Non-linear wave-number interaction in near-critical two-dimensional flows

Abstract

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

Traveling waves in the complex Ginzburg-Landau equation

“…The equation has been derived for many, mainly hydrodynamical, nonlinear stability problems, such as (binary) fluid convection [30], [34], Poiseuille flow [35], chemical reactions [26], and so on. In DiPrima et al [11] and Newell [29] it has been shown that the Ginzburg-Landau equation generically appears as a modulation, or amplitude/envelope, equation for general systems at the onset of pattern formation. Recently, some important steps have been made toward a rigorous proof of its mathematical validity as a modulation equation ([23], [10], [19]).…”

Periodic and quasi-periodic solutions of degenerate modulation equations

“…for a bifurcation/control parameter A close to a A~ at which a stationary laminar solution becomes linearly unstable) has been developed in refs. [2,15,17]. However (1.1) is only the G(1) part of a more complicated equation.…”

Slow time-periodic solutions of the Ginzburg-Landau equation

“…The equation has been derived by many authors, in various ways. The earliest papers in which the Ginzburg-Landau equation is derived are: Newell and Whitehead [11], Stewartson and Stuart [12], DiPrima, Eckhaus and Segel [13]. In these papers the authors start the analysis by investigating the linear stability of a basic (laminar) solution as a control parameter R is varied.…”