System Of Equations For Perturbations Research Articles

Modeling of the heat conduction process in composite media

In this work we proposed a two-channel model of heat conduction processes in composite media on the basis of the general transfer theory. The obtained system of equations can be reduced to both the generalized Fourier equation and the system of Cattaneo equations describing fast processes. The stability of the processes is considered. A system of equations for perturbations is obtained. The calculation of the perturbations’ spectrum is given depending on the parameters of the model. From this fact it follows that in a linear system the stationary process of heat conduction is stable with respect to any small perturbations. If we take into account nonlinearity even in the simplest cases it leads to the appearance of various instabilities and the excitation of heat waves.

Radiative instability of a barotropic jet flow in a stratified rotating atmosphere

The problem of the stability of a jet flow with a piecewise linear velocity profile in a stratified rotating atmosphere is considered. The linearized system of equations for perturbations is reduced to a single equation with respect to the amplitude of the longitudinal velocity component containing the turning points. In terms of Airy functions, an asymptotic solution of the equation is constructed that is valid for small values of the Rossby number. It is shown that the flow becomes unstable due to radiation of inertial-gravitational waves. An analytical expression is obtained for the growth rate of perturbations.

No-boundary proposal for braneworld perturbations

We propose a novel approach to the problem of cosmological perturbations in a braneworld model with induced gravity, which leads to a closed system of equations on the brane. We focus on a spatially closed brane that bounds the interior four-ball of the bulk space. The background cosmological evolution on the brane is now described by the normal branch, and the boundary conditions in the bulk become the regularity conditions for the metric everywhere inside the four-ball. In this approach, there is no spatial infinity or any other boundary in the bulk space since the spatial section is compact, hence, we term this setup as a no-boundary proposal. Assuming that the bulk cosmological constant is absent and employing the Mukohyama master variable, we argue that the effects of nonlocality on brane perturbations may be ignored if the brane is marginally closed. In this case, there arises a relation that closes the system of equations for perturbations on the brane. Perturbations of pressureless matter and dark radiation can now be described by a system of coupled second-order differential equations. Remarkably, this system can be exactly solved in the matter-dominated and de Sitter regimes. In this case, apart from the usual growing and decaying modes, we find two additional modes that behave monotonically on super-Hubble spatial scales and exhibit rapid oscillations with decaying amplitude on sub-Hubble spatial scales.

Analysis of scalar perturbations in cosmological models with a non-local scalar field

We develop the cosmological perturbations formalism in models with a single non-local scalar field originating from the string field theory description of the rolling tachyon dynamics. We construct the equation for the energy density perturbations of the non-local scalar field in the presence of arbitrary potential and formulate the local system of equations for perturbations in the linearized model when both simple and double roots of the characteristic equation are present. We carry out the general analysis related to the curvature and entropy perturbations and consider the most specific example of perturbations when important quantities in the model become complex.

Dark energy perturbations and cosmic coincidence

While there is plentiful evidence in all fronts of experimental cosmology for the existence of a nonvanishing dark energy (DE) density ${\ensuremath{\rho}}_{D}$ in the Universe, we are still far away from having a fundamental understanding of its ultimate nature and of its current value, not even of the puzzling fact that ${\ensuremath{\rho}}_{D}$ is so close to the matter energy density ${\ensuremath{\rho}}_{M}$ at the present time (i.e. the so-called ``cosmic coincidence'' problem). The resolution of some of these cosmic conundrums suggests that the DE must have some (mild) dynamical behavior at the present time. In this paper, we examine some general properties of the simultaneous set of matter and DE perturbations $(\ensuremath{\delta}{\ensuremath{\rho}}_{M},\ensuremath{\delta}{\ensuremath{\rho}}_{D})$ for a multicomponent DE fluid. Next we put these properties to the test within the context of a nontrivial model of dynamical DE (the $\ensuremath{\Lambda}\mathrm{XCDM}$ model) which has been previously studied in the literature. By requiring that the coupled system of perturbation equations for $\ensuremath{\delta}{\ensuremath{\rho}}_{M}$ and $\ensuremath{\delta}{\ensuremath{\rho}}_{D}$ has a smooth solution throughout the entire cosmological evolution, that the matter power spectrum is consistent with the data on structure formation, and that the ``coincidence ratio'' $r={\ensuremath{\rho}}_{D}/{\ensuremath{\rho}}_{M}$ stays bounded and not unnaturally high, we are able to determine a well-defined region of the parameter space where the model can solve the cosmic coincidence problem in full compatibility with all known cosmological data.

Control of Flexible Aircraft Executing Time-Dependent Maneuvers

A new generation of aircraft, namely, unmanned aerial vehicles (UAVs), and, in particular, autonomous UAVs, are expected to carry out very critical maneuvers, well in excess of what pilots are able to tolerate. A newly developed theory for the dynamics and control of maneuvering flexible aircraft is ideally suited for the analysis and design of such aircraft. The state equations for maneuvering flexible aircraft are nonlinear and of high dimension. With due consideration to the way aircraft are flown, the problem can be separated into one for the generally large aircraft motions on a given flight path and another one for small perturbations from the flight path and elastic vibration, where the second problem receives inputs from the first. The case in which the flight path represents a time-dependent maneuver, so that the system of perturbation equations is time varying is studied. Several developments designed to facilitate the treatment of time-varying systems are provided. In particular, included are 1) an explicit derivation of the matrices defining the state equations, 2) a new approach to the control of the perturbations from the flight path and the elastic vibration, and 3) expressions for the implementation of the controls in discrete time. Computer solutions for the system response can be carried out conveniently by the use of MATLAB® and MATHEMATICA. A numerical example illustrates the approach for the case of a flexible aircraft executing a pitch maneuver.

Laminar‐turbulent transition of the incompressible swept attachment‐line

The linear and non‐linear stability of an incompressible swept attachment‐line boundary layer are analysed, within the Görtler‐Hämmerlin framework. The system of perturbation equations is solved using spectral collocation method based on Chebyshev polynomials. The global solution method utilised for solving the eigenproblem yields the full spectrum of the least damped waves. The influence of suction and blowing on critical Reynolds numbers is analysed. A comparison of the present paper linear solutions for both zero suction and a layer in which blowing and suction are applied with the Spalart's DNS solutions (obtained for small initial disturbances) shows very good agreement. The transition Reynolds numbers have been predicted using well known exp(n) method and results are compared with the experimental data of Powilleit.

Spatial stability of incompressible attachment-line flow

Linear stability analysis of incompressible attachment-line flow is presented within the spatial framework. The system of perturbation equations is solved using spectral collocation. This system has been solved in the past using the temporal approach and the current results are shown to be in excellent agreement with neutral temporal calculations. Results amenable to direct comparison with experiments are then presented for the case of zero suction. The global solution method utilized for solving the eigenproblem yields, aside from the well-understood primary mode, the full spectrum of least-damped waves. Of those, a new mode, well separated from the continuous spectrum is singled out and discussed. Further, relaxation of the condition of decaying perturbations in the far-field results in the appearance of sinusoidal modes akin to those found in the classical Orr-Sommerfeld problem. Finally, the continuous spectrum is demonstrated to be amenable to asymptotic analysis. Expressions are derived for the location, in parameter space, of the continuous spectrum, as well as for the limiting cases of practical interest. In the large Reynolds number limit the continuous spectrum is demonstrated to be identical to that of the Orr-Sommerfeld equation.

Effects of solar tides on the zonal mean circulation in the lower thermosphere: solstice condition

Effects of momentum deposition due to solar diurnal and semi-diurnal tidal waves on the zonal mean circulation in the mesosphere and lower thermosphere for a solstice condition are discussed. In the present model, the system of zonally averaged equations and the system of perturbation equations are integrated simultaneously, so that the propagation of tidal waves is affected not only by the basic mean fields but also by the induced zonal mean fields due to the momentum deposition. Results for two different vertical eddy diffusion profiles are presented. It is shown that the solar tides make a significant contribution to the generation of the mean zonal winds in the upper mesosphere and the lower thermosphere. Below 120 km the main contribution is due to propagating diurnal tides, while above 120 km it is due to semidiurnal tides.

On forced oscillations in the boundary layer at frequencies near the upper branch of the neutral curve

Perturbations introduced into the boundary layer (BL) of an incompressible liquid by a harmonic oscillator in the form of a moving section of the surface are examined. Outside the oscillating part, the streamlined solid is a plane plate. It is assumed that the Reynolds number is large and the oscillation frequency, corresponds, in order of magnitude, to the asymptotic form of the upper branch of the neutral stability curve (NSC). The system of equations for perturbations, at small amplitudes of the oscillator, is linearized and is solved by the Fourier method. In addition, for each Fourier component, the flow field is divided into five sublayers. The amplitude of a Tollmin-Schlichting wave (TS) is calculated and separated from the perturbed background downstream of the oscillator. If the oscillator frequency exceeds the neutral value at the upper branch of NSC with the given Reynolds number, the TS wave amplitude decays. For frequencies below neutral, the wave amplitude increases exponentially downstream. In the final example, the parameters of the TS wave fall within the unstable region, between two NSC branches.

Perturbations des couches diffuses—I. Équations théoriques de base

Using the general theory of irreversible phenomena, we derive some general equations relating to cross-phenomena of diffusion-transport in a fluid system in movement. In the case of isothermal diffusion of a binary electrolyte in very dilute solution, we define a system of adimensional transport equations in which the basic variables are the conductivity and the volume charge density. We display the symmetry properties of these equations and establish very general relationships between conductivity and charge density. Finally we define the general system of equations for perturbations of a diffuse layer; we shall give some analytical solutions and experimental applications elsewhere.