Assume I have a trough of water 5mm deep. If I cut a slot (oval) into that trough what is the maximum size/area (length x width) that slot can be without leaking?
This question is similar to Max. Radius of a hole 47021 but this is a slot.
From the equation provided in 47021, I don't believe it would be reasonable to calculate the area of the round hole and carry it over to my slot situation. As the slot could mathematically be very thin and very long.
There is no limit to the length of your slot if it is narrow enough. The circumference of the slot determines the total force available to hold the liquid in place - so as long as the ratio of circumference divided by area is above a critical value, you can keep the water in. That ratio scales with radius for a circular hole - but once you allow elliptical / rectangular holes, there is no limit.
There is no answer for the total area of the slot. However we can calculate the width of the slot, and as you say this is done with the same method used in How to find out the maximum radius of a hole that can keep water stay in a container by water viscosity?.
If we have a cylindrical meniscus then the pressure difference it produces is:
$$ \Delta P = \frac{\gamma}{r} $$
where $r$ is the radius of the cylinder:
So for a given pressure the maximum radius before the water starts flowing out it:
$$ r = \frac{\gamma}{\Delta P} $$
And the slot width is just $2r$.
The reason there is no maximum area is that this equation does not contain the length of the slot (actually it assumes the slot has an infinite length as it ignores end effects). So you can make the area arbitrarily large by making the slot arbitrarily long.
