The Kramers Problem for Quantum Fermi Gases with Velocity--Dependent Collision Frequency and Diffusive Boundary Conditions

Source: arXiv

ABSTRACT The classical Kramers problem of the kinetic theory is analytically solved.
The Kramers problem about isothermal sliding for quantum Fermi gases is
considered. Quantum gases with the velocity-dependent collision frequency are
considered. Diffusive boundary conditions are applied. Dependence of isothermal
sliding on the resulted chemical potential is found out.

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    ABSTRACT: A new method for treating boundary-value problems in gas-kinetic theory has been developed. The new method has the advantage of reproducing the bulk or asymptotic flow properties accurately whilst giving a realistic description of the behaviour of the molecular distribution function in the neighbourhood of a wall. As an example, the Kramers, or slip-flow, problem is solved for a general specular-diffuse boundary condition and some new expressions for the slip coefficient, flow speed and molecular distribution function at the surface are derived.
    Journal of Fluid Mechanics 03/1969; 36(01):145 - 159. · 2.29 Impact Factor
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    ABSTRACT: The method of elementary solutions previously introduced for treating problems in linearized kinetic theory of gases is extended to a general class of models with velocity-dependent collision frequency.The previous treatment based on the Bhatnagar, Gross, and Krook model is a particular case of the present theory.The paper is devoted to steady shear flow problems. As an application the Kramers problem is exactly solved in terms of quadratures without specifying the model. As a consequence a formula for the dependence of the slip coefficient on the collision frequency is given.
    Annals of Physics - ANN PHYS N Y. 01/1966; 40(3):469-481.
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    ABSTRACT: The slip coefficient for a gas is computed for certain typical forms of the collision frequency. The computations are made by a numerical quadrature based on an exact formula and also by a variational procedure. There is a good agreement between the results obtained by both methods and between them and previously known results. Our calculations confirm that the slip coefficient is only slightly model dependent. It is also shown that, for any form of the collision frequency, the slip-coefficient dependence on the accommodation coefficient for tangential momentum can be found analytically, in terms of the value corresponding to complete accommodation. Si calcola il coefficiente di slittamento di un gas per particolari forme della frequenza degli urti intermolecolari. I calcoli sono fatti sia mediante una quadratura numerica basata su una formula esatta che con un procedimento variazionale. C’è un buon accordo tra i risultati ottenuti con i due metodi e risultati noti in precedenza. I nostri calcoli confermano che il coefficiente di slittamento dipende poco dal modello. Si mostra anche che, per ogni forma della frequenza d’urto, la dipendenza del coefficiente di slittamento dal coefficiente di accomodamento della quantità di moto trasversale si può determinare analiticamente, in termini del valore corrispondente a un accomodamento completo. Вычисляется козффициент скольжения для гаэа для некоторых типичных форм частоты соударений. Проводятся вычисления посредством численной квадратуры, основанной на точной формуле, и также посредством вариационной процедуры. Получается хорощее согласие между реэультатами, полученными обоими методами и между зтими реэультатами и предварительно иэвестными реэультатами. Нащи вычисления подтверждают, что козффициент скольжения очень слабо эависит от модели. Также покаэывается, что для любой формы частот соударений эависимость козффициента скольжения от козффициента аккомодации для тангенциального момента может быть найдена аналитически в терминах величины, соответствуюшей полной аккомодации.
    Il Nuovo Cimento B 01/1968; 57(2):297-306.

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May 28, 2014