The bottom line: a new paper shows how to compute mathematically airtight limits on how well a moving fluid can cool a heated surface, using tools straight out of the optimization toolbox — duality and semidefinite programming — rather than running endless physical simulations to guess at the best possible design.
Picture a system with a cold boundary and hot spots inside it, like a chip with heat sources on a cooled casing, or a room with a radiator and an open window. Stirring the fluid moves heat around and improves cooling, but stirring costs energy. The natural design question is: given a fixed energy budget, how much cooling can you possibly achieve, no matter how cleverly you stir? Answering that with brute-force simulation over all possible flow patterns is hopeless — the space of possible flows is infinite-dimensional. The paper instead asks for a provable ceiling on performance, a bound that no flow, however exotic, can beat.
The setup: fluid moves inside a two-dimensional domain with cold boundaries and internal heat sources and sinks. The flow's kinetic energy budget is measured by a nondimensional Péclet number, $\mathrm{Pe}$, which is a stand-in for how strongly the fluid is allowed to circulate relative to how fast heat spreads by conduction alone. The quantity to maximize is a cooling efficiency $\mathcal{E}(\mathrm{Pe})$, defined so that it's inversely proportional to the mean square gradient of the temperature field — a smoother, flatter temperature field means better cooling. So the underlying problem is: choose a fluid velocity field, subject to $\|\mathbf{u}\|$ bounded in a way governed by $\mathrm{Pe}$, to minimize the mean-square temperature gradient given the advection-diffusion equation coupling flow and temperature.
This is exactly the kind of problem where duality earns its keep. Rather than solving the original ("primal") maximization directly, which is a nonconvex, infinite-dimensional variational problem over admissible flows and temperature fields, the authors construct a dual problem whose feasible solutions each certify an upper bound on $\mathcal{E}(\mathrm{Pe})$ — much like how a dual LP gives a certified bound on a primal LP's optimal value without needing to solve the primal itself. $$\mathcal{E}(\mathrm{Pe}) \le \inf \{\text{dual objective} : \text{dual constraints hold}\}.$$ Any feasible point of the well-posed dual gives a valid, if not necessarily tight, ceiling on cooling performance; the smallest such ceiling found is the sharpest bound obtained.
Because the dual problem is still posed over function spaces, it can't be solved by hand or exactly on a computer either — so the authors approximate it numerically using a hierarchy of semidefinite programs (SDPs) obtained by discretizing the underlying functions and imposing positivity conditions as linear matrix inequalities. This is the same general 'background method' machinery that Fantuzzi and Wynn used to bound stress-driven shear flows via SDP, applied here to a different physical setting. Crucially, the hierarchy is convergent: as the discretization is refined, the SDP bounds are proven to approach the true dual optimum, so the numerical results come with a guarantee rather than being one-off simulation outputs.
The authors test this machinery on two concrete geometries — a square domain and an annulus (a ring) — and show how exploiting the symmetry of these shapes reduces the size of the SDPs that need to be solved, which matters practically because SDP solve times grow fast with problem size. That kind of symmetry reduction is a standard but essential trick in applied SDP work: without it, even modestly refined discretizations can become computationally intractable.
Beyond the numerical bounds, the paper derives new closed-form analytical bounds that hold for any two-dimensional domain and any arrangement of heat sources and sinks: $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^2$. For the more specific and physically common case of disks and annuli where the heat distribution has a positive azimuthal average, they tighten this to $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^2/\ln^2 \mathrm{Pe}$ — a logarithmic improvement that matches and generalizes earlier results Ian Tobasco derived specifically for a disk. That log-squared factor isn't just a mathematical nicety: it tells engineers that cooling efficiency gains from pumping in more kinetic energy have sharply diminishing returns, growing more slowly than the raw energy budget would naively suggest.
For the optimization community, the interesting story here isn't really about fluid dynamics — it's a reminder that duality and SDP relaxation hierarchies, tools developed for finite-dimensional convex optimization, extend cleanly to infinite-dimensional variational problems in continuum physics when the underlying structure is right. Bounding worst-case or best-case performance without ever finding the optimal solution itself is a recurring pattern across optimization applications, from combinatorial relaxations to control theory, and this paper is a clean, rigorously convergent example of that pattern applied to a genuinely open physical design question.
Sources: Anonymous authors, 'Optimizing bounds for energy-constrained optimal cooling problems in two dimensions', arXiv:2608.14334 · Ian Tobasco, 'Optimal cooling of an internally heated disc', arXiv:2110.13291 · Ian Tobasco, 'Optimal cooling of an internally heated disc', Phil. Trans. R. Soc. A, 2022, https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2021.0040 · G. Fantuzzi and A. Wynn, 'Optimal bounds with semidefinite programming: an application to stress driven shear flows', arXiv:1512.05615
