5 research outputs found
Axial Compression of a Thin Elastic Cylinder: Bounds on the Minimum Energy Scaling Law
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141757/1/cpa21704.pd
Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets
Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles—their wavelength—is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film
On the optimal design of wall-to-wall heat transport
We consider the problem of optimizing heat transport through an
incompressible fluid layer. Modeling passive scalar transport by
advection-diffusion, we maximize the mean rate of total transport by a
divergence-free velocity field. Subject to various boundary conditions and
intensity constraints, we prove that the maximal rate of transport scales
linearly in the r.m.s. kinetic energy and, up to possible logarithmic
corrections, as the rd power of the mean enstrophy in the advective
regime. This makes rigorous a previous prediction on the near optimality of
convection rolls for energy-constrained transport. Optimal designs for
enstrophy-constrained transport are significantly more difficult to describe:
we introduce a "branching" flow design with an unbounded number of degrees of
freedom and prove it achieves nearly optimal transport. The main technical tool
behind these results is a variational principle for evaluating the transport of
candidate designs. The principle admits dual formulations for bounding
transport from above and below. While the upper bound is closely related to the
"background method", the lower bound reveals a connection between the optimal
design problems considered herein and other apparently related model problems
from mathematical materials science. These connections serve to motivate
designs.Comment: Minor revisions from review. To appear in Comm. Pure Appl. Mat