For those who do not know what a saxon bowl is, it is similar to a water clock except its functions exactly opposite. Instead of using the rate of outflow of water to measure time, a bowl with a hole at the bottom is placed on a container of water to measure how long it takes for it to fully submerge. Is there any physics equation or model such as Bernoulli's Equation or something else that can describe a correlation between the diameter of the hole and how long it takes for the saxon bowl to fully submerge?
This question is ultimately a fluid mechanics question, which means a seemingly simple question would be quite difficult to answer fully. A naive expectation would be that the rate of flow through the hole would increase with $r^2$, as that is how the cross sectional area increases with $r$. But due to viscosity effects, for small holes the rate of flow actually increases with $r^4$. For intermediate holes, there would have to be an interpolation between those two. Further the rate of flow will be dependent on viscosity and the weight of the bowl (which will indirectly determine the pressure). There are a lot of factors at play, and I don't know if there are any straight-forward equations that would give you a good approximation for how long the bowl would take to submerge.