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A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows
Author(s) -
Herron I. H.,
Maslowe S. A.,
Melkonian S.
Publication year - 1987
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1987763265
Subject(s) - amplitude , instability , heat equation , physics , scaling , mathematical analysis , diffusion equation , perturbation (astronomy) , mathematics , classical mechanics , mathematical physics , mechanics , geometry , quantum mechanics , economy , economics , service (business)
A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u t + uu y = u xx + u yy , has an exact solution U ( y ) = −2tanh y . It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple‐scaling methods to derive a nonlinear evolution equation for a long‐wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat‐conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.