  The linear stability of buoyant parallel flow in a vertical porous layer with
an annular cross-section is investigated. The vertical cylindrical boundaries
are kept at different uniform temperatures and they are assumed to be
impermeable. The emergence of linear instability by convection cells is
excluded on the basis of a numerical solution of the linearised governing
equations. This result extends to the annular geometry the well-known Gill's
theorem regarding the impossibility of convective instability in a vertical
porous plane slab whose boundaries are impermeable and isothermal with
different temperatures. The extension of Gill's theorem to the annular domain
is approached numerically by evaluating the growth rate of normal mode
perturbations and showing that its sign is negative, which means asymptotic
stability of the basic flow. A concurring argument supporting the absence of
linear instability arises from the investigation of cases where the
impermeability condition at the vertical boundaries is relaxed and a partial
permeability is modelled through Robin boundary conditions for the pressure.
With partially permeable boundaries, an instability emerges which takes the
form of axisymmetric normal modes.
