Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x ₀, then the expansion coefficients of the velocity perturbations are even functions about x ₀ for even powers of the wave number and odd functions about for x ₀ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.     

Localized instabilities of vortex rings with swirl

Axisymmetric vortex rings with swirl in an inviscid incompressible fluid are considered and it is demonstrated how the geometrical optics method can be used for investigating their stability. The evolution of rapidly oscillating initial data is studied and it is shown that the corresponding rings are unstable if the transport equations associated with the wave which is advected by the flow have unbounded solutions. It is shown that unbounded solutions grow either exponentially or algebraically. By means of the analysis of the corresponding transport equations effective stability conditions for general vortex rings with swirl are obtained. © 1993 John Wiley & Sons, Inc.     

Spectral transverse instability of solitary waves in Korteweg fluids

The motion of Korteweg fluids is governed by the Euler-Korteweg model, which admits planar solitary waves for nonmonotone pressure laws such as the van der Waals law below critical temperature. In an earlier work with Danchin, Descombes and Jamet, it was shown by variational arguments and numerical computations that some of these solitary waves are stable in one space dimension. The purpose here is to study their stability with respect to transverse perturbations in several space dimensions. By Evans functions techniques and Rouché's theorem, it is shown that transverse perturbations of large wave length always destabilize solitary waves in the Euler-Korteweg model, whereas energy estimates show that perturbations of short wave length tend to stabilize them.     

Recurrence formulas for long wavelength asymptotics in the problem of shear flow stability

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for two-dimensional viscous incompressible shear flows with a nonzero average. It is shown that the critical eigenvalues are odd functions of the wave number, while the critical values of the viscosity are even functions. If the deviation of the velocity from its period-average value is an odd function of spatial variable, the eigenvalues can be found exactly.     

The estimation of M4 processes with geometric moving patterns

There are many parameters in multivariate maxima of moving maxima processes—or M4 processes. However, the more parameters there are, the more difficult it is to estimate them. It is not just an issue of numerical stability, of course. The statistical precision of the estimates will be poor if the number of parameters is too large. We consider asymmetric geometric structures which correspond to special moving patterns of extreme observations in observed time series. We study the model identifiability and propose parameter estimators. All proposed estimators are shown to be consistent and asymptotically joint normal. Simulation study and real data modeling of North Sea wave height data are illustrated.     

On the stability and transition for the Navier-Stokes-alpha model

The main aim of this paper is to investigate the stability and transition of the Navier-Stokes-alpha model. By using the continued-fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non-dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio-temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid-axis of the channel.     

Nonlinear temporal dynamics in magnetic fluids between undulatory parallel walls

The effects of undulatory parallel walls and a normal magnetic field on the stability of weakly nonlinear waves at a horizontal interface of two magnetic inviscid fluids are investigated. We assumed that the walls have a weak sinusoidal undulation. The frequency of the main waves is similar to a problem having smooth boundaries. The breaker surface tension and the breaker magnetic field are obtained. The stability analysis concerns the interaction of two propagation wave numbers satisfying the resonance condition imposed by the periodicity of the sinusoidal walls. The first-resonance case occurs whenever the wall wave number is nearly equal to twice the propagation wave number while the second-resonance case occurs whenever the two kinds of wave numbers are nearly equal. When the wave number of the undulation is far from the propagation wave number, the sinusoidal walls have the same effect of the smooth walls on the stability criterion. The stability conditions and the transition curves in the two resonance cases are treated away from the critical state. The existence conditions and stability of Stokes waves near the critical state are discussed. Numerous illustrations and graphs amplify the work.     

Stability of the planar rarefaction wave to two-dimensional Navier-Stokes-Korteweg equations of compressible fluids

This study is concerned with the large time behavior of the two-dimensional compressible Navier-Stokes-Korteweg equations, which are used to model compressible fluids with internal capillarity. Based on the fact that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws is nonlinearly stable to the one-dimensional compressible Navier-Stokes-Korteweg equations, the planar rarefaction wave to the two-dimensional compressible Navier-Stokes-Korteweg equations is first derived. Then, it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are suitably small perturbations of the planar rarefaction wave. The proof is based on the delicate energy method. This is the first stability result of the planar rarefaction wave to the multi-dimensional viscous fluids with internal capillarity.     

The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer

The effect of vertical vibration on the onset of Marangoni convection in a horizontal layer of a viscous incompressible uniform liquid with a free surface and a hard (solid) or soft (impermeable and stress-free) wall is investigated. In the case of harmonic vibration, a dispersion relation is constructed in explicit form using continued fractions. From this, equations are obtained for determining the critical values of the parameters for all three main types of loss of stability. Neutral curves of the monotonic and oscillatory instability are constructed, for fixed frequency and amplitude of the vibration, in the form of a graph of the Marangoni number against the wave number. The regions of parametric resonances, corresponding to synchronous and subharmonic modes are determined. The frequency values for which a high-frequency asymptotic form is reached are obtained. The long-wave Marangoni oscillatory instability is investigated, and it is shown that in this case the Marangoni numbers are negative and depend only on the Prandtl and Biot numbers.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

Convective Instabilities in Anisotropic Porous Media

This paper addresses the problem of the onset of Rayleigh-Bénard convection in a porous layer using Brinkman's equation and anisotropic permeability. The critical Rayleigh number and wave number at marginal stabilities are calculated for both free and rigid boundaries. In both cases, it is noted that there exist ranges for which the stability criteria is intermediate to the low porosity Darcy approximation and to high porosity single viscous fluid. The permeability anisotropy is found to select the mode of instability.     

Exponential stability of a kind of wave equation with boundary feedback control

The problem of exponential stability of a kind of wave equation with damping and boundary output feedback control is investigated. The spectral structure of the system operator is analyzed and it is shown that the c0-semigroup generated by the system operator is exponential stable if only the coefficients viscous damping and boundary feedback control are not zeros simultaneously.     

On the nonlinear stability of plane waves for the ginzburg-landau equation

I consider the nonlinear stability of plane wave solutions to a Ginzburg-Landau equation with additional fifth-order terms and cubic terms containing spatial derivatives. I show that, under the constraint that the diffusion coefficient be real, these waves are stable. Furthermore, it is shown that the radial component of the perturbation decays at a faster rate than the phase component of the perturbation as t → ∞. The result is also applicable to the classical Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.     

The effect of boundary conditions on the onset of chaos in Rayleigh–Bénard convection using energy-conserving Lorenz models

The influence of 16 boundary conditions on linear and nonlinear stability analyses of Rayleigh–Bénard system is reported. A Stuart–Landau amplitude equation for the Rayleigh–Bénard system between stress-free, isothermal boundary conditions is derived and the procedure used in this derivation serves as guidance for constructing an appropriate Fourier–Galerkin expansion for the other 15 boundary conditions to derive a generalized Lorenz model. The influence of the boundary conditions comes within the coefficients of the generalized Lorenz model. It is shown that the obtained generalized Lorenz model is energy conserving and for certain boundary conditions it retains features of the classical Lorenz model. Further, the principle of exchange of stabilities is shown to be valid for the present problem and hence it is the steady-state, linearized version of the generalized Lorenz model which yields an analytical expression for the Rayleigh number. On minimizing this expression with respect to wave number the critical Rayleigh number at which the onset of regular convective motion occurs in the form of rolls is determined for all 16 boundary conditions. It is found that these values are in good agreement with those of previous investigations leading to the conclusion that the chosen minimal Fourier–Galerkin expansion is a valid one. Exhibition of chaotic motion in the generalized Lorenz system at the Hopf Rayleigh number is studied. The phase-space plots which indicate a clear-cut visualization of the transition from regular convective motion to chaotic motion in the generalized Lorenz system are presented. Further, existence of a developing region for chaos (mildly chaotic motion) and windows of periodicity are captured using the bifurcation diagrams. It is concluded from the phase-space plots and the bifurcation diagrams that the generalized Lorenz model for certain sets of boundary conditions retains all the features of the classical Lorenz model. Such a conclusion cannot be made by a linear stability analysis and the need thus for a nonlinear analysis is highlighted in the paper.     

Halting combustion waves with a fire break

A simple model for the propagation of an exothermic reaction is proposed in which the reactant ignites at a specified temperature. It is shown that the model supports waves of permanent form and the speed and stability of such waves is determined. Finally the behavior of this wave as it encounters a break in the fuel is considered. In particular results are obtained for the size of the break required to halt the wave.     

Stability of contact discontinuity for the Boltzmann equation

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and Navier-Stokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation corresponding to the contact discontinuities for the Euler equations, we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate. In our analysis, an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates.     

The Stability of Two, Three, and Four Wave Interactions of a Prototype System

The stability of plane wave interactions of coupled nonlinear Schrödinger (CNLS) equations can be analyzed within a bisymplectic framework. This framework is a generalization of the Hamiltonian formulation. The current study considers a family of CNLS equations that are used as a prototype system for studying the combined interaction of unstable and stable component waves in optics. This popular family has a drawback when cast into a bisymplectic framework: the determinant controlling various types of fiber regime is zero. To solve this problem, it is proposed that a limit is taken from a more general CNLS family to the family in question. This method is then bench-marked against known stability results for the simple two plane wave interactions when amplitudes are equal and are found to agree. It is then applied to two wave interactions with unequal amplitudes as well as three and four wave interactions. The latter interactions for this particular system are not spectrally stable. By suggesting a slightly larger family of CNLS equations, it is illustrated that spectral stability can occur. This adapted prototype system may be of use in optics; in particular, to show that long-wave stability is possible given a judicious choice of parameter values.     

Existence of Neutral Rayleigh Waves in the Hagen–Poiseuille Flow Through a Pipe of Circular Cross Section

The inviscid neutral stability of Hagen–Poiseuille flow through a circular pipe is studied using both analytical and numerical techniques. A zero phase shift is applied across the critical surface to represent the effects of strong nonlinearity. Using a form of Sturm's comparison theorem it is possible to prove that no neutral solutions exist if a combination of the axial and azimuthal wave numbers of the perturbation exceeds a critical value. As a consequence, the physical problem admits only neutral solutions for an azimuthal wave number of unity.     

Tile patterns in excitable media subject to non-solenoidal flow fields

The propagation of spiral waves in excitable media subject to a non-solenoidal advective field which satisfies the no-penetration condition on the boundaries of the domain is studied numerically, and it is shown that, depending on the amplitude and spatial frequencies of the velocity field, the spiral wave may be distorted highly, break up into a number of smaller spiral waves, or exhibit polygonal shapes or tile patterns. These patterns reflect the symmetry/asymmetry of the velocity field and are characterized by thick regions of high concentration at stagnation points where the velocity gradient is largest, and thin ones which are parallel to the velocity vector. It is also shown that the advective field distorts the spiral wave by decreasing its thickness where the velocity is largest due to the stretching of the wave, and by increasing it at the stagnation points where the curvature of the wave is largest.     

Propagation of Normal Waves in an Isolated Porous Fluid-Saturated Biot Layer

For a porous fluid-saturated Biot layer with boundaries free from stresses and pressure, the wave field is found and dispersion equations are derived. The roots of the dispersion equations and the dependence of the phase velocities of the normal waves on the wave number are investigated by analytic methods. It is shown that the phase velocities of most of the normal waves decrease with increasing wave number. Special investigations are conducted in the case of bend and plate waves and their phase velocities for high and low frequencies. It is also shown that on the boundary of a porous Biot half-space, the Rayleigh wave does not always originate, and conditions for the existence of such a wave are established. Bibliography: 7 titles.