Continuum Field Theory for the Deformations of Planar Kirigami.

Mechanical metamaterials exhibit exotic properties that emerge from the interactions of many nearly rigid building blocks. Determining these properties theoretically has remained an open challenge outside a few select examples. Here, for a large class of periodic and planar kirigami, we provide a coarse-graining rule linking the design of the panels and slits to the kirigami's macroscale deformations.

Optimal cooling of an internally heated disc.

Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange.

Optimal Wall-to-Wall Transport by Incompressible Flows.

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|^{2}⟩≤Pe^{2} we construct steady two-dimensional flows that transport at rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large enstrophy limit. Combined with the known upper bound Nu(u)≲Pe^{2/3} for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy Nu∼Pe^{2/3} up to possible logarithmic corrections.