Internally heated convection beneath a poor conductor

Abstract

We consider convection in an internally heated layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable ($R_L$), and the energy analysis yields a Rayleigh number below which it is globally stable ($R_E$). For various boundary conditions on the velocity, exact expressions for $R_L$ and $R_E$ are found using long-wavelength asymptotics. Each $R_E$ is strictly smaller than the corresponding $R_L$ but is within 1%. The lower bound on the mean temperature is proven for no-slip velocity boundary conditions using the background method. The bound guarantees that the mean temperature of the fluid, relative to that of the top boundary, grows with the heating rate ($H$) no slower than $H^{2/3}$.

arXiv comment20 pages, 5 figures

References in corpus (3)

• Large scale dynamics in turbulent Rayleigh-Benard convection
• A Comparison of Turbulent Thermal Convection Between Conditions of Constant Temperature and Constant Flux
• Convectively driven shear and decreased heat flux

Cited by in corpus (14)

• Transition to the ultimate regime in a radiatively driven convection experiment
• Competing forces in liquid metal electrodes and batteries
• Bounds for convection between rough boundaries
• The background method: Theory and computations
• Bounds on heat transport for convection driven by internal heating
• On the role of the Prandtl number in convection driven by heat sources and sinks
• Bounds for internally heated convection with fixed boundary heat flux
• Analytical bounds on the heat transport in internally heated convection
• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Pressure-driven flows in helical pipes: bounds on flow rate and friction factor
• Optimal cooling of an internally heated disc
• Internal heating profiles for which downward conduction is impossible
• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Dynamics and Scaling of Internally Cooled Convection