A simple proof of the linear instability of rotating liquid drops

We give a simplified proof of the linear instability of equilibrium figures of rotating liquid based on energy estimates.      

On the effect of electron-mass variation in numerical simulations of the Farley-Buneman instability

The Farley-Buneman instability is the plasma instability in the E region of the Earth’s ionosphere. Many studies on instability simulations use algorithms based on the particle method, which has many shortcomings. In particular, the solutions obtained with this method involve numerical noises owing to a finite number of the particles, since the electron mass is increased in order to reduce the computational complexity of the algorithm. In this study, the effect of an increase in the electron mass on the qualitative and quantitative characteristics of the instability development process are considered. For this purpose, a software package developed on the basis of a mathematical model consisting of the Poisson equation for the electric potential, the fluid equation for electrons, and the kinetic equation for ions is used. The equations are solved numerically, which makes it possible to avoid the problems inherent in the particle method. An important result obtained in this paper is that even a slight increase in the electron mass leads to a considerable variation of the instability development parameters. This variation manifests itself as an increase in the wavelength of plasma fluctuations and a decrease in the strength of the turbulent electric field.     

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn. Received: July 25, 2002; revised: April 16, 2003     

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].     

Truncated Painleve expansion and reduction of sine-Poisson equation to a quadrature in collisionless cold plasma

The relativistic nonlinear self-consistent equations for a collisionlesscold plasma with stationary ions is considered. The resultingsystem of equations for stationary and non-stationary equationsin two-dimensions can be reduced to the sine-Poisson equation.The truncated Painlevé expansion and reduction of thepartial differential equation to a quadrature problem (RQ method)are described and applied to obtain the travelling wave solutionsof the sine-Poisson equation describing the charge density equilibriumconfiguration model.     

Nonlinear instability in Vlasov type equations around rough velocity profiles

In the Vlasov-Poisson equation, every configuration which is homogeneous in space provides a stationary solution. Penrose gave in 1960 a criterion for such a configuration to be linearly unstable. While this criterion makes sense in a measure-valued setting, the existing results concerning nonlinear instability always suppose some regularity with respect to the velocity variable. Here, thanks to a multiphasic reformulation of the problem, we can prove an “almost Lyapounov instability” result for the Vlasov-Poisson equation, and an ill-posedness result for the kinetic Euler equation and the Vlasov-Benney equation (two quasineutral limits of the Vlasov-Poisson equation), both around any unstable measure.     

Interactions of nonlinear ion-acoustic waves in a collisionless plasma

In the present work, based on a one-dimensional model, the interaction of two solitary waves propagating in opposite directions in a collisionless plasma is investigated by use of the extended Poincaré–Lighthill–Kuo (PLK) method. It is shown that bi-directional solitary waves are propagated and the head-on collision of these two solitons occur. The phase shifts and the trajectories of these two solitons after the collision are obtained.     

Electrostatic wave solutions for a nonlinear coupled system in an inhomogeneous collisional dusty magnetoplasma

In an inhomogeneous collisional dusty magnetoplasma, a new coupled (3 + 1)-dimensional nonlinear system is derived for the low-frequency electrostatic waves considering the collision between ions and neutrals. It is demonstrated that due to the collision, the scaling symmetry of the system is destroyed. By means of the classical Lie group approach, two types of exact similarity waves are obtained which show three important features. First, various waves can be constructed from these solutions, such as solitary waves, shock waves and periodic waves. Second, these waves may have shears that are time-dependent, linear and nonlinear. Third, the electrostatic potential and the parallel electron velocity possess more freedoms in the sense that they can have either same or different wave forms.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

A note on the growth of velocities in a collisionless plasma

We consider the initial value problem for the classical Vlasov–Poisson system with smooth compactly supported initial data. In the electrostatic case, we show that the size of the velocity support of the distribution function grows at most like t^2/7+bfε for any ε> 0 . Copyright © 2010 John Wiley & Sons, Ltd.     

Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth. For any positive number there exists a radially symmetric stationary solution with free boundary . The system depends on a positive parameter , and for a sequence of values there also exist branches of symmetric-breaking stationary solutions, parameterized by , small, which bifurcate from these values. In particular, for near the free boundary has the form where is the spherical harmonic of mode . It was recently proved by the authors that the stationary solution is asymptotically stable for any , but linearly unstable if , where if and if ; . In this paper we prove that for each of the stationary solutions which bifurcates from is linearly stable if and linearly unstable if . We also prove, for , that the point is a Hopf bifurcation, in the sense that the linearized time-dependent problem has a family of solutions which are asymptotically periodic in .

     

Modulational Stability of Travelling Waves in 2D Anisotropic Systems

The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion system and to the weak electrolyte model for electroconvection in nematic liquid crystals.      

Modulational instability in a full‐dispersion shallow water model

We propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.     

The spatial patterns through diffusion-driven instability in a predator-prey model

Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.     

Characterization of spatial patterns produced by a Turing instability in coupled dynamical systems

We quantify the degree of spatial order of patterns at fixed time generated by lattices of coupled dynamical systems, using correlation-based and recurrence-based numerical diagnostics. These patterns are obtained through numerical integration of differential equations describing the interplay between activator and inhibitor species generating Turing patterns. We consider different types of coupling: linear (diffusive) interaction with nearest-neighbors, global (all-to-all) coupling and intermediate (nonlocal) coupling. Numerical simulations are performed in one and two spatial dimensions. The effects of noise are briefly discussed. We introduce a recurrence-based quantity (recurrence-rate matrix) to characterize two-dimensional spatial patterns.     

Rayleigh-Taylor instability of compressible finitely conducting rotating fluid in the presence of a magnetic field

The character of the equilibrium of a non-viscous, compressible finitely conducting rotating fluid in the presence of a vertical magnetic field along the direction of gravitational field has been investigated. It is shown that the solution is characterised by a variational principle. Based on the existence of variational principle, an approximate solution has been derived for the case of a fluid having exponentially varying density in the vertical direction. Due to finite resistivity of the medium it is found that potentially stable or unstable configuration retains its character. Further the growth rate of disturbance has been obtained corresponding to short and long wavelengths and it is found that electrical resistivity suppresses the growth rate for large wavelengths but it increases the same for small wavelengths. It is further shown that magnetic field has a destabilizing influence for large wavelengths and a stabilizing influence for small wavelengths.     

Evans functions and bifurcations of standing wave fronts of a nonlinear system of reaction diffusion equations

Consider the following nonlinear system of reaction diffusion equations arising from mathematical neuroscience $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u]-w,~ \frac{\partial w}{\partial t}=\varepsilon(u-\gamma w).$ Also consider the nonlinear scalar reaction diffusion equation $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u].$ In these model equations, $\alpha>0$, $\beta>0$, $\gamma>0$, $\varepsilon>0$ and $\theta>0$ are positive constants, such that $0<2\theta<\beta$. In the model equations, $u=u(x,t)$ represents the membrane potential of a neuron at position $x$ and time $t$, $w=w(x,t)$ represents the leaking current, a slow process that controls the excitation.\\indent The main purpose of this paper is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing wave fronts) and Evans functions (complex analytic functions) to establish the existence, stability, instability and bifurcations of standing wave fronts of the nonlinear system of reaction diffusion equations and to establish the existence and stability of the standing wave fronts of the nonlinear scalar reaction diffusion equation.