Decay Of Small Perturbations Research Articles

A kinetic reaction model: Decay to equilibrium and macroscopic limit

We propose a kinetic relaxation-model to describe a generation-recombination reaction of two species.The decay to equilibrium is studied by two recent methods [9,13] for proving hypocoercivity of the linearized equations.Exponential decay of small perturbations can be shown for the full nonlinear problem.The macroscopic/fast-reaction limit is derived rigorously employing entropy decay, resulting in a nonlinear diffusion equation for the differenceof the position densities.

Time-to-ignition in a tubular reactor operated adiabatically

Even a simple exothermic reaction can show multiple stationary states in flow through a tubular reactor. Discontinuous jumps of ignition and extinction can occur in response to continuous changes in such typical control parameters as flow-rates or inlet-temperatures (expressed by Damköhler or Péclet numbers, Da or Pe ). Under marginally supercritical conditions, runaway develops expo­nentially but the time-constant becomes very large as criticality is approached. We show that the dependence can be written: t ign = M /[( Da / Da cr ) - 1] ½ . The validity of this form and the predicted size of the coefficient M are checked by numerical computation for the case of a single, exothermic reaction proceeding under adiabatic conditions. Similar expressions describe the decay of small perturbations to stationary states under marginally subcritical circumstances. At the point of criticality, these exponential growths or decays are replaced by hyperbolic forms.

Parametric instabilities of internal gravity waves in boussinesq fluids with large reynolds numbers

Abstract Previous studies of parametric instabilities of a finite-amplitude internal gravity wave in dissipationless Boussinesq fluids predict the largest growth rates for perturbations with vanishingly small wavelengths, leading to the practical question of which perturbations are most unstable in the presence of molecular dissipation. At high Reynolds numbers, the amplitude of the gravity wave decays on spatial and temporal scales which are large compared to the wavelength and period respectively. Therefore, the interaction equations describing the growth or decay of small perturbations can be solved by a conventional expansion procedure yielding a leading term in the form of a Floquet solution. Numerical calculations show that the competitive processes of parametric growth and viscous decay produce instabilities with maximum growth rates at wavelengths which may be similar or even longer than the wavelength of the basic gravity wave. Two transformations leading to a significant reduction of computation...

The Approach to Stationary States of Autocatalytic Systems in the Continuous‐flow Stirred Tank Reactor

AbstractThe relaxation times characterizing the growth or decay of small perturbations from a stationary‐state established under continuous‐flow, well‐stirred conditions are derived for the autocatalytic sequences A + n B → (n + 1) B, with n = 1 or 2. The results show how as the flow‐rate is varied, relaxation times may gradually become very long compared with the mean residence‐time of the system. This lengthening is accompanied by a change in the mathematical form, from an exponential to an inverse dependence on time. This is not, however, a critical phenomenon, nor are infinite values possible.

On the stability of plane shock waves

The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.