Transient Behavior of Unstable Nonlinear Systems with Applications to the Bénard and Taylor Problems

Abstract

A nonlinear initial value problem containing two parameters, $\varepsilon $ and $\lambda $, is considered for a function $u(t)$. For each t, $u(t)$ is a vector in a Hilbert space. When $\varepsilon = 0$ the problem has a steady state solution $u_0 $ for any value of $\lambda $. This solution is assumed to be linearly stable for $\lambda < \lambda _c $ and linearly unstable for $\lambda > \lambda _c $ for some value $\lambda _c $ . This means that the linear problem for the derivative $u_\varepsilon (t)$ at $\varepsilon = 0$ has exponentially growing solutions for $\lambda > \lambda _c $ but not for $\lambda < \lambda _c $. The solution $u(t)$ is found for $\varepsilon $ small and $\lambda $ slightly larger than $\lambda _c $. It is found that u contains one unstable mode which initially increases exponentially but ultimately approaches a constant, while all other modes decay, up to terms of order $\varepsilon ^2 $. To this order $u(t)$ approaches a steady state solution $u(\infty )$ different from $u_0 $, as t increases. The method of analysis is the two-time perturbation method. The method and results are illustrated by applying them to the Bénard problem of convective heat transfer through a horizontal layer of viscous fluid heated from below, and the Taylor problem of the instability of the Couette flow of a viscous fluid between coaxial rotating circular cylinders.