It is fun to spin a hard-boiled egg like a top, and because it is nice to spin I spin it on everywhere. It is quite intriguing when it met a shallow pool of water, several milimeters deep, the water unexpectedly climbs up the shell of the egg until it is about half way up it sprays out horizontally like a sprinkler.

At a glance this fountain effect seems very mysterious, but later I found out that the main idea behind it can be explained easily using basic physics. To avoid complexity, we can neglect any interaction with air and coriolis force, and also assume that the egg is spinning without precession. Thus the remaining forces responsible for this effect are adhesion force between water and the egg shell, centrifugal force, and gravity. When the egg is turning, the water tends to fly away due to centrifugal force but it also pulled by adhesion force toward the surface of the egg. The resulting motion will be the water climbing up the shell. The water rises until a certain point where the adhesion force no longer able to withstand the outward force, then water droplets sprinkle outward. The following is the quantitative explanation of this effect:

Where:

is the normal component of adhesion force

is the tangential component of adhesion force (acts like friction)

is the maximum radius of projectile

is the cross section radius where the water break up into water drops

We consider the forces acting on an infinitesimal portion of water with mass in a rotating reference frame. By applying Newton’s second law, we can write the following equations:

(1)

(2)

The water can climbs up if is positive. Assuming that the condition to stay on the surface is satisfied, all we need is a sufficiently large angular velocity.

The water can stay on the surface if . However if the water will still remains in equilibrium because the normal force begin to act. Therefore the water will leave the surface if , it will happen when exceed the maximum value of .

If the egg is spinning very fast, will reach the value of zero when . Above that point is negative, the water tends to flow back down. Therefore the water will accumulate around , thus the mass in equation (2) will be replaced with a much larger mass, the value non-inertial force is increased. The water will continue to accumulate until the adhesion force no longer strong enough to hold. As a result, we can predict that for large and adhesion force, the dripping will most likely to occurs at the point where the cross section radius of the egg is the largest. The dripping rate will balance the accumulation rate, so that the accumulated mass doesn’t change.

However this is not the case because egg’s adhesion with water is stronger than water’s cohesion. If the adhesion force is much stronger than cohesion, the mass will accumulate so much that the force responsible for maintaining the water is no longer adhesion, it is the cohesion between water. The water which is in contact with the egg shell remains there, while the outer part of the water drip away.

To prove these theories another experiment using wax coated egg(adhesion<cohesion) was conducted, unfortunately only small amount of water was attracted to spin together with the egg , that means there exist a coriolis force that was directed opposite to the centrifugal force, effectively it’s like the egg was spinning slower. So we need a higher angular velocity to observe the dripping effect, however it is quite difficult to reach high enough angular velocity using bare hand. I hope someday I can find a tool to provide high angular velocity….

If the dripping occurs when the cross section radius of egg is , The drops will leave the shell with tangential velocity . The drops will follow parabolic trajectories, the time of flight is

During this time, the drops will travel a distance tangentially. Then by using a simple geometry we can get the radius of the water fountain