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Suppose I have a thin membrane sealed onto a flat surface along its perimeter, and there is water between the two surfaces with a certain pressure $P$. I would like to know, if I made a circular hole in the membrane, what the minimum hole size (diameter $D$) would be to cause leakage of water out of the membrane. Assume that the size of hole is << than the size of membrane. What would I need to determine experimentally (e.g. surface tension) to have an approximate solution?

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marked as duplicate by John Rennie, user191954, Kyle Kanos, sammy gerbil, Jon Custer Sep 11 '18 at 13:08

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Ideally, you should be able to determine the minimum size of the hole using Laplace's formula, $\Delta p=\frac {2\gamma} R$, but in practice that may not work.

This is, among other reasons, because this formula does not take into account the interaction between the water and the membrane, which would depend on the material of the membrane, its thickness and the exact profile of the hole.

For instance, as water is coming out of the hole, it can spread out, wetting the surface of the membrane and forming an inverted dome around the hole. The radius of the dome would be greater than the radius of the hole and the cohesion forces keeping it from falling down will be aided by the adhesion forces between the water and the membrane. Naturally, the equation for this scenario would be more complicated than the equation for a stand-alone droplet of water.

In that light, it is understandable why you are looking for an experimental approach.

It is hard to come up with an exact recipe without knowing how reliable your system has to be and how much various conditions (temperature, pressure, water quality, etc.) could vary, but, if you want to guarantee the flow, the size of the hole has to be greater than the minimum size at which the water can flow under typical conditions.

So, I would perform a series of tests with different hole sizes, under different conditions, and, based on the results, pick the size that works under the worst case conditions.

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