Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid

TL;DR

This paper rigorously analyzes the boundary layer behavior of a steady Boussinesq fluid flow in a vertical channel, extending classical asymptotic analysis with new mathematical proofs involving Painlevé transcendents.

Contribution

It provides a rigorous proof of boundary layer asymptotics for Boussinesq flow, connecting nonlinear differential equations to fluid dynamics and nonlinear analysis conjectures.

Findings

Established non-degeneracy of a Painlevé-I solution.

Linked boundary layer problem to nonlinear Schrödinger ground states.

Related results to the Lazer-McKenna conjecture.

Abstract

Using a perturbation approach, we make rigorous the formal boundary layer asymptotic analysis of Turcotte, Spence and Bau from the early eighties for the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work. A key point in our proof is to establish the non-degeneracy of a special solution of the Painlev\'{e}-I transcendent. To this end, we relate this problem to recent studies for the ground states of the focusing nonlinear Schr\"{o}dinger equation in an annulus. We also relate our result to a particular case of the well known Lazer-McKenna conjecture from nonlinear analysis.