  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.
