Bounds on heat transport in Bénard-Marangoni convection

For Pearson’s model of Bénard-Marangoni convection, the Nusselt number Nu is proven to be bounded as a function Marangoni number Ma according to $\text{Nu}\le 0.838\times {\text{Ma}}^{2/7}$ for infinite Prandtl number and according to $\text{Nu}\lesssim {\text{Ma}}^{1/2}$ uniformly for finite Prandtl number. The analysis is also used to raise the lower bound for the critical Marangoni number for energy stability of the conduction solution from 56.77 to 58.36 when the Prandtl number is infinite.

Figure 1

Functions $f_k\left(z\right)$ for $k=1,10,100$.

Figure 2

Background profile $\tau \left(z\right)$ for Marangoni convection. The diagonal line is the conduction temperature profile.

Figure 3

Nusselt vs Marangoni number. Numerical data [9] are plotted together with the rigorous upper bounds.