Long-wave Perturbations Research Articles

Cosmological density perturbations in a conformal scalar field theory

We consider a scenario in which primordial scalar perturbations are generated when a complex conformal scalar field rolls down its negative quartic potential. Initially, these are perturbations of the phase of this field, which are then converted into adiabatic perturbations of the density. The existence of perturbations in the radial field direction, which have a red power spectrum, is a potentially dangerous feature of this scenario. But we show that in the linear order in the small parameter, the self-coupling, the infrared effects are completely nullified by an appropriate field redefinition. We evaluate the statistical anisotropy inherent in the model because of the presence of the long-wave perturbations of the radial field component. In the linear order in the self-coupling, the infrared effects do not affect the statistical anisotropy. They are manifested only at the quadratic order in the self-coupling, weakly (logarithmically) enhancing the corresponding contribution to the statistical anisotropy. The resulting statistical anisotropy is a combination of a large term, which decreases as the momentum increases, and a momentum-independent nonamplified term.

Nonstationary heat conduction in one-dimensional models with substrate potential

The paper investigates nonstationary heat conduction in one-dimensional models with substrate potential. To establish universal characteristic properties of the process, we explore three different models: Frenkel-Kontorova (FK), phi4+ (φ(4)+), and phi4- (φ(4)-). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with the parabolic Fourier equation of heat conduction and clearly demonstrates the necessity for hyperbolic corrections in the phenomenological description of the heat conduction process. The crossover wavelength decreases with an increase in the average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for the temperature field is justified. In all models investigated, the relaxation of thermal perturbations is exponential, contrary to a linear chain, where it follows a power law. The most popular lowest-order hyperbolic generalization of the Fourier law, known as the Cattaneo-Vernotte or telegraph equation, is also not valid for the description of the observed behavior of the models with the substrate potential, since the characteristic relaxation time in an oscillatory regime strongly depends on the excitation wavelength. For some of the models, this dependence seems to obey a simple scaling law.

The long-wave instability of the front of planar waves of impact ionization in semiconductors

Theoretical analysis of the transversal stability of a planar front of a impact ionization wave in semiconductors having a finite thickness and doping concentrations, has been carried out. The formulas were developed for an increment s of an increase in long-wave perturbation in both the presence and the absence of background charge carriers ahead of the front. These formulas are applicable at arbitrary relationship between the wave vector of perturbation k and a distance L between the front and the electrode towards which the impact ionization wave propagates, if the thickness of a region in which the volume charge of electrons and holes is large, is far less than min(1/k, L). In the limiting cases, when kL ≪ 1 and kL ≫ 1, the theory leads to the results obtained previously. The practical result of the formulas developed is that at all actual values L perturbation with a wave vector k < 1/L decrease or increase much slower than short-wavelength perturbations do. Therefore it is just evolution of short-wave perturbations that should result in current pinching and limit the reliability of avalanche voltage sharpeners.

Long-wave instability of an advective flow in an inclined fluid layer with perfectly heat-conducting boundaries

We investigate the stability of the steady convective flow in a plane tilted layer with ideal thermal conductivity of solid boundaries in the presence of uniform longitudinal temperature gradient. Analytically found the stability boundary with respect to the long-wave perturbations, find the critical Grashof number for the most dangerous among them of even spiral perturbation.

Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating

This paper investigates the stability of a thin incompressible viscoelastic fluid designated as Walters’ liquid B″ during spin coating. The long-wave perturbation method is proposed to derive a generalized kinematic model of the film flow. The method of normal mode is applied to study the linear stability. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. Using the multiple scales method, the weakly nonlinear stability analysis is studied for the evolution equation of a film flow. The Ginzburg–Landau equation is determined to discuss the threshold conditions of the various critical flow states. The study reveals that the rotation number and the radius of the rotating circular disk generate the destabilizing effects. Moreover, the viscoelastic parameter k indeed plays a more significant role in destabilizing the film flow than a thin Newtonian fluid during spin coating [27].

Mechanism of stability of the shear flow of a two-layer system of viscous liquids

171 The evolution of the morphology of multicompo� nent liquid media in the course of mechanical treat� ment is related, first of all, to the development of hydrodynamic instability at the interface. In the initial stage of the shear flow, this is caused by an exponential increase in the amplitude A(t) = A0e kΔct of capillary waves over a wide range of wavenumbers k. At positive values of the rate of change in the perturbation ampli� tude , the flow is unstable, while at negative values of this rate, it is stable. It was shown (1) that the Cou� ette and Poiseuille shear flows can be destabilized by applying weak longwave perturbations to the inter� face of a twolayer system of incompressible Newto� nian liquids at certain values of the viscosity ratio m and the thickness ratio n of the layers and at arbitrarily low Reynolds numbers. This qualitatively distin� guishes the instability of this type from the transition to turbulent flow. It was demonstrated (2, 3) that the change in the amplitude of short and long waves is caused by a phase shift of the velocity perturbation wave induced by inertial liquid transfer. At the same time, the mechanism of instability of shear flow at arbitrary wavelengths remained unclear. Solving this question requires involving numerical methods of analysis of the equations of momentum transfer of media with nonuniform viscosity distributions. In this work, for the first time, we performed a direct numerical modeling of the simple shear flow instability development of a twolayer system of New� tonian liquids over a wide range of wavelengths. We determined the conditions for the transition between the asymptotic solutions corresponding to longand Δ c shortwave perturbations. We also showed that the inertial shift of vortices of flow rate perturbations is proportional to the growth rate of the interface pertur� bation amplitude.

Transient two-layer thin-film flow inside a channel

The flow of two superimposed newtonian layers in a channel is investigated numerically in this study. The two-layer film flows inside a long channel due to a pressure gradient. The scaled conservation equations for two-layer incompressible newtonian film flow are first introduced. The weighted residual approach first proposed by Amaouche et al. [Phys. Fluids 19, 084106 (2007)] is used for finding the suitable weight functions before depth averaging. Subsequently, a linear stability analysis of thin-film equations for two-layer Poiseuille flow is carried out. The formulas which give the asymptotic stability with respect to long-wave perturbations obtained with Navier-Stokes equations [J. Non-Newtonian Fluid Mech. 71, 1 (1997)] are recovered with our averaging equations. In order to mimic the disturbance effect on the coextrusion flow, the steady flow, which is simply the uniform flow of two layers of fluid inside a channel, is then perturbed at inception. Following a finite difference based scheme, two types of boundary conditions are considered at the channel inception, namely a Dirac-type pulse and periodic forcing. The perturbation takes the form of a wave packet which may or may not be amplified as it moves downstream, depending on the values of the parameters involved in the problem. Furthermore, Gaster's relation is used to calculate the spatial growth rate of perturbation. The values obtained from this relation are in good quantitative agreement with those coming from the numerical simulations of thin-film equations. Then, our averaging equations also describe the nonlinear behavior of the interfacial instabilities occurring for the Poiseuille flow of two thin layers.

Investigation of the Conservative System of Equations for a Vertically Flowing Liquid Film

A new system of equations for modeling the dynamics of long-wave perturbations on the surface of a viscous liquid film flowing down a vertical plane was investigated. It is shown that the system obtained agrees with a known system derived by the variable changing method but unlikely has a conservative form that is perfect for designing of numerical efficient conservative difference schemes. The system is reduced to a single conservative equation for the function analogous to the hydrodynamic stream function with the appropriate boundary conditions. For the case of moderate Reynolds number it is noted that the equation with boundary conditions coincide with the well known Shkadov’s model under the assumption of self-similar profile of longitudinal velocity. At small Reynolds numbers typical to the condition of microgravity it was proved that the system is reduces to one evolution equation for the film thickness.

The wind instability and the resonant interaction of the elastic vibration of a thin plate with a supersonic gas stream

In a supersonic flow, the longwave perturbations running along the stream are stable, whereas the most rapidly growing waves propagate at an angle to the flow velocity. It has been shown that the wind instability of a nonuniform compressible gas flow having a velocity profile can give rise to elastic vibrations of a thin plate at a characteristic wavelength. The dispersion equation and the instability increment showing a maximum in the longwave band have been obtained.

Sufficient conditions for linear long-wave instability of steady-state axisymmetric flows of an ideal liquid with a free boundary in an azimuthal magnetic field

The problem of linear stability of steady-state axisymmetric shear jet flows of an inviscid ideally conducting incompressible liquid with a free surface and “frozen-in” azimuthal magnetic field is analyzed. The sufficient conditions for theoretical (on semi-infinite time intervals) and practical (on finite time intervals) instability of these flows relative to small axisymmetric long-wave perturbations are obtained by the direct Lyapunov method. An a priori lower estimate indicating (at least) an exponential increase with time of small perturbations under investigation is constructed in the case when these conditions are valid for theoretical as well as practical instability. In addition, an illustrative analytic example of steady-state flows under investigation and small axisymmetric long-wave perturbations superimposed on them is constructed (according to our estimate, these perturbations increase with time).

Hydromagnetic Stability Analysis of a Film Coating Flow Down a Rotating Vertical Cylinder

ABSTRACTThe influence of both the Rossby number and the Hartmann number on the hydromagnetic stability of a thin liquid film flowing down along the surface of a vertical cylinder is investigated. The long-wave perturbation method is employed to solve for generalized nonlinear kinematic equations with a free film interface. The normal mode approach is used to compute the stability solution for the film flow. The modeling results indicate that the stability of the liquid film is enhanced by increasing the strength of the magnetic field or reducing the speed at which the cylinder rotates. By contrast, the flow becomes relatively more unstable as the cylinder radius is increased at larger values of the Rossby number. Notably, this finding is the opposite of that observed for film flows along a stationary vertical cylinder.

Comparisons of predicted bore evolutions by the Benjamin-Davis-Ono and Navier-Stokes equations for idealized mesopause thermal ducts

[1] Numerical simulations are performed employing two numerical models to contrast nonlinear bore evolutions predicted by the Benjamin-Davis-Ono (BDO) equation with evolutions described by the Navier-Stokes (NS) equations. The first model is a simple one-dimensional solver of the BDO equation; the second describes the nonlinear two-dimensional dynamics of the NS equations. Both utilize the Boussinesq approximation. Owing to their simpler, horizontally isotropic nature, only isolated thermal ducts are considered in this study. Simulations assume an initial long-wave perturbation and address the influences of perturbation amplitude and wavelength, viscosity, and nonzero background stability on the resulting evolutions. Results indicate that the BDO equation provides reasonable predictions of bore character and evolution for conditions that satisfy its underlying assumptions. BDO predictions fail to describe bore character and evolution in cases where either the initial perturbations or the thermal environment differs significantly from BDO assumptions. Predictions employing the NS equations will thus provide more realistic guidance in the interpretation and understanding of bores observed in the mesopause region for general environments.

On the stability of jetlike magnetohydrodynamic flows

The linear stability problem is under study for steady axisymmetric translational flows of a density-homogeneous nonviscous incompressible ideal conducting fluid with free surface and “frozen-in” poloidal magnetic field. By the direct Lyapunov method, some sufficient conditions are obtained for the stability of these flows under small long-wave perturbations with the same symmetry. These stability conditions have partial converses; and, for unstable stationary flows, an a priori exponential lower bound is constructed on the growth of small perturbations under consideration, while the increment of the appearing exponent serves as an arbitrary positive parameter. An illustrative analytical example is given of steady flows with superimposed small long-wave axisymmetric perturbations growing in time in accordance with the estimate.

Interaction of Turing and Hopf Modes in the Superdiffusive Brusselator Model Near a Codimension Two Bifurcation Point

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by @ fi =@ j»j fi , 1 < fi < 2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, fi, is a measure of the rate of superdiffusion, which, in general, can be different for each of the two components. A weakly nonlinear analysis is used to derive two coupled amplitude equations describing the slow time evolution of the Turing and Hopf modes. We seek special solutions of the amplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed mode solution, and analyze their stability to long-wave perturbations. We find that the stability criteria of all three solutions depend greatly on the rates of superdiffusion of the two components. In addition, the stability properties of the solutions to the anomalous diffusion model are different from those of the regular diffusion model. Numerical computations in a large spatial domain, using Fourier spectral methods in space and second order Runge-Kutta in time are used to confirm the analysis and also to find solutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime. Specifically, we find a large number of steady state patterns consisting of a localized region or regions of stationary stripes in a background of time periodic cellular motion, as well as patterns with a localized region or regions of time periodic cells in a background of stationary stripes. Each such pattern lies on a branch of such solutions, is stable and

Convection in a Two-Layer System with a Deformable Interface Under Low Gravity Conditions

The onset of thermal convection in a system of two horizontal layers of immiscible liquids of similar densities is studied under low gravity conditions. A constant heat flux is prescribed at both rigid boundaries. A generalized Boussinesq approach that allows correct accounting for the interface deformation is used. The long-wave perturbations emerge under low-gravity conditions; either monotonic or oscillatory modes are critical depending on the problem. Moreover, two different modes of the monotonic instability exist. For the first instability mode, the convection dominates, whereas the interface remains nearly undeformable. The second monotonic instability mode is substantially related to interface deformations. The system of non-linear amplitude equations describing the behavior of long-wave regimes at finite-amplitude interface deflection and finite supercriticalities is obtained. The analytical and numerical investigations of these equations show that the stable non-trivial stationary solutions are absent, and after a transient at least one of the layers is split into the areas not connected to each other. The nonlinear regimes of cellular convection are studied numerically by the Level Set method.

Nonstationary heat conduction in one-dimensional chains with conserved momentum

This Rapid Communication addresses the relationship between hyperbolic equations of heat conduction and microscopic models of dielectrics. Effects of the nonstationary heat conduction are investigated in two one-dimensional models with conserved momentum: Fermi-Pasta-Ulam (FPU) chain and chain of rotators (CR). These models belong to different universality classes with respect to stationary heat conduction. Direct numeric simulations reveal in both models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with parabolic Fourier equation of the heat conduction. The crossover wavelength decreases with increase in average temperature in both models. For the FPU model the lowest-order hyperbolic Cattaneo-Vernotte equation for the nonstationary heat conduction is not applicable, since no unique relaxation time can be determined.

Competition between kinematic and dynamic waves in floods on steep slopes

We present a theoretical stability analysis of the flow after the sudden release of a fixed mass of fluid on an inclined plane formally restricted to relatively long time scales, for which the kinematic regime is valid. Shallow-water equations for steep slopes with bed stress are employed to study the threshold for the onset of roll waves. An asymptotic solution for long-wave perturbations of small amplitude is found on background flows with a Froude number value of 2. Small disturbances are stable under this condition, with a linear decay rate independent of the wavelength and with a wavelength that increases linearly with time. For larger values of the Froude number it is shown that the basic flow moves at a different scale than the perturbations, and hence the wavelength of the unstable modes is characterized as a function of the plane-parallel Froude numberFrpand a measure of the local slope of the free-surface height φ by means of a multiple-scale analysis in space and time. The linear stability results obtained in the presence of small non-uniformities in the flow, φ &gt; 0, introduce substantial differences with respect to the plane-parallel flow with φ = 0. In particular, we find that instabilities do not occur at Froude numbersFrcrmuch larger than the critical value 2 of the parallel case for some wavelength ranges. These results differ from that previously reported by Lighthill &amp; Whitham (Proc. R. Soc.A, vol. 229, 1955, pp. 281–345), because of the fundamental role that the non-parallel, time-dependent characteristics of the kinematic-wave play in the behaviour of small disturbances, which was neglected in their stability analyses. The present work concludes with supporting numerical simulations of the evolution of small disturbances, within the framework of the frictional shallow-water equations, that are superimposed on a base state which is essentially a kinematic wave, complementing the asymptotic theory relevant near the onset. The numerical simulations corroborate the cutoff in wavelength for the spectrum that stabilizes the tail of the dam-break flood.

Weakly Nonlinear Stability Analysis of a Thin Magnetic Fluid during Spin Coating

This paper investigates the stability of a thin electrically conductive fluid under an applied uniform magnetic filed during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. After linearizing the nonlinear evolution equation, the method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The Ginzburg-Landau equation is determined to discuss the necessary conditions of the various critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that the rotation number and the radius of the rotating circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.

Experimental investigation of the onset of instability in a radial Hele-Shaw cell

The initial stage of interface instability upon radial displacement of a fluid in a Hele-Shaw cell is investigated. An air-silicone oil system is analyzed. The critical radii of stability relative to long-wave perturbations are determined. It is found that, in the investigated range of parameters, instability most often begins by a translational mechanism. It is ascertained that in the overwhelming majority of cases the critical radii of instability are smaller than the values predicted by the linear stability theory and external effects make this difference even greater. The obtained results are discussed and compared with the existing theories.

Effect of electric field on the stability of an oscillatory contaminated film flow

The stability of the viscous liquid film on an oscillating plane is investigated in the presence of both insoluble surfactant on the film surface and uniform electric field, acting normal to the plane. In this problem main motivation is to study the combine effect of surfactant and electric field on the stability of the liquid film. Here liquid is treated as a perfect conductor and the air above the liquid film is also treated as a perfect dielectric. The linear stability analysis is performed using the long-wave perturbation method based on Floquet theory. It is observed that two Floquet modes exist due to the presence of surfactant and both modes can be unstable. The growth rate corresponding to the Floquet modes increase with the presence of an electric field and decrease with the presence of surfactant.