The role of exponentially-small terms in transition to turbulence.

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In a conventional asymptotic analysis of high-Reynolds-number flows, only one or two terms of the asymptotic expansions are calculated. It is assumed that the asymptotic expansion converges to the Navier-Stokes solution as the Reynolds number increases. However, over the last decade or so, examples have been furnished where this is not the case, e.g. in the case of high-Reynolds-number unsteady flows that are susceptible to Rayleigh instabilities. The aim of this project is to place the understanding of this phenomenon on a firmer mathematical basis. In particular, it is speculated that this is a Stokes phenomenon arising from the turn-on of exponentially-small terms. The first part of the project will be to demonstrate that this is the case for a one-dimensional model problem that captures the essentials, before extending the analysis to more realistic flows.