Internally heated convection beneath a poor conductor

TL;DR

This paper analytically investigates the stability and temperature bounds of internally heated convection beneath a poor conductor, deriving thresholds and bounds that describe the onset of instability and mean temperature growth.

Contribution

It provides exact stability thresholds and a lower bound on mean temperature growth for internally heated convection with various boundary conditions, using purely analytical methods.

Findings

Linear stability threshold $R_L$ derived for different boundary conditions.

Energy stability threshold $R_E$ found, slightly below $R_L$.

Lower bound on mean temperature growth proportional to $H^{2/3}$.

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 $H^{2/3}$.