Internal heating profiles for which downward conduction is impossible

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 $⟨\delta T{⟩}_h\ge \sigma R^{-1/3}-\mu$, where $⟨\delta T{⟩}_h$ is the average temperature difference between the bottom and top plates, $R$ is a `flux' Rayleigh number and the constants $\sigma ,\mu >0$ depend on the geometric properties of the internal heating. This result implies that mean downward conduction (for which $⟨\delta T{⟩}_h<0$) 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 $⟨\delta T{⟩}_h$ is known to be positive for all $R$.

13 pages, 4 figures

References in corpus (11)

• Radiative heating achieves the ultimate regime of thermal convection
• Optimal wall-to-wall transport by incompressible flows
• Internally heated convection beneath a poor conductor
• Relationship between the methods of bounding time averages
• The background method: Theory and computations
• Bounds on heat transport for convection driven by internal heating
• Transition between Boundary-Limited Scaling and Mixing-Length Scaling of Turbulent Transport in Internally Heated Convection
• Bounds for Rayleigh-Bénard convection between free-slip boundaries with an imposed heat flux
• Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks
• Bounds for internally heated convection with fixed boundary heat flux
• Bounds on heat transfer by incompressible flows between balanced sources and sinks

Cited by in corpus (1)

• Internally heated convection with rotation: bounds on heat transport