Modeling film flows down inclined planes

A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q, and a new variable that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier-Stokes equations. Received: 16 April 1998 / Revised: 29 June 1998 / Accepted: 2 July 1998     

Statics and dynamics of a single ferrofluid-peak

We consider a single peak of a ferrofluid resulting from the Rosensweig instability for a small fluid container. Minimizing the total energy of the system by a variational method we determine the shape of the peak in a static field as well as the characteristics of the subcritical bifurcation leading to its formation. The latter are in very good agreement with experiment. Generalizing the approach to dynamic situations we qualitatively reproduce the complicated subharmonic response of the peak to an oscillating part in the external magnetic field found in recent experiments. Received 14 December 1999 and Received in final form 31 May 2000     

An analytical estimate of the period for the delayed logistic application and the Lotka-Volterra system

We first introduce a simple and new method for the quantitative analysis of some nonlinear oscillating systems. It is shown that if the dynamics of the system reduces to piecewise exponential growth and exponential damping phases, then the amplitude and period of the motion can be computed with accuracy in the nonlinear regime without invoking linear stability arguments or perturbative expansions. This method is then successfully applied to the delayed logistic application and to the Lotka-Volterra prey-predator model. For both of these systems, we provide an accurate analytical expression for the period of the oscillations in the nonlinear regime. Received: 27 April 1998 / Received in final form: 25 June 1998 / Accepted: 29 June 1998     

Nonlinear-driven instability of dynamical plasma in solids: Emergence of spatially self-organized order and chaotic-like behavior

We analyze in detail the nonlinear kinetics of a carrier system in a photoinjected plasma in semiconductors under the action of constant illumination with ultraviolet light. We show that the spatially homogeneous steady-state becomes unstable, and a charge density wave emerges after a critical intensity of the incident radiation is achieved. It is shown that this instability can only follow in doped p-type materials. In bulk systems the critical intensity was found to be too high making the phenomenon not observable under realistic experimental conditions. However, a more efficient electron excitation can be obtained in low dimensional p-type systems, like some molecular and biological polymers, where the interaction may follow by chemical interaction with the medium. We show that for intensities beyond the critical threshold an increasing number of modes provide further contributions (subharmonics) to the space inhomogeneity. It is conjectured that this process could lead the system to display chaotic-like behavior. Received 8 July 1998 and Received in final form 6 May 1999     

Spatio-temporal patterns in the thermoconvection of a planar nematic layer: I. Weakly nonlinear models

We study theoretically the formation of convection patterns in a laterally extended planar nematic layer heated from below, in the linear and weakly nonlinear regimes. By reformulating the viscous coupling terms of the basic nematohydrodynamic equations, a simple interpretation of the flow effects on the director dynamics can be proposed. A detailed linear analysis of the problem is presented. A systematic method to investigate nonlinear mechanisms is developed, and exemplified by the study of the nonlinear saturation in rolls. The extension of the roll amplitude equation with the envelope formalism is used to characterize the dynamics of the roll modulations near threshold. Coupled envelope equations are shown to describe the structure of the point defects in zig-zags observed experimentally. Finally the bifurcation to the bimodal varicose is studied. The secondary wavevector in the bimodal appears to be selected by a rotation of the director in the horizontal plane. Quantitative predictions concerning the amplitude of this rotation are given. Received: 1st December 1997 / Revised: 25 May 1998 / Accepted: 2 June 1998     

Interface dynamics in liquid crystals

We have experimentally observed the pattern instabilities of an Ising wall formed in a nematic or cholesteric liquid crystal layer. We have deduced an envelope equation, relevant close to the Fréedericksz transition, from which we derived an equation for the dynamics of the interface in the vicinity of its bifurcation. In the case of the zig-zag instability, this model is characterized by a conservative and variational order parameter whose gradient satisfies a Cahn-Hilliard equation. We have also investigated the influence of slightly broken symmetries on the dynamical behaviour of the system. The disappearance of the interface translational invariance or of the reflection symmetry along the wall axis may induce new interfacial patterns which have been both experimentally and theoretically pointed out. Received 5 August 1999 and Received in final form 13 September 1999     

A self-adaptative oscillator

The dynamics of a system where a mass is free to slide on a vibrating string is investigated as the excitation frequency is varied. One degree of freedom is thus added to the system studied by Helmholtz in which a mass was fixed on a vibrating string. This new system exhibits a specific dynamics characterized by the existence of a self- adaptative behaviour. When the driving frequency falls into wide and well defined frequency bands, a long transient is observed by which the mass adjusts its position so that the whole system becomes resonant. In the gaps between these bands, bifurcations give other equilibrium positions. A theoretical model is proposed. It accounts for all the experimental results. In the case where two masses are present on the string, two degrees of freedom are added and the set of equilibrium positions would be expected to be infinite. However, in the experiment, the two masses are observed to go to positions where they are symmetrical with respect to the middle of the string. A selection mechanism due to the string stretching is pointed out. Received 15 October 1998     

Heteroclinic behavior in rotating Rayleigh-Bénard convection

We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven fluid layer with stress-free top and bottom boundaries, a square horizontal periodicity with a small aspect ratio, and rotation at low to moderate rates about a vertical axis. The Prandtl number is 6.8. If the rotation is not too slow, the skewed-varicose instability leads from stationary rolls to a stationary mixed-mode solution, which in turn loses stability to a heteroclinic cycle formed by unstable roll states and connections between them. The unstable eigenvectors of these roll states are also of the skewed-varicose or mixed-mode type and in some parameter regions skewed-varicose like shearing oscillations as well as square patterns are involved in the cycle. Always present weak noise leads to irregular horizontal translations of the convection pattern and makes the dynamics chaotic, which is verified by calculating Lyapunov exponents. In the nonrotating case, the primary rolls lose, depending on the aspect ratio, stability to traveling waves or a stationary square pattern. We also study the symmetries of the solutions at the intermittent fixed points in the heteroclinic cycle. Received 10 June 1999     

Experimental characterization of intermittency regimesin the Couette-Taylor system

We investigate experimentally the properties of spatio-temporal intermittency states (turbulent bursts and spiral turbulence) in the counter-rotating Couette-Taylor system. The mean turbulent fraction increases continuously from turbulent bursts to spiral turbulence and depends on the angular velocity of both cylinders. The axial velocity of turbulent spirals, which depends only on the outer cylinder, is smaller than the azimuthal component. Received 15 December 1999 and Received in final form 19 October 2000     

Coloured noise influence on system evolution

Within the power-law approach for noise amplitude dependence on stochastic variables, we present a picture of noise-induced transitions in systems affected by coloured multiplicative noise. The governed equations for main statistical moments are obtained and investigated in detail. We show that a reentrant noise-induced transition is realized within a window of the control parameter. Received 15 October 2001 / Received in final form 8 July 2002 Published online 17 September 2002     

Statistical analysis of the transition to turbulence in plane Couette flow

We argue on general grounds that the transition to turbulence in plane Couette flow is best studied experimentally at a statistical level. We present such a statistical analysis of experimental data guided by a parallel investigation of a simple coupled map lattice model for spatiotemporal intermittency. We confirm that this generic type of spatiotemporal chaos is relevant in the context of plane Couette flow, where the linear stability of the laminar regime at all Reynolds numbers insures the necessary local subcriticality. Using large ensembles of similar experiments, we show the existence of a well-defined threshold Reynolds number above which a unique, turbulent, intermittent attractor coexists with the laminar flow. Furthermore, our data reveals that this transition to spatiotemporal intermittency is discontinuous, i.e. akin to a first-order phase transition. Received: 10 April 1998 / Revised: 22 June 1998 / Accepted: 24 June 1998     

On the stability of the inertial circulation in the 11/2-layer, quasi-geostrophic model

The effect of the time-dependent interface, separating an inertial quasi-geostrophic upper fluid layer from the quiescent abyss, on the non-linear stability of a steady circulation that takes place in this layer is explored. The analysis resorts to the method of Arnol'd's invariant resulting in a conditional stability criterion, which proves the stabilizing effect of the interface with respect to the single-layer case. The uniqueness of the stable basic flow field follows. Finally, non-linear and linear analyses are compared in the special case of a channeled flow with a fluctuating interface, the latter leading to an unconditional stability statement, whose meaning is clarified by resorting to the previously obtained nonlinear criterion. Received 8 June 2000 and Received in final form 18 September 2000     

Inverse problem of the variational calculus for higher KdV equations

It is shown that the usual Hamilton's variational principle supplemented by the methodology of the integer-programming problem can be used to construct expressions for the Lagrangian densities of higher KdV fields. This is demonstrated with special emphasis on the second and third members of the hierarchy. However, the method is general enough for applications to equations of any order. The expressions for Lagrangian densities are used to calculate results for Hamiltonian densities that characterize Zakharov-Faddeev-Gardner equation. Received 27 January 2002 / Received in final form 6 May 2002 Published online 24 September 2002     

Flame spreading over liquid ethanol

Flame spreading over liquid ethanol has been experimentally characterized for ethanol for subflash temperatures, in two different channels. Three different spreading regimes have been observed. A uniform region (with flame velocities close to 10 cm/s) appears for values of the initial surface liquid temperature above a critical value . For values an oscillatory regime occurs. For very low temperatures, , a new uniform regime appears with slow propagation velocities (close to 1 cm/s). The critical point has been described as a Hopf bifurcation, while resembles a homoclinic connection. Received 16 October 1998 and Received in final form 23 June 1999     

Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures

The Eckhaus stability boundaries of travelling periodic roll patterns arising in binary fluid convection is analysed using high-resolution numerical methods. We present results corresponding to three different values of the separation ratio used in experiments. Our results show that the subcritical branches of travelling waves bifurcating at the onset of convection suffer sideband instabilities that are restabilised further away in the branch. If this restabilisation is produced after the turning point of the travelling-wave branch, these waves do not become stable in a saddle node bifurcation as would have been the case in a smaller domain. In the regions of instability of the uniform travelling waves we expect to find either transitions between states of different wave number or modulated travelling waves arising in these bifurcations.     

Micro-transitions or breathers in L-alanine?

Within mean field approximation, a procedure is elaborated to consider noise induced phase transitions with arbitrary relations between the noises of different degrees of freedom. The proposed approach is applied to investigate effects of cross correlation between noises in the generalized synergetic model of Lorenz type. This cross correlation is shown to induce phase transitions of the dynamical system under consideration. Additionally, we find the correlation between noises transforms a synergetic behavior to a thermodynamic one. Received 13 November 2002 Published online 11 April 2003 RID="a" ID="a"e-mail: dikh@sumdu.edu.ua     

Convective and absolute instabilities in the subcritical Ginzburg-Landau equation

We study the nature of the instability of the homogeneous steady states of the subcritical real Ginzburg-Landau equation in the presence of group velocity. The shift of the absolute instability threshold of the trivial steady state, induced by the destabilizing cubic nonlinearities, is confirmed by the numerical analysis of the evolution of its perturbations. It is also shown that the dynamics of these perturbations is such that finite size effects may suppress the transition from convective to absolute instability. Finally, we analyze the instability of the subcritical middle branch of steady states, and show, analytically and numerically, that this branch may be convectively unstable for sufficiently high values of the group velocity. Received 17 December 1998     

Discrete scale invariance in viscous fingering patterns

We study viscous fingering patterns in a lifting Hele-Shaw cell, where a non-Newtonian fluid (oil paint) is displaced by air. The lengths of the air fingers are measured and their cumulative distribution is seen to follow a power law with log-periodic oscillations indicating the presence of discrete scale invariance. Received 21 May 1999     

Electric Nusselt number characterization of electroconvection in nematic liquid crystals

We develop a characterization method of electroconvection structures in a planar nematic liquid crystal layer by a study of the electric current transport. Because the applied potential difference has a sinusoidal time dependence, we define two electric Nusselt numbers corresponding to the in-phase and out-of-phase components of the current. These Nusselt numbers are predicted theoretically using a weakly nonlinear analysis of the standard model. Our measurements of the electric current confirm that both numbers vary linearly with the distance from onset until the occurence of secondary transitions. A systematic comparison between our theoretical and experimental results, using no adjusted parameters, demonstrates moderate agreement, but discrepancies remain. Electric transport measurements during electroconvection represent a quantitative test of the standard model completely independent from optical probes. Thus, the technique described here can be a useful complement to traditional structural measurements. Received 21 January 2001 and Received in final form 1st February 2002