Internal heating driven convection at infinite Prandtl number

We derive an improved rigorous lower bound on the space and time averaged temperature ⟨T⟩ of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries driven by uniform internal heating. A singular stable stratification is introduced as a perturbation to a non-singular background profile yielding ⟨T⟩ ⩾ 0.419[R log R]−1/4 where R is the heat Rayleigh number. The analysis relies on a generalized Hardy-Rellich inequality that is proved in the Appendix.