Variational principles for nonpotential operators

One presents numerous approaches for the construction of variational principles for equations with operators which, in general, are nonpotential. One considers separately linear and nonlinear ordinary differential equations, partial and integropartial differential equations. One constructs and investigates both extremal and stationary variational principles and one gives applications of these principles in theoretical physics and in analytic mechanics. A series of unsolved problems are indicated. The survey is intended for mathematicians, physicists, working in both theoretical and applied areas, as well as for graduate students of physics and mathematics.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 40, pp. 3–176, 1992.     

Refinement of equations describing longitudinal-radial vibrations of a circular cylindrical viscoelastic shell

Moscow Institute of Construction Engineering. Translated from Prikladnaya Mekhanika, Vol. 26, No. 2, pp. 63–71, February, 1990.     

Periodic solution of a linear functional equation with variable coefficients

Translated from Issledovaniya po Prikladnoi Matematike, No. 17, pp. 46–50, 1990.     

The equicontinuity condition for sequences of solution spaces

The verification of an appropriately stated equicontinuity condition for a sequence of solution spaces is one of the two key points in the theory of the Cauchy problem for equations with singular right-hand sides. We obtain a related sufficient condition. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 416–423, March, 1997. Translated by V. E. Nazaikinskii     

Superheating instability in a pulsed xenon discharge

It is shown that previously obtained conditions for superheating instability substantially vary if we take into account secondary xenon ionization. Instability completely vanishes if the density of heavy particles in the discharge is kept constant and whenever a discontinuous time variation of temperature T in a restricted region between 15 · 10³ ° K and 20 · 10³ ° K is possible for a constant effective pressure. The development of instability is studied numerically by a ranging method. Stationary temperature distributions possessing a high contrast as a local temperature passes through a given range of instability with constant pressure are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 27–30, January–February, 1976.     

Pressure pulsations on an obstacle in a jet

At the present time, much attention is devoted to auto-oscillations that arise from the interaction between a supersonic underexpanded jet and an obstacle that it encounters at right angles [1, 2]. There are far fewer data on the pressure pulsations on an obstacle in the absence of auto-oscillations [3–6]. However, in many cases the highest total levels of the pressure pulsations are observed when the barrier is situated at fairly large distances from the nozzle opening and the pressure pulsations have a random nature. We have investigated the pressure pulsations on a plate normal to a supersonic strongly underexpanded jet. The pulsation characteristics were measured for an arrangement of the obstacle when auto-oscillations are absent. We have established dependences that generalize the results of measurement of the pulsation characteristics at both subsonic and supersonic velocities on the jet axis directly in front of the obstacle. We have also investigated the correlation between the pressure pulsations on the plate and external acoustic noise. We have obtained the dependence of the level of the acoustic noise on the value of the maximal pressure pulsations on the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–167, January–February, 1980.     

Pressure fluctuations and heating of a gas by the inflow of a supersonic jet into a cylindrical cavity

The formation of pressure fluctuations at the inflow of a jet into a cavity (or the so-called resonance tube) was first observed by Hartmann. Further investigations showed that at the same time there is a heating of the gas in the cavity [1, 2]. It was established in [1, 2] that at subsonic and slightly supersonic velocities (M < 2.0) the cavity air can be heated up to 500–700 °K. Further investigations [4, 6] showed that by using monatomic gases inside the cavity one can reach even higher temperatures (T 800–900 °K). The resonance tubes find an application as powerful sound sources. There is also a possibility of their use in thermochemistry, and for the plasma production [6], In the literature, there is an absence of data on the resonance tube characteristics for large Mach numbers. In the present work we investigate the resonance tubes for M = 3.2–4.0. These investigations have shown that pressure oscillations can occur at these Mach numbers with the peak-to-peak amplitude of P 0.4·P_o, where P_o is the total pressure in the inflowing jet. Depending on the clearance between the nozzle and the cavity, both low- and high-frequency oscillations can be set up. It is established that the most intense shock-wave heating of the gas takes place at high-frequency fluctuations, although their amplitude is smaller in comparison with the low-frequency ones. It is shown that the cold air inside the cavity can be heated by means of the fluctuations up to T 1600 °K or more.Translated from Izvestiya Akamemii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–111, September–October, 1977.