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 $H^{2/3}$.
Analytical bounds on the heat transport in internally heated convection
我们得到了一个解析边界的平均垂直对流热通量$\langle w T \rangle$之间的两个平行的边界驱动的均匀内部加热。我们考虑两种配置，一个与两个边界保持在相同的恒定温度，和另一个与保持在恒定温度的顶部边界和一个完全绝缘的底部边界。对于第一种构型，Arslan等人（J. Fluid Mech.919：A15，2021）最近提供了数值证据，证明如果经典背景方法用最小值原理扩充，说明流体的温度不小于顶部边界的温度，则对唯一已知的严格边界$\langle w T \rangle \leq 1/2$的瑞利数相关修正是可证明的。在这里，我们证实了这一事实，严格的两种配置，证明界限$\langle wT \rangle$，接近$1/2$指数从下面的瑞利数增加。获得这些边界的关键是内边界层的背景场与一个特定的逆功率缩放，这可以控制在谱约束使用Hardy和Rellich不等式。这些允许在分析中进行标准构造无法获得的质量改进。
