Prototype nonlinear differential equation for nonperiodic instabilities
Abstract
A nonlinear ordinary differential equation that describes the temporal evolution of nonperiodic instabilities, such as those that arise in convection phenomena, is developed. The role played by this prototype equation in the description of nonperiodic instabilities is the counterpart of that played by the van der Pol equation in the description of periodic instabilities. The equation is solved by singular perturbation methods and convergence problems that arise in the use of standard (Poincare) perturbation methods are indicated. The significance of terms of the form tau exp (tau) in the approach to equilibrium is noted, these terms being analogous to the secular terms that appear in the periodic case. The prototype equation can be used as a guide in evaluating phenomena that are of possible importance as nonlinear saturation mechanisms in nonperiodic instabilities. The use of the equation is illustrated with the wellknown Benard thermal instability and the inappropriate use of the conventional Landau analysis in earlier treatments of nonperiodic instabilities is indicated.
 Publication:

Maryland Univ. College Park Report
 Pub Date:
 August 1975
 Bibcode:
 1975umd..rept.....G
 Keywords:

 Fluid Flow;
 Nonlinear Equations;
 Nonuniform Flow;
 Convective Flow;
 Partial Differential Equations;
 Perturbation Theory;
 Fluid Mechanics and Heat Transfer