# Diameter to prevent water flow into a closed-end tube?

Imagine I drill three holes of different diameter into a large block of plastic but the holes don't go all the way through. (They form three close-ended tubes, see the image.) I then submerge the block into a bucket and fill it with water. I want to know which holes/indents/tubes will fill with water.

Water will not flow through these tubes (they're closed on one end), and if the diameter is narrow enough, water will not fill it completely either. Based on this information, it seems that wettability, capillary action, and viscosity are the important factors determining water ingress. Wettability will be constant across all three holes (they're made of the same material and are submerged in the same liquid), as will viscosity. The only difference is the radius of the entrance, and I believe in the equation for calculating capillary action, the radius is in the denominator.

Is there a way to calculate the largest diameter of the hole/tube that will prevent water from flooding it? (And by flooding it, I mean removing all air from the tube.) Assume a constant, arbitrary height of water in the cup, say 1 meter.

• When do you consider the water to be flooding in? – acarturk May 23 at 0:10
• You haven't mentioned the height of water above the holes, as the pressure above the holes also matters. – Drew May 23 at 0:25
• Sorry, I edited the question to address your concerns. Flooding in would be when all the air in the tube is removed and replaced with water. The height is arbitrary, so let's just say it is 1 meter. – Andrew May 23 at 0:49

## 1 Answer

Water will flow down into that tube which supports something called two-phase flow, in which the water can enter the tube at the same time the air already there is leaving the tube.

Two-phase flow can occur when the surface tension of the liquid is low and the diameter of the tube is large. In this case, the fluid and air can bulge past one another in opposite directions. It cannot occur when the surface tension is large and the diameter of the tube is small, in which case the fluid and air occupy separate segments of the pipe without interpenetration (which is called "slug flow").

The Weber number, which is (density of fluid)x(velocity squared)x(characteristic length)/(surface tension), is used to estimate when surface tension forces will exceed inertial forces in a pipe (thereby mode-locking the pipe into slug flow) or when inertial forces are dominant over surface tension effects (supporting two-phase flow instead). The characteristic length in this case is taken to be the pipe diameter.

• What is the velocity though, in this situation if nothing has started moving yet? – Andrew May 24 at 15:07
• it's very small. If you assume it's equally small for two conditions of surface tension and diameter, and look at the ratio of the resulting weber numbers, the density and velocity terms cancel. – niels nielsen May 24 at 16:46