Invariant transformations in non-steady gasdynamics and magneto-gasdynamics

Zusammenfassung Es werden adjungierte Beziehungen für gewisse nichtstationäre gasdynamische und magnetogasdynamische Strömungen angegeben. Auch eine allgemeine Formulierung solcher Transformationen wird beschrieben. Dadurch werden neue Transformationen erzeugt.     

Reciprocal relations between cyclotomic fields

We describe a reciprocity relation between the prime ideal factorization, and related properties, of certain cyclotomic integers of the type ?n(c−ζm) in the cyclotomic field of the m-th roots of unity and that of the symmetrical elements ?m(c−ζn) in the cyclotomic field of the n-th roots. Here m and n are two positive integers, ?n is the n-th cyclotomic polynomial, ζm a primitive m-th root of unity, and c a rational integer. In particular, one of these integers is a prime element in one cyclotomic field if and only if its symmetrical counterpart is prime in the other cyclotomic field. More properties are also established for the special class of pairs of cyclotomic integers q(1−ζp)−1 and p(1−ζq)−1, where p and q are prime numbers.     

General relations for waves in one-dimensional elastic systems

Common features inherent in waves propagating in one-dimensional elastic systems are pointed out. Local laws of energy and wave momentum transfer when the Lagrangian of an elastic system depends on the generalized coordinates and their derivatives up to the second order inclusive are presented. It is shown that in a reference system moving with the phase velocity, the ratio of the energy flux density to the wave momentum flux density is equal to the phase velocity. It is established that for systems, the behaviour of which is described by linear equations or by nonlinear equations in the unknown function, the ratio of the mean values of the energy flux density to the wave momentum density is equal to the product of the phase and group velocities of the waves.     

Reciprocal relations of Bernoulli and Euler numbers/polynomials

By means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921–930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149–173.     

Integral methods in rarefied gasdynamics

The question of convergency in integral methods for rarefied gasdynamics is examined through the example of low-speed flow over a sphere. The four-moment solution and the six-moment solution both compare favorably with experimental data for drag and heat transfer over the entire range of gas density. Discrepancy between the two solutions show a maximum of 0.6% in drag coefficient and a difference of the order of Mach number squared in heat transfer.

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Zusammenfassung Die Konvergenzfrage bei Integralmethoden für die Dynamik verdünnter Gase wird untersucht anhand des Beispiels der Kugelumströmung bei niedrigen Geschwindigkeiten. Four-moment und six-moment Lösungen sind in guter Übereinstimmung mit Versuchsdaten für Widerstand and Wärmeübergang im gesamten Dichtebereich. Die Diskrepanz zwischen beiden Lösungen ist maximal 0.6% für den Widerstandsbeiwert und von der Grössenordnung des Quadrats der Machzahl für den Wärmeübergang.

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This paper is a token contribution in honor of Professor N. Rott who has been a teacher to the author beyond rarefied gasdynamics.     

High frequency waves in quasi-linear inviscid gasdynamics

The lack, in non-linear systems, of a principle of superposition has motivated the development of the method of the multiphase asymptotic waves [1]. This technique permits one to resolve a complete initial value problem i. e. to study the evolution of an initial perturbation along all the characteristic rays. In this paper a concrete application to the inviscid isentropic gas is carried out up to the second order of approximation. The main effects of non-linear superposition are pointed out. Excellent agreement is found between the approximated solutions and the exact solutions obtainable in some particular cases.

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Riassunto La mancanza, nei sistemi non lineari, di un principio di sovrapposizione ha motivato lo sviluppo del metodo delle onde asintotiche a più fasi [1]. Tale metodo permette di risolvere un problema ai valori inziali completo, cioè di studiare l'evoluzione d'una perturbazione iniziale lungo tutti i raggi caratteristici. In questo lavoro viene presentata un'applicazione al gas isentropico non viscoso in cui gli sviluppi asintotici sono troncati al secondo ordine. Vengono inoltre messi in evidenza gli effetti della sovrapposizione non lineare. Un confronto tra le soluzioni approssimate e le soluzioni esatte che si hanno in due casi particolari conferma la validità del metodo.

     

Weakly nonlinear interaction of two waves in radiative gasdynamics

The interaction of two weakly nonlinear waves in radiative gasdynamics is investigated following the perturbation analysis developed by He and Moodie. Explicit solutions are given and the two waves interactions are graphically shown.     

Exact solution of planar and nonplanar weak shock wave problem in gasdynamics

In the present paper, an analytical approach is used to determine a new exact solution of the problem of one dimensional unsteady adiabatic flow of planer and non-planer weak shock waves in an inviscid ideal fluid. Here it is assumed that the density ahead of the shock front varies according to the power law of the distance from the source of disturbance. The solution of the problem is presented in the form of a power in the distance and the time.     

Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis

In this paper, we explicitly characterize a class of solutions to the first order quasilinear system of partial differential equations (PDEs), governing one dimensional unsteady planar and radially symmetric flows of an adiabatic gas involving shock waves. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system, whose simple solutions provide non trivial solutions of the original system. It is interesting to remark that one of the special solutions obtained here, using this approach, is precisely the blast wave solution known in the literature.     

Reciprocal Polynomial Extrapolation

An alternative to the classical extrapolations is proposed. The stability and the accuracy are studied. The new extrapolation behaves better than the classical ones when there are problems of stability. This technique will be useful in those problems where the region of stability is very small and it forces to work with too fine scales.     

The non-steady flow in a magnetohydrodynamic shock-wave generator of finite length

Zusammenfassung Durch geeignete Transformationen der diese Vorgänge beschreibenden Grundgleichungen in differentieller Form, Impulssatz, Kontinuitätsgleichung und Energiesatz, können sogenannte Verträglichkeitsbedingungen erhalten werden. Diese gelten entlang den Charakteristiken beziehungsweise den Lebenslinien. Letzteres System von gewöhnlichen, nichtlinearen Differentialgleichungen darf linearisiert werden, sofern die elektromagnetischen Wechselwirkungen mit der Strömung schwach sind. Elementare Integrationen führen dann zu relativ übersichtlichen und einfachen Resultaten. Insbesondere sind die unbekannten Funktionen an drei ausgezeichneten Stellen des -t-Feldes explizite zu bestimmen und die Funktionswerte im übrigen Strömungsfeld lassen sich leicht graphisch ermitteln. Die stromaufwärts reflektierten Druckschwankungen, verursacht durch die elektromagnetische Wechselwirkung im Generator, können nach einer von den Betriebsbedingungen abhängigen Anlaufstrecke zu einem Stoss auflaufen. Die exakte Bestimmung dieser Anlaufstrecke ist im Rahmen der Linearisierung auch möglich. Es ist erstaunlich, wie rasch, gemessen an der Zeit, die die Stosswelle zum Durchlaufen des Generators braucht, die Strömung im Generator stationär wird.     

A non-steady flow of liquid in a porous pipe with variable permeability

LetB denote the infinitesimal operator of a strongly continuous semigroup S(t), with resolvent R_λ, on Banach space L. We define related operators P and V so that λR_λf = Pf + λVf + o(λ), as λ → 0⁺. For α, η > 0 and possibly unbounded, linear operator A, we let U_{α, η}(t) represent a strongly continuous semigroup generated by αA + ηB. We show that under appropriate simultaneous convergence of α and η, U_{α, η}(t) converges strongly to a strongly continous semigroup U(t), having infinitesimal operator characterized through PA(VA)^rf where r =min{j ? 0, PA(VA)^j ≠ 0}. We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations.