I came across this picture earlier today: https://i.stack.imgur.com/lMrkU.jpg and it left me kind of baffled. Can anyone explain the mathematics behind the reason why this is happening?
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Further, is there a way to determine a pattern that would be given by a particularly shaped cup? Or better yet, be able to determine the shape of a cup required to create a given pattern? – Michael Smith Jun 26 '18 at 16:16
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I can only guess that the fluid traveling along the flat parts gains altitude from the inertia when it encounters the curve, and then rebounds center-ward in some kind of angle-of-incidence-equals-angle-of-reflection. Just behind the curve, there is low pressure compared to what is on the curve so there is a low channel in the liquid which isn't filled because the high-liquid behind 'it' is going as much center-ward from rebound as it is in the direction of rotation. I wonder what you would find if you stacked and taped identical cookie cutters in a bowl, filled them with liquid and stirred. – poetasis Jun 26 '18 at 16:49
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Had the cup been circular we would see vortex shaped upper surface deeper at the center than at edges with polar circular symmetry as there is no radial component. When boundary is a regular polygon of $n$ sides we can see standing wave interference patterns caused by vibration frequency due to radial and circumferential component streamline flow interfering to form $n$ spiral arms.
An attempt at modeling should include the two interactions.
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If such a cup is placed and rotated sufficiently fast on a turntable I suspect it may give rise to some similar pattern, that is if viewing at all is possible.
Narasimham
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Any insight into what might happen if the boundary is not a regular polygon? – Michael Smith Jun 26 '18 at 17:32
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Perhaps a smaller Swastika with irregularly spaced and unequal length spiral arms. – Narasimham Jun 27 '18 at 06:52
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The shape which is formed must have $90 \deg$ rotational symmetry equivalent to the shape of the cup.
The shape will depend chaotically lots of factors (viscosity, exact shape, speed at which it is stirred, shape of the spoon which is stirring the liquid) hence it is almost impossible to mathematically show that the shape is as observed. I doubt whether the question as it stands can be answered at all.
Agile_Eagle
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If we're able to identify all of the relevant variables, I think a simulation might be doable that could give an answer. I would disagree with the chaotic nature you're suggesting, since it did form a shape that didn't appear to be emergent (spiral arms in our galaxy are emergent, for instance, they don't last, they're just now) – Michael Smith Jun 26 '18 at 17:10
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@MichaelSmith Spiral arms are density waves instead of actual structures. – Agile_Eagle Jun 26 '18 at 17:12