One-dimensional Perturbations Research Articles

Specificities of one-dimensional dissipative magnetohydrodynamics

One-dimensional dynamics of a plane slab of cold (β ≪ 1) isothermal plasma accelerated by a magnetic field is studied in terms of the MHD equations with a finite constant conductivity. The passage to the limit β → 0 is analyzed in detail. It is shown that, at β = 0, the character of the solution depends substantially on the boundary condition for the electric field at the inner plasma boundary. The relationship between the boundary condition for the pressure at β > 0 and the conditions for the electric field at β = 0 is found. The stability of the solution against one-dimensional longitudinal perturbations is analyzed. It is shown that, in the limit β → 0, the stationary solution is unstable if the time during which the acoustic wave propagates across the slab is longer than the time of magnetic field diffusion. The growth rate and threshold of instability are determined, and results of numerical simulation of its nonlinear stage are presented.

Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated.A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions.

О поведении дискретного спектра при одномерном возмущении

О поведении дискретного спектра при одномерном возмущении

Effect of sediment particle size on bed wavelength in closed conduits

The problem of the stability of the sand bed surface in a rectangular closed conduit with respect to spatially one-dimensional perturbations is formulated. The bed-form stability problem is solved using an analytical formula of bed load which takes into account the effect of bottom pressure on bed-load transport. An analytical dependence of the bed wavelength on hydrodynamic flow parameters and sediment particle diameter is obtained and compared with experimental data.

Controlling photoluminescent emission from polymer membranes through the creation of local deformation zones.

Understanding and optimizing light propagation and extraction in light-emitting systems, such as fluorescent chemical sensors, is important for the production of more efficient sensors. We apply Monte Carlo ray tracing to model the effects of one-dimensional perturbations of film thickness on the luminescent emission (spatial, directional, spectral) of a freestanding transparent polymer film embedded with luminescent chromophores. Such modification not only enhances light extraction but also allows its location and direction to be controlled. Optimization of the deformation geometry allows for a 3.6-fold increase in intensity for a far-field detector.

An elastic half-space under the action of one-dimensional time-dependent diffusion perturbations

A one-dimensional problem of elastic diffusion for a one-component half-space is considered. A locally static geometrically linear model of elastic diffusion is used, which contains mass transfer equations and a coupled system of the motion equations of an elastic body. To construct the solution, the integral Fourier transform with respect to the spatial coordinate and the integral Laplace transform with respect to time are applied. The problem of inverting Laplace transformants is reduced to inverting a rational function, while the inverse Fourier transform is implemented numerically. A fundamental solution of the problem is constructed. An example for the case where the diffusion flux on the boundary is constant is considered. The obtained results are a theoretical basis for an analysis of the stress-strain state in aeronautical and space constructions operating in conditions of multifactorial external influences.

Dynamic density functional theory with hydrodynamic interactions: theoretical development and application in the study of phase separation in gas-liquid systems.

Building on recent developments in dynamic density functional theory, we have developed a version of the theory that includes hydrodynamic interactions. This is achieved by combining the continuity and momentum equations eliminating velocity fields, so the resulting model equation contains only terms related to the fluid density and its time and spatial derivatives. The new model satisfies simultaneously continuity and momentum equations under the assumptions of constant dynamic or kinematic viscosity and small velocities and/or density gradients. We present applications of the theory to spinodal decomposition of subcritical temperatures for one-dimensional and three-dimensional density perturbations for both a van der Waals fluid and for a lattice gas model in mean field theory. In the latter case, the theory provides a hydrodynamic extension to the recently studied dynamic mean field theory. We find that the theory correctly describes the transition from diffusive phase separation at short times to hydrodynamic behaviour at long times.

One-dimensional perturbations of unbounded selfadjoint operators with empty spectrum

We study spectral properties of one-dimensional singular nonselfadjoint perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a Volterra operator.

The scattering problem for nonlocal potentials

We solve the direct and inverse scattering problems for integro- differential operators which are one-dimensional perturbations of the self- adjoint second derivative operator on the half-axis. We also describe the scattering data for this class of operators. Bibliography: 28 titles.

A quasi-one-dimensional model of thermoacoustics in the presence of mean flow

In thermoacoustic regenerators, the interaction of thermo-viscous boundary layers and axial temperature gradients causes a conversion from thermal energy to acoustic power or vice versa. In this paper, an improved analytical model for thermoacoustic boundary layer effects in the presence of mean flow is derived and analyzed. Previous formulations of the thermo-acoustic effect take into account effects of mean flow on acoustic propagation only implicitly, i.e. in as much as mean flow influences the mean temperature field. The new model, however, includes additional terms in the perturbation equations, which describe explicitly the interaction between steady mean flow and acoustics. For a parallel plate pore the three-dimensional thermoacoustic equations are derived and reduced to a transversally averaged system of differential equations by applying Green׳s function technique and suitable assumptions. The resulting one-dimensional perturbation equations are then solved numerically for two sets of boundary conditions to obtain the linear scattering matrix coefficients. The solutions, generated for a wide range of frequencies, can be applied in a low-order “network model” context to study the stability of thermoacoustic devices. The impact of mean flow on the thermoacoustic interaction is investigated and validated against full computational fluid dynamics simulations of laminar, compressible flow for one specific configuration. It is shown that at low frequencies (Womersley number <1) the new formulation predicts the acoustic behavior more accurately than the earlier formulations. Finally, the ideas and benefit of further improved and more complex models for higher Mach numbers are discussed.

Обратная спектральная задача для самосопряженного дифференциального оператора при одномерном возмущении

Обратная спектральная задача для самосопряженного дифференциального оператора при одномерном возмущении

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.

Back-reaction instabilities of relativistic cosmic rays

We explore streaming instabilities of an electron–ion plasma with relativistic and ultra-relativistic cosmic rays in the background magnetic field using the multi-fluid approach. Cosmic rays can be both electrons and ions. The drift speed of cosmic rays is directed along the magnetic field. In equilibrium, the return current of the background plasma is taken into account. One-dimensional perturbations parallel to the magnetic field are considered. The dispersion relations are derived for transverse and longitudinal perturbations. It is shown that the back-reaction of magnetized cosmic rays generates new instabilities one of which has a growth rate that can approach the growth rate of the Bell instability. These new instabilities can be stronger than the cyclotron resonance instability. For unmagnetized cosmic rays, the growth rate is analogous to the Bell one. We compare two models of the plasma return current in equilibrium with three and four charged components. Some differences between these models are demonstrated. For longitudinal perturbations, an instability is found in the case of ultra-relativistic cosmic rays. The results obtained can be applied to investigation of astrophysical objects such as the shocks by supernova remnants, galaxy clusters, intracluster medium and so on, where interaction of cosmic rays with turbulence of the electron–ion plasma produced by them is of great importance for cosmic-ray evolution.

A Dirichlet-to-Neumann Approach for The Exact Computation of Guided Modes in Photonic Crystal Waveguides

This work deals with one-dimensional infinite perturbation---namely, line defects---in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. Contrary to existing methods, this one is exact, but there is a price to be paid: the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. © 2013, Society for Industrial and Applied Mathematics

Direct and inverse problems for an operator with nonlocal potential

The spectrum of a selfadjoint operator which is a one-dimensional perturbation of the second derivative operator on a finite interval is analysed. It is shown that all the components of the one-dimensional perturbation can be recovered from two spectra up to complex conjugation.Bibliography: 13 titles.

Stability of non-isothermal phase transitions in a steady van der Waals flow

This paper is concerned with the multidimensional stability of non-isothermal subsonic phase transitions in a steady supersonic flow of van der Waals type. For the sake of seeking physical admissible planar waves, the viscosity–capillarity criterion (Slemrod in Arch Ration Mech Anal 81(4):301–315, 1983) is chosen to be the admissible criterion. By showing the Lopatinski determinant being non-zero, we prove that subsonic phase transitions are uniformly stable in the sense of Majda (Mem Am Math Soc 41(275):1–95, 1983) under both one-dimensional and multidimensional perturbations.

Non-local effects in radial heat transport in silicon thin layers and graphene sheets

We explore non-local effects in radially symmetric heat transport in silicon thin layers and in graphene sheets. In contrast to one-dimensional perturbations, which may be well described by means of the Fourier law with a suitable effective thermal conductivity, two-dimensional radial situations may exhibit a more complicated behaviour, not reducible to an effective Fourier law. In particular, a hump in the temperature profile is predicted for radial distances shorter than the mean-free path of heat carriers. This hump is forbidden by the local-equilibrium theory, but it is allowed in more general thermodynamic theories, and therefore it may have a special interest regarding the formulation of the second law in ballistic heat transport.

Effect of disperse phase on instability in systems with combustion

The dynamics of weak perturbations with finite amplitude in a two-phase homogeneous medium (gas with suspended solid particles) featuring a nonequilibrium chemical reaction has been studied. Using an asymptotic approach, a weakly nonlinear model of the evolution of one-dimensional perturbations is developed that takes into account the kinetic wave interactions and dissipative properties including the interphase exchange of heat and momentum. Conditions for the loss of stability of the homogeneous state of the system are analyzed. Numerical solutions of the evolution equation are obtained in the form of established self-sustained oscillations. The stabilizing effect of the inert disperse phase is described.

Pulsating regime of magnetic deflagration in crystals of molecular magnets

The stability of a magnetic deflagration front in a media of molecular magnets, such as ${\mathrm{Mn}}_{12}$ acetate, is considered. It is demonstrated that stationary deflagration is unstable with respect to one-dimensional perturbations if the energy barrier of the magnets is sufficiently high in comparison with the release of Zeeman energy at the front; their ratio may be interpreted as an analog to the Zeldovich number, as found in problems of combustion. When the Zeldovich number exceeds a certain critical value, a stationary deflagration front becomes unstable and propagates in a pulsating regime. Analytical estimates for the critical Zeldovich number are obtained. The linear stage of the instability is investigated numerically by solving the eigenvalue problem. The nonlinear stage is studied using direct numerical simulations. The parameter domain required for experimental observations of the pulsating regime is discussed.

Magnetothermoelastic analysis of functionally graded hollow spherical structures under thermal and mechanical loads

The exact solution for the one-dimensional steady-state magnetothermoelastic stresses and perturbation of magnetic field vector in a FGM (functionally graded material) hollow sphere is presented. The material properties are assumed to depend on the variable r and they are expressed as power functions of r. The values used in this study are arbitrary chosen to demonstrate the effect of inhomogeneity on magnetothermoelastic stresses and perturbation of magnetic field vector distributions of the FGM hollow sphere. The direct method is used to solve the heat conduction and Navier equations. The results carried out may be used as a reference to solve other magnetothermoelastic problems in FGM hollow spherical structures and to design optimum FGM hollow spherical structures.