Optimizing bounds for energy-constrained optimal cooling problems in two dimensions
Abstract
We study optimal control problems for incompressible fluids in two-dimensional domains with a cold boundary and internal heat sources and sinks. Given a kinetic energy budget, measured by the square of a nondimensional Péclet number Pe, the goal is to maximize a cooling efficiency inversely proportional to the mean square gradient of the fluid's temperature. Using Lagrange duality, we formulate a well-posed dual problem whose solution yields an upper bound on the maximum cooling efficiency ℰ(Pe). We then numerically approximate the dual problem using a convergent hierarchy of semidefinite programs obtained via discretization. We illustrate this computational approach on optimal cooling problems in a square and in an annulus, explaining also how problem symmetries can be exploited to reduce computational complexity. Finally, we construct admissible points for the dual problem to obtain new analytical upper bounds on the optimal cooling efficiency ℰ(Pe). Specifically, we prove that ℰ(Pe) ≲ Pe2 for arbitrary domains and heat distributions, and that ℰ(Pe) ≲ Pe2/ ln2Pe for cooling flows in disks and annuli with heat source/sink distributions with a uniformly positive azimuthal average. These results generalize and improve known efficiency bounds for energy-constrained cooling flows in a disk.
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BibTeX
@misc{braga2026optimizing,
title = {{Optimizing bounds for energy-constrained optimal cooling problems in two dimensions}},
author = {Pedro Blöss Braga and Giovanni Fantuzzi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.14334},
eprint = {2608.14334},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14334},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimizing bounds for energy-constrained optimal cooling problems in two dimensions
AU - Pedro Blöss Braga
AU - Giovanni Fantuzzi
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.14334
UR - https://arxiv.org/abs/2608.14334
AB - We study optimal control problems for incompressible fluids in two-dimensional domains with a cold boundary and internal heat sources and sinks. Given a kinetic energy budget, measured by the square of a nondimensional Péclet number Pe, the goal is to maximize a cooling efficiency inversely proportional to the mean square gradient of the fluid's temperature. Using Lagrange duality, we formulate a well-posed dual problem whose solution yields an upper bound on the maximum cooling efficiency ℰ(Pe). We then numerically approximate the dual problem using a convergent hierarchy of semidefinite programs obtained via discretization. We illustrate this computational approach on optimal cooling problems in a square and in an annulus, explaining also how problem symmetries can be exploited to reduce computational complexity. Finally, we construct admissible points for the dual problem to obtain new analytical upper bounds on the optimal cooling efficiency ℰ(Pe). Specifically, we prove that ℰ(Pe) ≲ Pe^2 for arbitrary domains and heat distributions, and that ℰ(Pe) ≲ Pe^2/ ln^2Pe for cooling flows in disks and annuli with heat source/sink distributions with a uniformly positive azimuthal average. These results generalize and improve known efficiency bounds for energy-constrained cooling flows in a disk.
ER -