Carlo Cercignani

Variational Approach to Boundary‐Value Problems in Kinetic Theory

A general variational principle applying to kinetic models is introduced. Then the principle is applied to three typical problems, i.e., Kramers problem, plane Couette flow, and plane Poiseuille flow. With extremely simple trial functions, very accurate results are found.

The method of elementary solutions for kinetic models with velocity-dependent collision frequency

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.

Flow of a Rarefied Gas past an Axisymmetric Body. II. Case of a Sphere

The drag exerted by the flow on a sphere is explicitly calculated by using the relation between the drag and a certain functional. A comparison of the results with the experimental data of Millikan gives excellent agreement.

Comparison of kinetic theory analyses of linearized heat transfer between parallel plates

Abstract Linearized heat transfer between two parallel plates is considered for inverse Knudsen numbers ranging from 0 to 10. The Bhatnagar, Gross and Krook model is used and transformed into a couple of integral equations for density and temperature. These equations are solved numerically. Besides a variational calculation of the solution is made by introducing simple trial functions in a suitable variational principle. The results obtained for the heat flux through the two methods are compared and found in strict agreement (discrepancy less than 0·5 per cent). For the limiting case of the half-space problem, the temperature jump coefficient is evaluated both by a numerical and a variational procedure. The latter gives a value differing by about 0·5 per cent from the value given by the former procedure and of about 1 per cent from the value previously obtained by Welander. Comparisons are made with the results of the non-linear analysis of Willis, the linearized four moment solution and Takaos experimental data.

Influence of the accommodation coefficient on the heat transfer in a rarefied gas

Abstract Linearized heat transfer between parallel plates and concentric cylinders is considered for inverse Knudsen numbers ranging from 0 to 10, and arbitrary accommodation coefficients. The Bhatnagar, Gross and Krook model is used and transformed into a system of integral equations. These equations are solved by a variational method, previously applied to simpler situations. Comparisons are made with experimental results and previously available calculations.

Unsteady solutions of kinetic models with velocity-dependent collision frequency☆

Abstract The method of elementary solutions recently extended to treat steady problems with kinetic models with velocity-dependent collision frequency is now extended to cover also time-dependent problems. The theory is somewhat different from the previous one holding for the Bhatnagar, Gross, and Krook model, since the continuous spectrum of space transients covers now a two-dimensional region of the complex plane. Consequently, in order to solve explicitly half-space problems we are not faced with one-dimensional singular integral equations but two-dimensional equations with integrable kernel. This circumstance does not make it possible to use the Muskhelishvili techniques which were used in previous papers. However, a connection of the present theory with the theory of generalized analytic functions is shown, and, on this basis, an analytic method of solving the relevant equations is constructed. The present theory would be easily applied to typical time-dependent problems, such as the Rayleigh problem, the oscillating wall, and the evolution of an initial discontinuity.

New boundary conditions in the transition régime

Starting from the Boltzmann equation, new boundary conditions are derived to be matched with the Navier—Stokes equations, that are supposed to hold in the main body of a gas. The idea upon which this method is based goes back to Maxwell and Langmuir. Since the distribution function is supposed to be completely determined by the Navier—Stokes equations, this new set of boundary conditions extends in some sense the validity of the macroscopic equations to the transition and free molecular regimes. In fact, it is shown that the free molecular and slip flow regimes are correctly described by this method; the latter is also supposed to give a reasonable approximation for the complete range of Knudsen numbers. The new procedure is applied to different problems such as plane Couette flow, plane and cylindrical Poiseuile flow, heat transfer between parallel plates and concentric cylinders. Results are obtained and compared with the exact numerical solutions for the above-mentioned problems.

Linear Representation of Spinors by Tensors

A linear representation of spinors in n‐dimensional space by tensors is proposed. In particular, in three‐dimensional space a set composed by a scalar and a vector is associated to any two‐component spinor, while in four‐dimensional space the set corresponding to a four‐component spinor is composed by a scalar, a pseudoscalar, a vector, a pseudovector, and an antisymmetrical tensor of second order. The resulting formalism is then applied to Schrodingers and Diracs equations. In three‐dimensional space it turns out that the proposed procedure automatically assigns an intrinsic magnetic moment to an electron in a magnetic field without introducing any relativistic ideas or ad hoc assumptions. In four‐dimensional space we can write the Dirac equation in a generally covariant fashion, without introducing new concepts with respect to the usual tensor analysis. The zero‐mass Dirac equation splits into two sets of equations, describing respectively the neutrino and the photon. The possible bearing of the propo...

Wall and collision effects in plasma capacitors

SummaryThe problem of finding the electric field within a plasma-filled plate condenser, upon which an alternating e.m.f. is imposed, is considered. The adopted model implies the solution of the Boltzmann-Vlasov system of equations with a single relaxation time collision term, matched by suitable boundary conditions at the walls of the condenser. The plasma is assumed to be completely ionized and the frequency large enough to consider negligible the ion motions. In order to solve the Boltzmann-Vlasov system the method of separating the variables is used. Firstly a general mathematical theory of such solutions is developed, then applications to the plasma capacitor are considered. Both diffusing and reflecting walls are considered. In the limiting cases of large and small separation of the plates the effective permittivity is evaluated.RiassuntoSi considera il problema di valutare il oampo elettrico che si stabilisce tra le armature di un condensatore a facca piane parallels, pieno di plasma, quando una f.e.m. alternata viene applicata alle facce. II modello adottato richiede la soluzione del sistema di Boltzmann-Vlasov con termine di collisione a rilassamento unico, accompagnato da opportune condizioni al contorno. Si suppone che il plasma sia completamente ionizzato e la frequenza della f.e.m. abbastanza grande cosicché si possa trascurare il moto degli ioni. Per risolvere le equazioni di Boltzmann-Vlasov si adotta il metodo della separazione delle variabili. Innanzitutto si sviluppa una teoria matematica generale delle soluzioni elementari; quindi se ne considerano le applioazioni al condensatore a plasma. Si prendono in considerazione entrambi i casi di pareti diffondenti e riflettenti. Nei casi limite in cui la distanza tra le armature è molto grande o molto piccola si valuta la permittività efficace.

Diffusion of a velocity discontinuity according to kinetic theory

SommarioSi risolve analiticamente per mezzo del modello linearizzato di Gross, Bhatnagar e Krook dellequazione di Boltzmann, il problema dellevoluzione temporale di una discontinuità iniziale di velocità.La soluzione che così si ottiene viene confrontata con precedenti soluzioni approssimate e con la soluzione ottenuta dalle equazioni di Navier-Stokes che descrivono macroscopicamente il fluido.SummaryThe problem of describing the smoothing out and diffusion of an initial velocity discontinuity is solved analytically by means of the linearized Bhatnager, Gross and Krook model of the Boltzmann equation. The solution is compared with previous approximate solutions and with the solution arising from a continuum treatment based on Navier-Stokes equations.