Internal heating profiles for which downward conduction is impossible
arxiv(2024)
Abstract
We consider an internally heated fluid between parallel plates with fixed
thermal fluxes. For a large class of heat sources that vary in the direction of
gravity, we prove that ⟨δ T ⟩_h ≥σ R^-1/3 - μ,
where ⟨δ T ⟩_h is the average temperature difference between
the bottom and top plates, R is a `flux' Rayleigh number and the constants
σ,μ >0 depend on the geometric properties of the internal heating.
This result implies that mean downward conduction (for which ⟨δ T
⟩_h< 0) is impossible for a range of Rayleigh numbers smaller than a
critical value R_0. The bound demonstrates that R_0 depends on the heating
distribution and can be made arbitrarily large by concentrating the heating
near the bottom plate. However, for any given fixed heating profile of the
class we consider, the corresponding value of R_0 is always finite. This
points to a fundamental difference between internally heated convection and its
limiting case of Rayleigh-Bénard convection with fixed flux boundary
conditions, for which ⟨δ T ⟩_h is known to be positive for
all R.
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