Gravity-induced branch selection and no-shear theorems for compressible pure-swirl vortices

We classify steady axisymmetric pure-swirl solutions of the compressible Navier–Stokes–Fourier equations for a calorically perfect ideal gas under uniform gravity, with no distributed volumetric heating or azimuthal body forcing. Radial and vertical momentum balance impose the compatibility conditions T=Θ(χ) and p=P(χ), where χ=gz−Ψ(r) and Ψ′(r)=Vϕ2/r. For constant dynamic viscosity and arbitrary positive conductivity k(T), the azimuthal momentum equation admits Vϕ=Ar+B/r, but the energy equation retains only the two pure branches: Ar with uniform temperature and B/r with k(T)Tχ=−2μ. The mixed branch AB≠0 is excluded. If viscosity depends on temperature, every sheared region must satisfy μ[T(χ)]=μ0eλχ. Power-law and Sutherland viscosity under the standard constant-Prandtl closure are incompatible with the remaining energy constraint; no steady sheared pure-swirl state then exists in this class. The constant-viscosity potential branch survives as an exact conductive solution on an annulus. Thus, each additional thermodynamic or transport constraint reduces the admissible pure-swirl solution set.