Near-resonance highly nonlinear gas oscillations in tubes

Near-resonance highly nonlinear ideal perfect gas oscillations in tubes are studied numerically for boundary conditions of various types. The oscillations are initiated by weak periodic perturbations at one end of the tube. As distinct from earlier studies [1–10], the oscillation amplitudes were not assumed to be small and the entropy increase at the shock waves formed was taken into account. Periodic flow regimes result as a limit of the solution of a Cauchy problem for one-dimensional time-dependent gasdynamic equations. The frequency responses of the oscillations under consideration are determined for boundary conditions of various types. It is shown that in specific cases the attainment of a periodic regime is accompanied by the appearance of long-wave modulations. The “repeated resonance” effect is revealed. This is due to the change in the tube's natural acoustic frequency, which takes place during the heating of the gas in the tube by the shock waves traveling in it. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–157, July–August, 1994.     

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRⁿ{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRⁿ{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .     

Modeling jet flows caused by the incidence of a shock wave on a profiled free surface

The problem of the incidence of a shock wave with a front-pressure amplitude of about 30 GPa at the profiled free surface of an aluminum sample is studied. It is shown that in the case of large perturbations (amplitude 1 mm and wavelength 10 mm), jet flows occur on the free surface. The data obtained are described using a kinetic fracture model that takes into account the damage initiation and growth in the material due to tensile stress and shear strain. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 16–23, January–February, 2007.     

A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

This paper is devoted to the persistence of periodic orbits under perturbations in dynamical systems generated by evolutionary equations, which are not smoothing in finite time, but only asymptotically smoothing. When the periodic orbit of the unperturbed system is non-degenerate, we show the existence and uniqueness of a periodic orbit (with a minimal period near the minimal period of the unperturbed problem) by using “modified” Poincaré methods. Examples of applications, including the perturbed hyperbolic Navier–Stokes equations, systems of damped wave equations and the system of second grade fluids, are given.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

Thermovibrational convective instability of mechanical equilibrium of an inclined fluid layer

The convective instability of mechanical equilibrium of an inclined plane layer of fluid developing under the action of a static gravity field and high-frequency vibration is studied. Configurations corresponding to four directions of the equilibrium temperature gradient — vertical, longitudinal, horizontal, and transverse — are considered for an arbitrary orientation of the vibration axis. The stability limits and the characteristics of the critical perturbations are determined. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 8–15, January–February, 1998. This investigation was carried out with partial support form RSA-NASA (contract No. 920/18 — 5208/96).     

Calculation of stagnation streamline quantities in hypersonic blunt body flows

An approximate method for the efficient calculation of stagnation-streamline quantities in hypersonic flows about spheres or cylinders is suggested. Based on the local similarity of the flow field the two-dimensional Navier-Stokes equations are simplified to a one-dimensional approximation for the stagnation streamline. These equations are solved with an implicit finite-volume scheme. Comparisons with fully two–dimensional Euler and Navier–Stokes calculations for flows about spheres are presented, that include perfect gas flows and flows in chemical non-equilibrium. Comparisons with a number of experiments conclude this report. Received 8 May 1996 / Accepted 31 October 1996     

Instability of helical magnetohydrodynamic flows

The problem of the linear stability of a single particular class of helical steady-state flows of an ideal incompressible infinitely-conducting fluid in a magnetic field is studied. A necessary and sufficient condition of stability of this class of flows with respect to perturbations of the same symmetry type is obtained by the direct Lyapunov method [1, 2]. A priori two-sided exponential estimates of the perturbation growth are derived, the corresponding exponents being calculated using the steady flow parameters and the initial data for the perturbations. A class of the most rapidly growing perturbations is identified and an exact formula for determining their growth rate is obtained. An example of steady-state flows and initial perturbations whose linear stage of development with time can be described by means of the estimates obtained is constructed. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–156, January–February, 1999. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-01771).     

Conditions for the unique solvability of the initial-value problem for linear second-order differential equations with argument deviations

We establish new conditions under which the initial-value problem for a system of linear second-order differential equations with argument deviations has a unique solution, which depends monotonically on additive perturbations of the problem. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 535–547, October–December, 2006.     

Spectral methods based on the least dissipative modes for wall bounded MHD flows

We present a new approach for the Spectral Direct Numerical Simulation (DNS) of Low-Rm wall-bounded magnetohydrodynamic (MHD) flows. The novelty is that instead of using bases similar to the usual Chebyshev polynomials, which are easy to implement but incur heavy computational costs to resolve the Hartmann boundary layers that arise along the walls, we use a basis made of elements that already incorporate flow structures such as anisotropic vortices and Hartmann layers. We show that such a basis can be obtained from the eigenvalue problem of the linear part of the governing equations with the problem’s boundary conditions. Since this basis is not always orthogonal, we develop a spectral method for non-orthogonal bases. We then demonstrate the efficiency of this method on the simple case of a laminar channel flow with periodic forcing. In particular, we show that this method eliminates the computational costs incurred this Hartmann layer, and this for arbitrary high magnetic fields B. We then discuss the application of our method to nonlinear, turbulent flows for which the number of modes required to resolve the flow completely decreases strongly when B increases, instead of increasing as in the case of currently employed Chebyshev-based methods.     

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.     

Stability of Creeping Flows of Maxwell Fluids

We derive rigorous criteria for the linear stability of viscoelastic flows under periodic boundary conditions. The results are based on recent work of R. Shvydkoy (Commun Math Phys 265:507–545, 2006).     

Plane analog of spontaneous swirling

A plane analog of the problem of spontaneous swirling—the occurrence of a free transverse flow due to disturbance of the initial plane-parallel flow—is considered. It is shown that in flows with circular streamlines between coaxial cylinders, loss of stability can result in the occurrence of axial flow that is axisymmetric on the average (averaging over the axial coordinate and the azimuthal angle) because of the countergradient transfer of the axial momentum component by Reynolds stresses. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 33–36, November–December, 1998.     

Thermodynamics of the activated state of materials

The thermodynamics of irreversible processes is extended to deformable materials whose state and behavior under nonequilibrium conditions are determined by the value and evolution of the additional parameter — the activation parameter. General thermodynamic relations are presented. The concept of the time of existence of a nonequilibrium state is introduced, and the phase coexistence conditions are generalized taking into account the properties of the interface. Methods are described to generalize the relations for irreversible flows, thermodynamic forces, and the equations of state. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 141–152, January–February, 2009.     

Wave Packets, Signaling and Resonances in a Homogeneous Waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e^−iωt + O(e^−ɛt ), for t → ∞, where Im ω₀ = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Breaking of waves of limiting amplitude over an obstacle

Homogeneous heavy fluid flows over an uneven bottom are studied in a long-wave approximation. A mathematical model is proposed that takes into account both the dispersion effects and the formation of a turbulent upper layer due to the breaking of surface gravity waves. The asymptotic behavior of nonlinear perturbations at the wave front is studied, and the conditions of transition from smooth flows to breaking waves are obtained for steady-state supercritical flow over a local obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 3–11, May–June, 2006.     

Calculation of the permeability and apparent permeability of three-dimensional porous media

In this study, creeping and inertial incompressible fluid flows through three-dimensional porous media are considered, and an analytical–numerical approach is employed to calculate the associated permeability and apparent permeability. The multiscale homogenization method for periodic structures is applied to the Stokes and Navier–Stokes equations (aided by a control-volume type argument in the latter case), to derive the appropriate cell problems and effective properties. Numerical solutions are then obtained through Galerkin finite-element formulations. The implementations are validated, and results are presented for flows through cubic lattices of cylinders, and through the dendritic zone found at the solid–liquid interface during solidification of metals. For the interdendritic flow problem, a geometric configuration for the periodic cell is built by the approximate matching of experimental and numerical results for the creeping-flow problem; inertial effects are then quantified upon solution of the inertial-flow problem. Finally, the functional behavior of the apparent permeability results is analyzed in the light of existing macroscopic seepage laws. The findings contribute to the (numerical) verification of the validity of such laws.     

Stability of a surface-charged viscous jet in an electric field

The stability of a surface-charged cylindrical jet in a longitudinal uniform electric field with respect to capillary pertubations is investigated in the linear approximation. The evolution of both axisymmetric and azimuthal-periodic perturbations is analyzed. In the latter case the first two modes among the azimuthal wavenumbers — bending and Bohr — are considered. Axisymmetric and bending instabilities lead to the transverse disintegration of the jet into individual drops and the Bohr mode to the longitudinal separation of the input jet into two parts. It is found that the axisymmetric and bending instabilities, respectively, can be completely suppressed and significantly attenuated by means of an external longitudinal field. In this case the role of the Bohr mode becomes more important leading under certain conditions to longitudinal longwave jet splitting. Events which can be interpreted as manifestations of longitudinal partition of the jet (dumbbell-like cross-section, branching nodes) are observed in experiments with evaporating polymer-solution microjets. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 29–40, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 97-01-00153).     

Special case of a traveling wave in one model of a multicomponent medium

The Kuropatenko model for a multicomponent medium whose components are polytropic gases is considered. It is assumed that, as x → ±∞, the multicomponent medium is in a homogeneous state with constant gas-dynamic parameters — velocity, pressure, and temperature. For the traveling wave flows, conditions similar to the Hugoniot conditions are obtained and used to uniquely determine the flow parameters for x → −∞ from the flow parameters x → +∞ and traveling wave velocity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 39–47, July–August, 2009.     

Equations of convective instability of a two-phase medium with modulation of the gravity force

Thermal convection in a heterogeneous medium consisting of a fluid and solid particles is studied under conditions of finite-frequency vibrations. Equations of convection are derived within the framework of the generalized Boussinesq approximation, and the problem of stability of a horizontal layer to infinitesimal perturbations under the condition of vertical vibrations is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 21–28, March–April, 2008.