Stable long-wavelength convection with Dirichlet thermal boundary conditions through a Batchelor-Nitsche instability

It is a well known fact that the onset of Rayleigh-B\'{e}nard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. In this work, we consider three-dimensional Rayleigh-B\'{e}nard convection in a cell of infinite extent in the $x$-direction and having vertical walls located at $y=0$ and $y=\delta$ and horizontal boundaries located at $z=0$ and $z=h$. We assume stress-free boundary conditions and thermally conducting walls. We put forth a set of values of the parameter $\delta$ for which we show the existence and stability of long-wavelength convection.