Abstract
Journal of Fluid Mechanics 998 (2024) A24 We consider the two-dimensional Rayeigh-BΓ©nard convection problem between Navier-slip fixed-temperature boundary conditions and present a new upper bound for the Nusselt number. The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result in Whitehead & Doering (2011) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (2022) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number and slip-length, our upper bounds indicate five possible scaling laws:ππ’ βΌ (L_(s)β»ΒΉπ
π)^(β
) ,ππ’ βΌ (L_(s)^(-β
)π
π)^((5/13)) ,ππ’ βΌ π
π^((5/12)) ,ππ’ βΌ ππ^(-β
) (L_(s)^(-(4/3))π
π)^(Β½)andππ’ βΌ ππ^(-β
) (L_(s)^(-β
)π
π)^(Β½)