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# The Mathematics of Wrinkles on Curved Surfaces

Posted May 20, 2015 7:49 AM by Bayes

An Unexpected Simplicity

Sometimes problems exist for years simply because they are thought to be too complicated to solve. Then a group of researchers come along, notice a pattern where others did not, and develop a theory that turns out to be far simpler than expected. This is what recently happened at MIT where some engineers noticed that the wrinkles forming on hollow silicone spheres as the air was sucked out of them looked a lot like the stripes and swirls that appear on oil when heated due to Rayleigh-Bénard convection. The researchers did some experiments and indeed found a simplified formula for describing how the wrinkles formed.

Seemingly unimportant discoveries like these can have an important impact in unexpected ways. It's hard to know at this moment where this formula might be useful. Certainly it is probably related to fingerprints and other wrinkled curved surfaces. One immediate benefit might be better control of fabricating microlens arrays.

Here's the article about how the MIT researchers developed the theory:

## The Fascinating Math of How Wrinkles Form

Pedro Reis, an engineer at the Massachusetts Institute of Technology, had long been interested in how things wrinkle. For example, a dimpled surface like that of a golf ball offers less air resistance than a smooth sphere. If a flying object could dimple or wrinkle on command, Reis thought, it could alter its own aerodynamics midflight. Reis constructed silicone test spheres and sucked air out of them. He noticed that under pressure, some of the spheres formed the dimples he wanted, but some formed squiggly, labyrinthine patterns instead. Some had both dimples and labyrinths. When a member of his group shared the puzzle with mathematicians at MIT, they were intrigued: The wrinkling patterns resembled the stripes and swirls that appear when you heat a thin layer of oil, a phenomenon called Rayleigh-Bénard convection. Those phenomena had simplified, calculable equations - so why shouldn't wrinkles have a simplified equation too? Earlier researchers had worked backwards from specific wrinkling effects to create simulations that worked in single cases, but nobody had simplified the full elastic equations from the ground up to describe all wrinkling behavior - there was not yet a universal theory of wrinkles. It had been unclear which of the many variables were important.

Reis and the mathematicians started to go over the detailed body of experiments that Reis's group had assembled. When they examined the data from the rubbery spheres, the researchers found that just two factors controlled the formation of patterns: the curvature of a lower layer as compared to the thickness of the wrinkling layer on top, and the stress applied to that wrinkling layer. Films over less-curved surfaces would quickly transition to hybrid or labyrinth forms when put under stress. Setups that were more curved with a thicker layer on top would form a hexagonal layout of dimples and then, if stressed enough (as when Reis pulled air from inside the spheres), would eventually go labyrinthine as well. Releasing the stress would transition the surface back. "What's interesting is not just that these two parameters are important, but that all the other parameters are not important," said Norbert Stoop, one of the MIT mathematicians. The researchers found that the stiffness of the wrinkling layer, for instance, has no effect on the outcome. "Our theory you could basically apply to the surface of the moon or Mars, or the surface of a grape."

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#1

### Re: The Mathematics of Wrinkles on Curved Surfaces

05/20/2015 8:43 AM

Nice Article, it was mentioned the co-efficiency factor improves and this can be applied in many industries.

I worked at a shipyard in the 90's where on one project I was talking to the designers and naval architects on a Proto-Type Special Operations Craft competition for the naval we contracted from the UK for their expertise. And they were talking about the these same things, only at the time the technology was not developed or excepted yet.

Yet the competitor's the boats for America's cup, they are trying similar surfaces, (scale-like features)

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#6

### Re: The Mathematics of Wrinkles on Curved Surfaces

07/01/2015 11:56 PM

A related thought/question:

Has anyone investigated how dimpling the surface of an aircraft fuselage affects air resistance/drag ..... presumably the size, depth and density of the dimples would be critical parameters determining the results.

A step further:

What about dimples on the wings? how would this affect lift/drag, aerodynamic performance, stall angle, etc.

What about dimples on the top surface and not on the bottom surface, or on the bottom surface and not on the top surface?

And yet further:

What about hydrodynamics has anyone investigated dimpling a ship's hull, or submarine's.... again I would expect that the size, depth and density of the dimples would be critical parameters determining the results ......

so many questions .... I believe that some forms of hull surface texturing has been tried on competition sailing hulls with some success, but I am not sure of the details.

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#7

### Re: The Mathematics of Wrinkles on Curved Surfaces

07/03/2015 8:38 AM

I'm going from my memory,..... You have been warned.

I have read similar things have been looked into, such as having penetration (micro penetrations I believe) in the wings. This breaks up the small turbulent flow boundary layers. There were a number of drawbacks, one of which the penetrations become plugged. But surface texture was and is also looked into, such as a scale like surfaces on boat hulls that are similar to sharks.....? I even question myself, Sharks have scales.

I see I did mention it.... :/

I worked at a ship yard, where there was research and development in stealth textures on submarines in an effort to reduce cavitation of which I was involved in. What I just said is available publicly.... But that's all that's available.

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#8

### Re: The Mathematics of Wrinkles on Curved Surfaces

07/03/2015 9:47 AM

The first thought into my head on this one: A very detailed calculation of boundary layer effects, Koanda effect, Reynolds numbers, Prandtl numbers, etc. must be done first on these modified surfaces. If this results in improved contact of flowing fluid over the surface (which now has more surface area by quite a bit, just as a wrinkly coastline has more linear distance of shore line), then would this also have a direct bearing on heat transfer designs as well?

If wing dimples help the same way as they do with a golf ball, it might put a completely new spin on things. (not that a spinning aircraft is a good thing).

If the boundary layer shows more interaction with these types of surfaces, would this not make a disc turbine even the more efficient? Would it help with metal warping problems at high temperatures?

You can see I have many questions, few answers.

Happy July 4th to all my fellow Americans, and God Save the Queen to our British counterparts, and Cannucks. You Aussies are spent for luck. God Bless America! Britannia rule the waves! By the way, when are the Brits going to invade and try to take back the colonies again, as they are totally out of control now? ROFLMAO.

If - the biggest little word in the English language.

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#2

### Re: The Mathematics of Wrinkles on Curved Surfaces

05/20/2015 2:09 PM

Thermal image of a hot cup of coffee showing convection cells...

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#3

### Re: The Mathematics of Wrinkles on Curved Surfaces

05/20/2015 2:20 PM

OK, now I'm checking prunes, raisins, and the wrinkles on my fingers when I've been in the water.

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#4

### Re: The Mathematics of Wrinkles on Curved Surfaces

05/20/2015 8:10 PM

Great! Now I can tell my girlfriend that mathematicians have equations that describe all those wrinkles on her neck.

Ummm. No, wait a sec. Maybe that's not such a good idea.

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#5

### Re: The Mathematics of Wrinkles on Curved Surfaces

05/21/2015 11:42 AM

Smart lad! Best to steer clear of telling the woman in your life such things!

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