Linear Stability Results Research Articles

Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

The linear stability theory is used to investigate analytically the effects of gravity modulation on solutal convection in the mushy layer of solidifying binary alloys. The gravitational field consists of a constant part and a sinusoidally varying part, which is synonymous to a vertically oscillating mushy layer subjected to constant gravity. The linear stability results are presented for both the synchronous and subharmonic solutions. It is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. Beyond the transition point, further increases in the frequency tend to destabilize the solutal convection, but gradually. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (η 0 ) is to destabilize the solutal convection.

Linear Stability and Convection in a Gravity Modulated Porous Layer Heated from Below: Transition from Synchronous to Subharmonic Solutions

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The linear stability results are presented for both the synchronous and subharmonic solutions and the exact point for the transition from synchronous to subharmonic solutions is computed. It is also demonstrated that increasing the excitation frequency rapidly stabilizes the convection up to the transition point from synchronous to subharmonic convection. Beyond the transition point, the effect of increasing the frequency is to slowly destabilize the convection.

Stability of Convection in a Gravity Modulated Porous Layer Heated from Below

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The gravitational field consists of a constant part and a sinusoidally varying part, which is tantamount to a vertically oscillating porous layer subjected to constant gravity. The linear stability results are presented for the specific case of low amplitude vibration for which it is shown that increasing the frequency of vibration stabilises the convection.

Linear stability of perturbed Hamiltonian systems: theory and a case example

In this paper we present a general theory, based on a Lyapunov?Schmidt reduction, for the linearized stability of a perturbed Hamiltonian system with a number of symmetries. The reduction leads to an eigenvalue problem in a small subspace of the original system, whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system. The Krein signature of the eigenvalues can also be obtained in this framework. The method is applied to the case example of the coupled nonlinear Schr?dinger equations giving excellent agreement with the full numerical linear stability results.

Rupture of a Surfactant-Covered Thin Liquid Film on a Flexible Wall

The rupture of a surfactant-covered thin liquid film on a flexible wall is studied in this paper. Evolution equations for the deflections of the air-liquid and wall-liquid interfaces and surfactant surface concentration are derived using lubrication theory, and their linear and nonlinear stability characteristics are investigated. Our linear stability results indicate that increases in the level of damping, the longitudinal wall tension, and the relative magnitude of Marangoni stresses have a stabilizing influence. Numerical simulations of the evolution equations are used to investigate the nonlinear characteristics of the instability. In all cases considered, the surfactant concentration decreases in the rupture region as rupture is approached, and the resulting Marangoni flows retard but do not prevent rupture. Self-similar rupture is examined and power-law scalings are extracted for different parameter values. These appear to be unchanged from those for rigid substrates, evidently because the van der Waals forces that drive the instability dominate the rupture dynamics.

Stability of multidimensional undercompressive shock waves

This paper is devoted to the study of linear and nonlinear stability of undercompressive shock waves for first order systems of hyperbolic conservation laws in several space dimensions. We first recall the framework proposed by Freistühler to extend Majda's work on classical shock waves to undercompressive shock waves. Then we show how the so-called uniform stability condition yields a linear stability result in terms of a maximal $L^2$ estimate. We follow Majda's strategy on shock waves with several improvements and modifications inspired from Métivier's work. The linearized problems are solved by duality and the nonlinear equations by mean of a Newton type iteration scheme. Finally, we show how this work applies to phase transitions in an isothermal van der Waals fluid.

Absolute and convective instability of a relaxational plane liquid jet

The stability of a plane relaxational liquid jet has been studied theoretically and experimentally through linear stability analysis and flow visualizations. The relaxational liquid jet is obtained by the outflow of a liquid from a plane channel with an upstream fully developed Poiseuille flow into an ambient stagnant gas. Linear spatial stability calculations show that there are five convectively unstable modes, three sinuous and two dilatational. The spatial stability calculations are compared to experimental results for wavenumber variation and the growth of waves found in the visualizations. These variations have been quantified with a wavelet transform and through a comparison with the spatial stability results the type of mode observed in the visualizations has been determined. For this type of mode the calculated wavenumber variation is in good agreement with the experimental results. Also, in the experiments the breakup of the jet has been observed when the Reynolds number reaches a certain value, and as the Reynolds number increases this breakup moves closer to the channel exit. This upstream movement of the breakup can be explained by the linear stability results. Finally the relaxational liquid jet is shown to be absolutely unstable for a certain parameter region. Close to the nozzle both a sinuous mode and a dilatational mode are shown to be absolutely unstable. As the jet profile relaxes to uniform, the sinuous mode is shown to be the only unstable mode. This occurs for Weber numbers , which is in agreement with the theory for liquid jets with uniform velocity profile. The frequency selection for the observed waves is believed to be related to the region of absolute instability located closest to the channel exit.

Radial source flows in porous media: Linear stability analysis of axial and helical perturbations in miscible displacements

Linear stability results are presented for axial and helical perturbation waves in radial porous media displacements involving miscible fluids of constant density. A numerical eigenvalue problem is formulated and solved in order to evaluate the relevant dispersion relations as functions of the Peclet number and the viscosity ratio. In contrast to the constant algebraic growth rates of purely azimuthal perturbations [C. T. Tan and G. M. Homsy, Phys. Fluids 30, 1239 (1987)], axial perturbations are seen to grow with a time-dependent growth rate. As a result, there exists a critical time up to which the most dangerous axial wavenumbers are larger, and beyond which the most dangerous azimuthal wavenumbers have higher values. This raises the possibility that early on, the smaller flow scales appear in the axial direction, whereas the later flow stages are dominated by smaller azimuthal features. By rescaling the axial wavenumber, the explicit appearance of time can be eliminated. The maximum growth rate of axial perturbations, as well as their most dangerous and cutoff wavenumbers, are seen to increase with the Peclet number and the viscosity ratio. The most dangerous wavenumber is observed to shift towards the lower end of the spectrum as the Peclet number increases. With increasing viscosity contrast, it first moves towards the lower part of the spectrum, only to shift towards the higher end later on. In the limit of large Pe, asymptotic solutions are obtained for the growth of axial disturbances. Numerical solutions of the full eigenvalue problem generally show good agreement with these asymptotic solutions for large Peclet numbers. Over the entire range of wave vector directions between the purely axial and azimuthal extrema, helical waves display an approximately constant maximum growth rate. The wavenumber of maximum growth as well the maximum growth rate of helical waves can be evaluated from the corresponding purely azimuthal and axial problems. This suggests that in three-dimensional flows the nature of the initial conditions plays an important role.

Analytical and numerical modeling of sandbanks dynamics

Linear and nonlinear behavior of large‐scale underwater bedform patterns like sandbanks are studied using linear stability analysis and numerical modeling. The model is based on depth‐integrated hydrodynamics equations with a quadratic bottom friction law and a bed load sediment transport model including a bottom slope effect. First, the linear stability analysis of a flat erodible bottom subject to a steady current is performed. The direction of the most amplified bedform is controlled by the current, whereas its wavelength is selected by the gravity driven sediment flow. The instability proved to be due to the velocity component parallel to the mean flow, whereas the transverse velocity and the bottom slope effect are damping processes. The growth rate is then related to the spatial phase‐lag between flow velocity and bathymetry. In order to validate the numerical model in this linear regime, the growth rate and the phase celerity of an infinitesimal bottom perturbation are computed as a function of its wavevector. A good agreement with linear stability results is found. Second, using the property that the growth rates for block‐function and steady current are the same, the nonlinear behavior of the instability is investigated for a simple tidal current. The saturation height of the theoretically most amplified mode is estimated using the numerical model. The analysis of the sediment fluxes gradients shows that the saturation is mostly due to the hydrodynamic processes, although the saturation height slightly depends on the value of the bottom slope effect coefficient. Third, a Landau equation, whose coefficients are computed from the previous results, is used to predict the temporal evolution of the bedform amplitude from the initial infinitesimal perturbation to the saturation. A comparison with the characteristics of continental shelf sandbanks shows that the model gives a reasonable estimation of the temporal dynamics of these large‐scale bedforms. However, the saturation height seems to be slightly overestimated. The limitations of the method are also discussed.

Application of triple-deck theory to the prediction of glaze ice roughness formation on an airfoil leading edge

A viscous–inviscid interaction triple-deck structure is developed to describe the thermomechanical interaction of an air boundary layer with ice sheets and liquid films. Linear stability results are compared with nonlinear triple-deck computations, and a number of nonlinear simulations of air–water–ice interactions are presented. An icing instability is encountered in regimes with simultaneous wall and air cooling that is believed to admit small scale and highly irregular surface roughness. The stabilization of the smallest scale icing disturbances is obtained through the Gibbs–Thomson relation. This local thermodynamic condition relates the freezing temperature of a pure substance to the surface tension and the mean curvature of the interface and provides a short scale stabilizing mechanism for icing instability modes. Comparison with available experimental data on glaze ice roughness diameters, accreted on NACA 0012 airfoil leading edges under glaze icing conditions, is provided. It is also found in all cases computed in this study that water beads can be formed on a wetted ice surface once the water film is locally ruptured by ice roughness elements.

Low-gravity sideways double-diffusive instabilities

Instabilities of a double-diffusive system in a slot are considered and relevance to the microgravity environment is identified. Formulation of the problem including the effects of g-jitter is given, and a two-time-scale analysis, valid in the limit of high-frequency modulation, is employed to derive the equations governing the mean field. Two-dimensional linear stability results show that the effective rapid gravity modulation is destabilizing, resulting in a dramatically wider range of parameters where this double-diffusive instability is expected to prevail in suitable space experiments.

The dynamics of a laminar flow in a symmetric channel with a sudden expansion

Bifurcation analysis, linear stability study, and direct numerical simulations of the dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long channel with a sudden expansion with right angles and with an expansion ratio D/d (d is the width of the channel inlet section and D is the width of the outlet section) are presented. The bifurcation analysis of the steady flow equations concentrates on the flow states around a critical Reynolds number Rec(D/d) where asymmetric states appear in addition to the basic symmetric states when Re [ges ] Rec(D/d). The bifurcation of asymmetric states at Rec has a pitchfork nature and the asymmetric perturbation grows like √Re − Rec(D/d). The stability analysis is based on the linearized equations of motion for the evolution of infinitesimal two-dimensional disturbances imposed on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric mode of disturbance exists at Rec(D/d) for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when Re < Rec(D/d) the symmetric states have an asymptotically stable mode of disturbance. However, when Re > Rec(D/d), the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when Re > Rec(D/d) the asymmetric states have an asymptotically stable mode of disturbance. The direct numerical simulations are guided by the theoretical approach. In order to improve the numerical simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to asymmetric states is demonstrated. The simulations clarify the relationship between the linear stability results and the time-asymptotic behaviour of the flow. The current analyses provide a theoretical foundation for previous experimental and numerical results and shed more light on the transition from symmetric to asymmetric states of a viscous flow in an expanding channel. It is an evolution from a symmetric state, which loses its stability when the Reynolds number of the incoming flow is above Rec(D/d), to a stable asymmetric equilibrium state. The loss of stability is a result of the interaction between the effects of viscous dissipation, the downstream convection of perturbations by the base symmetric flow, and the upstream convection induced by two-dimensional asymmetric disturbances.

Miscible porous media displacements in the quarter five-spot configuration. Part 3. Non-monotonic viscosity profiles

The influence of a non-monotonic viscosity–concentration relationship on miscible displacements in porous media is studied for radial source flows and the quarter five-spot configuration. Based on linear stability results, a parametric study is presented that demonstrates the dependence of the dispersion relations on both the Péclet number and the parameters of the viscosity profile. The stability analysis suggests that any displacement can become unstable provided only that the Péclet number is sufficiently high. In contrast to rectilinear flows, for a given end-point viscosity ratio an increase of the maximum viscosity generally has a destabilizing effect on the flow. The physical mechanisms behind this behaviour are examined by inspecting the eigensolutions to the linear stability problem. Nonlinear simulations of quarter five-spot displacements, which for small times correspond to radial source flows, confirm the linear stability results. Surprisingly, displacements characterized by the largest instability growth rates, and consequently by vigorous viscous fingering, lead to the highest breakthrough recoveries, which can even exceed that of a unit mobility ratio flow. It can be concluded that, for non-monotonic viscosity profiles, the interaction of viscous fingers with the base-flow vorticity can result in improved recovery rates.

Matched Asymptotic Expansions and the Closure Problem for Combustion Waves

The closure problem for combustion waves arises when applying the method of matched asymptotic expansions for large activation energy to many nonsteady combustion problems. The exponential nature of the dependence of the reaction rate on temperature and the large coefficient (activation energy) in the exponent lead to the first order correction for temperature appearing in the equations at leading order. Equations describing the first order correction involve the second order correction, and so on. These terms can be scaled away for steady solutions, but when considering nonsteady propagation, they remain for any constant scaling of temperature. The closure problem refers to the fact that the equations must be solved at all orders before the leading order solution can be determined. One traditional approach to alleviate the problem is to truncate the series. While sacrificing the distinction between scales of temperature variation ahead of and behind the flame, these methods allow the replacement of the distributed reaction with a delta function reaction rate model. We present an alternative procedure which uses generalized matched asymptotic expansions and an associated truncation-like restriction to allow formulation of a model which retains the scales of temperature variation at the expense of a simple delta function interpretation of the model. The procedure is demonstrated through the derivation of models for gasless combustion as well as the diffusional-thermal (constant density approximation) theory for gaseous combustion. Comparison with results using standard truncation methods reveals quantitative differences in linear stability results. Stability of planar uniformly propagating wave solutions for the diffusional-thermal model is shown to depend on both Lewis number (ratioof thermal and species diffusivities) and Zeldovich number (ratio of diffusive to reactive length scales). This procedure should be of interest to those exploring weakly nonlinear analysis of steady solutions to gasless and diffusional-thermal combustion problems and to those developing models for other combustion configurations such as porous media combustion.

Analysis of nonlinear vibrations in metal cutting

A nonlinear model of wave cutting is presented. Orthogonal cutting conditions are assumed. Effects of the cutting velocity on the mean friction coefficient are accounted for. An approximate analytical solution of the nonlinear problem is obtained. It is shown that for some supercritical conditions, corresponding to linear instability, the amplitude of vibrations saturates and decreases in the nonlinear analysis. A study of the influence of different parameters (friction, ploughing force, cutting velocity, etc.) on the linear and nonlinear stability results is developed.

Identification of a nonlinear compressor model

In this article, a technique for the identification of a nonlinear compressor performance model exhibiting bifurcation phenomena is presented. The approach consists of employing the modal form of the compressor flow equations along with derived linear and nonlinear stability results. When the identified steady-state model is used in the system, bifurcation analysis tools predict stable solutions in accord with experimental steady-state data.

Stability of Newtonian‐PTT coextrusion fiber spinning

AbstractLinear stability analysis has been applied to a coextrusion fiber spinning flow that consists of a Newtonian fluid as a core layer and a Phan‐Thien/Tanner (PTT) fluid as a skin layer. These two chosen fluids show entirely different rheological behaviors. The stability of this coextrusion system was affected by the choice of three characteristic parameters: the skin layer fraction (f), the extensional parameter (ϵ), and the shear thinning parameter (ξ) in the Phan‐Thien/Tanner model. The linear stability results indicate that the viscoelastic skin layer (PTT fluid) has a stabilizing effect that delays the onset of draw resonance. Under fixed compositions (f is fixed), the stability envelopes changed from upturned curves to flattened ones as extensional force dominated the system. The neutral stable curves closed to the horizontal line at a critical draw ratio of around 20, showing similar behavior to a Newtonian fluid where the system has a very high Deborah number or it is dominated by shear thinning effects.

Mathematical Study of a Model for Liquid–Vapour Phase Change in Porous Media

This paper is concerned with the mathematical study of a model describing the liquid–vapour phase change in porous media. Existence and uniqueness of equilibria for the stationary problem are proved. Linear and nonlinear stability results are obtained. In the one space dimensional case the global existence and uniqueness of the solution for arbitrary initial data are also proved.

Görtler Vortices with System Rotation

The steady primary instability of Gortler vortices developing along a curved Blasius boundary layer subject to spanwise system rotation is analysed through linear and nonlinear approaches, to clarify issues of vortex growth and wavelength selection, and to pave the way to further secondary instability studies. A linear marching stability analysis is carried out for a range of rotation numbers, to yield the (predictable) result that positive rotation, that is rotation in the sense of the basic flow, enhances the vortex development, while negative rotation dampens the vortices. Comparisons are also made with local, nonparallel linear stability results (Zebib and Bottaro, 1993) to demonstrate how the local theory overestimates vortex growth. The linear marching code is then used as a tool to predict wavelength selection of vortices, based on a criterion of maximum linear amplification. Nonlinear finite volume numerical simulations are performed for a series of spanwise wave numbers and rotation numbers. It is shown that energy growths of linear marching solutions coincide with those of nonlinear spatially developing flows up to fairly large disturbance amplitudes. The perturbation energy saturates at some downstream position at a level which seems to be independent of rotation, but that increases with the spanwise wavelength. Nonlinear simulations performed in a long (along the span) cross section, under conditions of random inflow disturbances, demonstrate that: (i) vortices are randomly spaced and at different stages of growth in each cross section; (ii) “upright” vortices are the exception in a universe of irregular structures; (iii) the average nonlinear wavelengths for different inlet random noises are close to those of maximum growth from the linear theory; (iv) perturbation energies decrease initially in a linear filtering phase (which does not depend on rotation, but is a function of the inlet noise distribution) until coherent patches of vorticity near the wall emerge and can be amplified by the instability mechanism; (v) the linear filter represents the receptivity of the flow: any random noise, no matter how strong, organizes itself linearly before subsequent growth can take place; (vi) the Gortler number, by itself, is not sufficient to define the state of development of a vortical flow, but should be coupled to a receptivity parameter; (vii) randomly excited Gortler vortices resemble and scale like coherent structures of turbulent boundary layers.

Interfacial wave stability of concurrent two-phase flow in a horizontal channel

A wave front perturbation method is applied to a one-dimensional two-phase flow in a horizontal duct to analyze the interfacial wave stability. The governing equations are thus reduced to a first-order ordinary differential equation (ODE). The linear solution of this ODE agrees with the conclusion from the small amplitude wave perturbation approach. The non-linear theory suggests that the linear stability result is sufficient for predicting instability in wavy flow, and may over-predict the critical gas velocity for large disturbances. Accordingly, a weak non-linear stability criterion is obtained. The type of unstable wave, under-cut wave or roll wave, also can be predicted analytically.