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BGK Model

TH eBGK model is used in molecular regime with the hypothesis of hard sphres for molecule. It is is a simple approach for vacuum calculations

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5 views15 pages

BGK Model

TH eBGK model is used in molecular regime with the hypothesis of hard sphres for molecule. It is is a simple approach for vacuum calculations

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jp.perin52
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VOLUME 94, NUMBER 3 MAY 1, 1954 A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems* P. L. Buamuaoans E. P. Gross,t ax M. Keoox Astrowmy Department, Hareord Unisersty, Combidge, Massachusds ond Laboratory for Insulation Research, ‘Massachusetts Intute of Fcinolgy, Cambridge, Massechuats (Received November 27,1953) ‘A kinetic theory approach to cllsion processes in ionized and ‘neural gases is presented. This approach is adequate for the uni fed treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen Init to the high-pressure Tint where the aerodynamic equations are valid. Itisalso possible to satisfy the correct. microscopic boundary” conditions. The ‘method consists in altering the eolison terms in the Boltzmann equation. The modified colison terms are constructed. so that ‘ac calisin conserves partiele number, moment, and eneTgy bother characteristics such as persistence of veloities and angular Gependence may be included. ‘The present article lustrates the technique for a simple model involving the seeumption of = Callison time independent of velocity this model i applied to the study of small amplitude ocilations of one-component ionized and neutral gases. The initial value problem fo unbounded space {s solved by performing a Fourier transformation on the spuce 4, INTRODUCTION pi nea sats oak Sten at qa eaeanas sre fe tae tribution function f(v,r,!) is defined by the condition Snr eia tet tarn i Pade iinet he Taal gi a teal sniaat tea I fae ay coins get atte he eat ze let vey Se EAS CS ane, SHRSElaprnta tnd G. Coming, The Molomai! Thay of owe Gases (Cbg Utara Pre Leon, 82), Bd en Comber variables and a Laplace transformation on the time vaviable. For uncharged gases there results the correct adiabatic limiting law for sound- ave propagation at high pressures and in adlton, fone obtains a theory of absorption and dispersion of sound for arbitrary pressures. For ionized gases the dference in the nature ofthe organization in the low-pressure plasma oscillations and in high-pressure sound-type osllations is studied. Two important ‘asta are dlstingwlahed. Ifthe wavelengths ofthe onilatons are Jong compared to either the Debye length or the mean fee path, ll change in frequency is obtained asthe ealson frequency varies from aero to infinity. The accompanying absorption is tlt reaches its maximm value wen the callisfon frequency fsquals the plasma frequency. The second case refers to waves Shorter than both the Debye length and the mean fee path; these waves are characterized by a very heavy absorption, which the macroscopic theory does not give a correct description, During recent years interest in such prob- Jems has increased considerably. An extreme example of the breakdown of the aerodynamic theory, vi. in the Knudsen region, has been known for some time; here the mean free path of a molecule is large compared to some linear dimension of the apparatus, and behavior at the boundaries becomes important. Other examples are provided by high-frequency sound waves ina rarefied gas and by plasma oscillations. ‘The solution of the Boltzmann equation is, in general, ‘4 matter of considerable difficulty even in cases corre- sponding to the physically simplest situations. Signifi ‘ant progress has been confined practically to the study of two limiting cases in which two different approxima- tion procedures can be applied. A criterion for the range of validity of the approximate methods is pro- vided by the comparison of some characteristic time 1 (or characteristic length 7.) for the relevant process with the average time 7. (or mean free path L,) be- tween molecular collisions. For high density (7>re or 12>L.) the Enskog- ‘Chapman (E-C) theory may be used. The frst approxi- mation of the theory consists in assuming local thermo- dynamic equilibrium and a common drift velocity for all molecular species. The next approximation corrects the distribution function by terms proportional to the first derivatives of temperature, velocity, and density; this corresponds to the aerodynamic equations with coeficients of heat conduction, viscosity, and diffusion, ‘The high-density region (72>r.) isin fact the range in which the aerodynamic equations provide an adequate description. Higher approximations of the E-C theory lead to correction terms proportional to higher deriva- sit 512 tives of 7, q, p. The successive approximations of the E-C theory correspond to an expansion of the distribu- tion function in powers of the mean free path L.. If we consider sound waves with wavelength Z>>L the first and second approximations are already sufficient 10 sive all significant features of the process, However, ‘when L becomes comparable with Z,, it is necessary to go to the third and even higher approximations to obtain adequate results; as discussed by Herzfeld? important dynamic contributions appear only in ap- proximations higher than the second. The third approxi- mation already involves formidable labor and has been used to solve only the simplest problems. Consideration of higher approximations is, in any case, of doubtful value as the entire procedure breaks down in just the range where the contributions from these higher-order terms become important. In addition, the boundary conditions in many problems cannot be specified properly within the scope of the E-C formalism. A different approach, using expansions in terms of Hermite polynomials in velocity space, has been given by Grad. He uses some moments of low order in addi tion to the usual ones representing p,q, and 7. The procedure involves a gain in simplicity over the E-C theory but is still quite complicated. Tt is basically a high-density theory and is capable of dealing only with boundary conditions which can be specified in terms of the moments appearing in the theory. The limitations of any theory based on the use of a finite number of ‘moments will be brought out in the discussions of this series of papers. ‘The opposite limiting case (rr, or L

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