the National Museum Act (administered by the Smithsonian Institution). 

Chiou J, Matsudaira P, Sonin A, Ehrlich D. Anal Chem 2001 73 2018-2021. 

---