Reentrant and whirling hexagons in non-Boussinesq convection

We review recent computational results for hexagon patterns in non-Boussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a vertical axis the transition from hexagons to rolls is replaced by a Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects they display defect chaos of the type described by the two-dimensional (2D) complex Ginzburg–Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and localized bursting of the whirling amplitude is found. In this regime the coupling of the whirling amplitude to (small) deformations of the hexagon lattice becomes important. For yet stronger non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly disordered states characterized by whirling and lattice defects are obtained.     

Stability of fronts separating domains with different symmetries in hydrodynamical instabilities

A generalization of the Swift-Hohenberg (SH) equation is used to study several stationary patterns that appear in hydrodynamical instabilities. The corresponding amplitude equations allow one to find the stability of planforms with different symmetries. These results are compared with numerical simulations of a generalized SH equation (GSHE). The transition between different symmetries, the hysteretic effects, and the characteristics of the defects observed in experiments are well reproduced in these simulations. The existence of steady fronts between domains with different symmetries is also analyzed. Steady domain boundaries between hexagons and rolls, and between hexagons and squares are possible solutions in the amplitude equation framework and are obtained in numerical simulations for a full range of coefficients in the GSHE.     

Defect chaos and bursts: hexagonal rotating convection and the complex Ginzburg-Landau equation

We employ numerical computations of the full Navier-Stokes equations to investigate non-Boussinesq convection in a rotating system using water as the working fluid. We identify two regimes. For weak non-Boussinesq effects the Hopf bifurcation from steady to oscillating (whirling) hexagons is supercritical and typical states exhibit defect chaos that is systematically described by the cubic complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the oscillations exhibit localized chaotic bursting, which is modeled by a quintic complex Ginzburg-Landau equation.     

Stability and fluctuations of multi-mode configurations near the convection instability

The exact solution of a Fokker-Planck equation yields a distribution function which governs the stability and fluctuations of various mode configurations (e.g. hexagons and roles) of the Bénard convection of fluids and related problems.     

Secondary instabilities of hexagonal patterns in a Bénard-Marangoni convection experiment

We have identified experimentally secondary instability mechanisms that restrict the stable band of wave numbers for ideal hexagons in Bénard-Marangoni convection. We use "thermal laser writing" to impose long wave perturbations of ideal hexagonal patterns as initial conditions and measure the growth rates of the perturbations. For epsilon=0.46 our results suggest a longitudinal phase instability limits stable hexagons at a high wave number while a transverse phase instability limits low wave number hexagons.     

Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation

Weakly nonlinear spatially periodic patterns coupled to a Goldstone (zero) mode of the phase-field crystal model are investigated. Rotationally invariant equations for the dynamics of the amplitudes of a hexagonal pattern are derived first, which then allows us to determine stability regions for stripes and hexagons. There are parameter regimes in which all periodic patterns become unstable as a result of long-wavelength instabilities generated by the zero mode.     

Nonequilateral hexagonal patterns

Rotational symmetry of pattern formation problems is the origin of a variety of patterns (rolls, squares, hexagons etc.) in convection and reaction-diffusion systems. Traditionally, only the patterns based on equilateral lattices in the Fourier space were considered. In the present paper, we develop an analysis of the patterns with slightly different lengths of the basic wave vectors. The analysis applies as well to systems with a broken rotational symmetry (convection in an inclined layer, etc.). We find, in the framework of the amplitude equations, existence and stability conditions for periodic nonequilateral patterns based on two and three wave vectors. In the latter case, special attention is paid to the case when the three amplitudes are coupled by the resonant interaction.     

Stability of steady thermal convection in a tilted rectangular cavity in low-mode approximation

The model system of ordinary differential equations [1, 2] governing the behavior of a non-uniformly heated fluid in a tilted cavity is used for studying the stability of steady regimes of thermal convection at arbitrary (not small) tilting of the rectangular cavity. The bifurcation curve is constructed, which separates the region of parameters (the Rayleigh number — the cavity tilting angle) into two regions — the internal and external ones. In the external region, the system has one stable steady solution, and in the internal region, it has three steady solutions. One of them is always unstable in a monotone way, and two others may be both stable and unstable. The neutral curves are constructed, which determine the boundaries of the incipience of the oscillatory and monotone instabilities. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 07-01-96070).     

Neural field dynamics with heterogeneous connection topology

Neural fields receive inputs from local and nonlocal sources. Notably in a biologically realistic architecture the latter vary under spatial translations (heterogeneous), the former do not (homogeneous). To understand the mutual effects of homogeneous and heterogeneous connectivity, we study the stability of the steady state activity of a neural field as a function of its connectivity and transmission speed. We show that myelination, a developmentally relevant change of the heterogeneous connectivity, always results in the stabilization of the steady state via oscillatory instabilities, independent of the local connectivity. Nonoscillatory instabilities are shown to be independent of any influences of time delay.     

Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

By using the mathematical formalism of absolute and convective instabilities we study the nature of unstable three-dimensional disturbances of viscoelastic flow convection in a porous medium with horizontal through-flow and vertical temperature gradient. Temporal stability analysis reveals that among three-dimensional (3D) modes the pure down-stream transverse rolls are favored for the onset of convection. In addition, by considering a spatiotemporal stability approach we found that all unstable 3D modes are convectively unstable except the transverse rolls which may experience a transition to absolute instability. The combined influence of through-flow and elastic parameters on the absolute instability threshold, wave number and frequency is then determined, and results are compared to those of a Newtonian fluid.     

Transient patterns of the convection instability: A model-calculation

Using a recently derived non-linear partial differential equation describing the temperature field we have performed computer calculations on the evolving convection patterns in different geometries. In this way we calculate the generation of various patterns e.g. of rolls or hexagons.     

Natural convection in porous annular domains: Mimetic scheme and family of steady states

Natural convection of the incompressible fluid in the porous media based on the Darcy hypothesis (Lapwood convection) gives an intriguing branching off of one-parameter family of steady patterns. This scenario may be suppressed in computations when governing equations are approximated by schemes which do not preserve the cosymmetry property. We consider the problem for the annular porous domain in polar coordinates and derive a mimetic finite-difference scheme. This scheme allows to compute the family of convective regimes accurately and to detect the instabilities on some parts of the family.     

Linear stability analysis on the onset of Soret-driven motion in a nanoparticles suspension

The onset of Soret-driven instability in binary mixture heated from above is analysed using the linear stability theory. The horizontal fluid layer placed between two plates is in a thermally stable state but the Soret diffusion can induce buoyancy-driven convection in the case of a negative Soret coefficient. It is well known that convective motion sets in from both boundaries if the Rayleigh number exceeds its critical value. For the case of highly unstable density stratification the buoyancy-driven motion sets in during the transient diffusion stage. The new stability equations are derived and are solved analytically and numerically. Here the stability limits which are related to the onset time of instabilities and wave number are presented as a function of the Rayleigh number, Lewis number and the separation ratio. The present stability criteria are compared with the existing experimental data.     

Investigations to the increase of the specific energy of a gasdynamic CO-laser

The input power density and hence the output power of electrically excited gasdynamic lasers is limited by instabilities of the glow discharge. The application of theoretical results, which have been obtained with respect of convection or flow lasers, to the discharge region of an electrically excited gasdynamic CO-laser shows, that especially thermal instabilities cause this glow collapse. An increased convection of local heat concentrations in front of the anode surface results in an improved stability behaviour. Input power densities of up to 100 W/cm³ are now accessible to operate the glow discharge and hence specific laser output powers of 57 kJ/kg are obtained.     

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.     

Reentrant hexagons in non-Boussinesq Rayleigh-Bénard convection: effect of compressibility

We present experimental studies of a new pattern sequence observed in non-Boussinesq convection in a compressible fluid near its gas-liquid critical point (CP). Besides the known hysteretic transitions among conduction state, hexagons, and rolls, another hysteretic transition from rolls to hexagons at higher values of the control parameter is found. This reentrance phenomenon is observed in a rather narrow range of about 60- to 100-microm cell heights and is attributed to large compressibility of a fluid near the CP.     

Secondary transverse instabilities in optical parametric oscillators

We investigate the effect of coupling between diffraction and walk-off on secondary instabilities in nondegenerate optical parametric oscillators. We show that traveling waves that propagate in the walk-off direction, which are generated at the onset of absolute instability, experience Eckhaus and zigzag phase instabilities. Each of these secondary instabilities splits into absolute and convective instabilities that modify the Eckhaus and zigzag instability boundaries. As a consequence, the stability domain of modulated traveling waves is enlarged and may coexist with uniform steady states. The predictions are consistent with the numerical solutions of the optical parametric oscillator model.     

Instabilities of thermocapillary-buoyancy convection in open rectangular liquid layers

This article presents the experimental investigation on instabilities of thermocapillary-buoyancy convection in the transition process in an open rectangular liquid layer subject to a horizontal temperature gradient. In the experimental run,an infrared thermal imaging system was constructed to observe and record the surface wave of the rectangular liquid layer. It was found that there are distinct convection longitudinal rolls in the flow field in the thermocapillary-buoyancy convection transition process. There are different wave characterizations for liquid layers with different thicknesses. For sufficiently thin layers, oblique hydrothermal waves are observed, which was predicted by the linear-stability analysis of Smith Davis in 1983. For thicker layers, the surface flow is distinct and intensified, which is because the buoyancy convection plays a dominant role and bulk fluid flow from hot wall to cold wall in the free surface of liquid layers. In addition, the spatiotemporal evolution analysis has been carried out to conclude the rule of the temperature field destabilization in the transition process.     

Evidence of low-dimensional chaos in magnetized plasma turbulence

We analyze probe data obtained from a toroidal magnetized plasma configuration suitable for studies of low-frequency gradient-driven instabilities. These instabilities give rise to field-aligned convection rolls analogous to Rayleigh-Benard cells in neutral fluids, and may theoretically develop similar routes to chaos. When using mean-field dimension analysis, we observe low dimensionality, but this could originate from either low-dimensional chaos, periodicity or quasi-periodicity. Therefore, we apply recurrence plot analysis as well as estimation of the largest Lyapunov exponent. These analyses provide evidence of low-dimensional chaos, in agreement with theoretical predictions. Our results can be applied to other magnetized plasma configurations, where gradient-driven instabilities dominate the dynamics of the system.     

On the stability of one-dimensional diffusion flames established between plane,parallel, porous walls

A one-dimensional, non-premixed flame stability analysis is undertaken.Oscillatory and cellular flame instabilities are identified by a careful studyof the numerically calculated eigenvalues of the linearized system of equations. The numerical investigation details the critical locations for changes in flame behaviour, as well as the critical values of variousparameters that affect flame stability. A critical Lewis number, greaterthan unity, is identified as the value where unstable oscillations maybegin to appear (Le?>?Le _c) and for which cellular flames can exist(Le?<?Le _c). Some prior discussions are clarified regarding theaforementioned critical values, as well as the role of convection inproducing flame instabilities. The methodology of the stability analysis isdiscussed in detail.