Scaling of nonuniform temperature and density components in the non-perturbative regime

Determine the scaling with the small parameter 𝜖 of the nonuniform components of the density and temperature for stationary finite-current states with current of order one in the non-perturbative regime.

Background

The paper studies whether a compressible, wall-heated fluid can sustain a stationary circulating current in zero gravity when the duct radius varies along the loop. Perturbative analysis in a small parameter 𝜖 yields a scaling condition requiring particularly weak viscous and thermal dissipation, while numerical calculations also reveal finite-current fixed points outside the perturbative regime.

For non-perturbative stationary states with current of order one, the authors observe that the nonuniform parts of the density and temperature remain small, but they do not establish how these quantities scale with 𝜖. Resolving this scaling is relevant because it determines whether the perturbative estimate of the heat-transfer contribution to entropy production, and hence the dissipation condition on the viscosity and thermal conductivity, remains valid beyond the perturbative regime.

References

It is interesting to note that, as illustrated in Fig. 5, the non-uniform component of ρ and T continues to be small also for J = O(1), even though the scaling in ǫ remains unclear, suggesting that the estimate for q in Eq. (14), and consequently also the condition on the scaling of γ and κ in Eq. (18), continue to hold in the non-perturbative regime.

— Zero-gravity convection in a closed duct as the realization of a thermal machine  (2608.21052 - Olla, 21 Aug 2026) in Page 5, paragraph following Figure 4