Water coming out of a capillary tube when height is reduced below initial level

Question

Consider a capillary tube of height $H_1$. Water raises to a height $h_1$ in this tube. Now the capillary tube is cut such that its new height is $H_2 < h_1$.

I've read in many textbooks and other online sources that water will rise to a height $H_2$ in this new capillary and NOT come out of it like a fountain, not even slowly.

However I cannot find a rigorous mathmetical proof for this claim, intuitively I feel that the water should actually come out (slowly perhaps). If anyone could help me with the proof or provide some hint or insight related to this, it would be very much appreciated.

Thank you.

Answer 1

When you see water rise to a height in an everyday (large) pipe or tube, it is because of pressure inside the fluid. So the level it rises is nearly independent of the tube. Removing the tube above the water level frees the pressure to push water out.

But in a thin capillary tube, an important amount of force is not just from the pressure in the fluid below but also from the tube "pulling" the water up. This adhesive force between the water and the tube is sufficient to bring the height further up the tube than it would be solely from the bulk fluid pressure.

When you remove that segment of the tube, you are some of the force which was pulling up the fluid. Lacking that force pulling it up, the water level drops.