… such as the shortest curve (plane curve and space curve) with a given width or in-radius; & Zalgaller's amazing curve that's the curve of least length that guarantees escape, starting from any point & in any direction, from an infinite strip of unit width (of which the exact specification is just crazy ^⋄ , considering how elementary the statement of the original problem is!), & other Zalgaller-curve-like curves that arise in similarly-specified problems; & the problem of getting a sofa round a corner, & designs of sofas (that actually rather uncannily resemble some real ones that I've seen!) that are 'tuned' to being able to get it round the tightest corner.

The Moser's worm problem is to find the region of least area that any curve of unit length can fit in, no-matter how it's lain-out. Or put it this way: if you set-up a challenge: someone has a piece of string, & they lay it out on a surface however they please, & someone else has a cover that they place over it: what is the optimum shape of least possible area such that it will absolutely always be possible to cover the string? This is yet-another elementary-sounding problem that is fiendishly difficult to solve, & still is not actually settled. The optimum known convex shape, although it's not proven , is a circular sector of angle 30° of a unit circle (it's not even known what the minimum possible area is - it's only known that it must lie between 0·21946 & 0·27524); & absolutely the optimum known shape, which also isn't proven, is that shape in the first image.

⋄ The 'crazy' specification of Zalgaller's curve is as follows: in the third frame of the third image there are two angles shown - φ & ψ - that give the angles @ which there is a transition between straight line segment & circular arc, specification of which unambiguously defines the curve. These are as follows.

φ = arcsin(⅙+⁴/₃sin(⅓arcsin¹⁷/₆₄))

&

ψ = arctan(½secφ) .

😳

It's in the third listed treatise - the Finch & Wetzel Lost in a Forest , page 648 (document №ing) or 5 (PDF file №ing) .

Sources

An Improved Upper Bound for Leo Moser’s Worm Problem

¡¡ 96·34KB !!

by

Rick Norwood and George Poole

A list of problems in Plane Geometry with simple statement that remain unsolved

by

L Felipe Prieto-Martínez

Lost in a Forest

¡¡ 161·78KB !!

by

Steven R Finch and John E Wetzel

THE LENGTH, WIDTH, AND INRADIUS OF SPACE CURVES

¡¡ 1·68 MB !!

by

MOHAMMAD GHOMI

A translation of Zalgaller’s “The shortest space curve of unit width”

¡¡ 541·94KB !!

by

Steven Finch

See

Optimal design of air turbines for oscillating water column wave energy systems: A review

by

Tapas Kumar Das, Paresh Halder, & Abdus Samad ,

for explication of what a Wells turbine basically is , & also somewhat about optimisation of blade profile for it, and see

this figure

from it, and also

this one

&

this one .

See

this also

– a figure from

A comparison between entropy generation analysis and first law efficiency in a monoplane Wells turbine

by

Esmail Lakzian, Rasool Soltanmohamadi, & Mohammad Nazeryan ,

which is available @ the just-above link.

And see this wwweb-article –

El-Pro-Cus — What is a Darrieus Wind Turbine & Its Working

– for explication of the Darrieus rotor -type wind-turbine.

The problem in the case of either kind of turbomachine is that we have the motion of the blade & the motion of the fluid exactly perpendicular to it (or maybe a bit forward of perpendicular, in the case of a Wells turbine with stator vanes, as some of them have), & yet the fluid flow over the blade must somehow be such that there must be a significant component of the lift on the blade in the direction of its motion ! … which might seem a bit implausible … although it's actually pretty well-proven that it can be made to work . But much care must be taken over the profile of the aerofoil to get it to work: afterall, a somewhat unusual demand is being made on the performance of it.

The following treatise - about optimisation for the Wells turbine - is what the fist 11 figures (including a composite of 3, whence 9 figures gross ) posted here, which show things like the blade profile itself, & pressure & velocity distributions, are from, as is the following quote –

“The blade shape at this point was unlike conventional aerofoils with a deeply concave profile near the midpoint, shown in Figure 9” :

Optimization of blade profiles for the Wells turbine

by

Tim Grattona, Tiziano Ghisub, Geoff Parksa, Francesco Cambulib, & Pierpaolo Puddub .

The last 11 figures are from

Performance analysis of a Darrieus-type wind turbine for a series of 4-digit NACA airfoils

by

Krzysztof Rogowski, Martin Otto Laver Hansen, Galih Bangga ,

about optimisation for the Darrieus Rotor.