Finite bandwidth, finite amplitude convection

Abstract

The main purpose of this work is to show how a continuous finite bandwidth of modes can be readily incorporated into the description of post-critical Rayleigh-Bénard convection by the use of slowly varying (in space and time) amplitudes. Previous attempts have used a multimodal discrete analysis. We show that in addition to obtaining results consistent with the discrete mode approach, there is a larger class of stable and realizable solutions. The main feature of these solutions is that the amplitude and wave-number of the motion is that of the most unstable mode almost everywhere, but, depending on external and initial conditions, the roll couplets in different parts of space may be 180° out of phase. The resulting discontinuities are smoothed by hyperbolic tangent functions. In addition, it is clear that the mechanism for propagating spatial nonuniformities is diffusive in character.

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

“…Going beyond the question of existence, an equally fundamental topic is stability, or "selection," of periodic patterns, and linear and nonlinear behavior under perturbation [E,NW,M1,M2,M3,S1,S2,DSSS,SSSU,JZ,JNRZ1,JNRZ2]. Here, two particular landmarks are the formal "weakly unstable," or small-amplitude, theory of Eckhaus [E] deriving the Ginzburg Landau equation as a canonical model for behavior near the threshold of instability in a variety of processes, and the rigorous linear and nonlinear verification of this theory in [M1, M2, S1] for the Swift-Hohenberg equation, a canonical model for hydrodynamic pattern formation.…”

Multi-bump, self-similar, blow-up solutions of the Ginzburg–Landau equation

“…It is derived in, for example, Rayleigh-Bénard convection, Taylor-Couette flow, nonlinear optics, models of turbulence, superconductivity, superfluidity and reaction-diffusion systems, see [18,3,23,8,9] and the review article [2]. The GL can be viewed as a normal form describing the leading order behaviour of small perturbations in 'marginally unstable' systems of nonlinear partial differential equations defined on unbounded domains, [16].…”

Finite bandwidth, long wavelength convection with boundary imperfectons: near-resonant wavelength excitation

“…For the case/g 1, we found that the solution (4.6) is stable for P) < -1. Of course, these stability conditions were determined based on upper bounding type approach, and it is possible that the solutions for N 1 and N 2 still remain stable if/l is bigger [3,4], where it is assumed that :r-axis is along these rolls The resulting amplitude system will then be a system of nonlinear partial differential equations where each equation is second order in derivative with respect to z, and fourth order in derivative with respect to /s Although the results for such a system will be reported elsewhere, it is of interest to note here that such a system can admit nonmodal solutions with kinks, different from those discussed in the present paper, and the resulting preferred patterns will be affected accordingly.…”

“…The present investigation of the continuous finite bandwidth of convection modes applies the method of approach due to Newell and Whitehead [4]. As was explained by Newell and Whitehead [4], continuous-modal analysis of convection leads to a wider class of solutions which can describe adequately the problem with the amplitude modulations which inevitably occur as a result of, for example, nonuniform boundary imperfections 172 D N. RIAHI Rees and Riley [5,6] investigated the effects of one-dimensional sinusoidal boundary imperfections on weakly nonlinear th=rmal onv=ction in a porous mdium and dtermined, in particular, the nonlin=ar system for the flow amplitudes, and the effects of the boundary modulations on the stability of different roll cells, and the evolution of the unstable rolls were studi=d Rs [7] investigated the effect of onedimensional long wavelength thermal modulations on th= onset of Convection in a porous medium and predicted, in particular, the preference of a mode in the form of rectangular cells for certain ranges of values of the modulation wave number.…”

“…In order to formulate the problem for a continuous finite bandwidth of modes, we follow the method of approach due to Newell and Whitehead [4] However, these authors formulated the problem with isothermal boundaries where they allowed an o(e) The amplitude functions A, satisfy the condition A A_, (3 6) where the asterisk indicates the complex conjugate The expressions for fl (z) and gx (z) are given in Riahi [3], and the details of the results for Ro and its minimum/ attained at c cc can also be found in Riahi [3]. …”