Why does spinning water lift its contents?

This used to be one of my favourite experiments in my childhood. When the water in a glass is put into a spin by a DC motor from the top, the things lying at the bottom gets lifted to the top. The effect was particularly interesting because it resembles tornadoes. (Click to see Video) What causes this lifting?

First of all, let's analyse the effect of the shaft in a particular circular cross-section. When the shaft of the motor rotates, it causes the water surrounding it to revolve around the shaft in a circular path. The flow rate of water near the shaft is more than that away from it. This is a consequence of viscosity of water.

According to the Bernoulli's equation: $$P+\rho gh+\frac 1 2\rho v^2=\text{constant}$$ where $$P$$ is the pressure, $$\rho$$ is the density of the fluid, $$g$$ is the acceleration due to gravity, $$h$$ is the height and $$v$$ is the speed of flow; at a fixed height when the speed of flow increases the pressure decreases. As we have discussed earlier, the speed of flow is larger near the centre and less towards the periphery. So, the pressure near the shaft of the motor is small when compared to the pressure near the periphery.

As an illustration, the pressure of a particular layer as viewed from the top looks something like the following image: Due to this pressure difference, a particle away from the axis experiences a force towards the centre. Now let's come to your question:

Why does spinning water lift its contents?

The reason is almost similar to why tornados lift cars, trucks, etc. In tornados the active fluid is mostly air, however here it's water.

The speed of flow of water decreases from the top to the bottom because of viscosity effects. Or in other words, water near the top flows faster than water near the bottom. Due to this speed difference, as discussed before, there exists a pressure difference along the vertical column. The pressure near the top is less than the pressure near the bottom. This pressure difference looks something like the following image: The pressure difference produces lift and hence causes the contents near the bottom to rise above. The object rises up until it's weight is balanced by the lift due to differential pressure on its top and bottom and the buoyant force.

The following diagram shows the order of pressures at two annular regions at two different heights: The contents near the bottom rise and due to fluid's rotation these gain some speed in the horizontal direction. The centripetal force required to maintain the circular motion is provided by the differential pressure in the cross section as we discussed in the beginning of the answer. This is why the objects execute an approximate circular motion.

• Great explanation, but I'd like to know how we can use Bernoulli's equation for this fluid when we also consider it as viscous. By the way, will the same effect happen, if the glass is placed on a rotating disc (as I had assumed in my answer)? Mar 27 '20 at 11:30
• @Krishna: Thanks! Of course, Bernoulli's equation is applicable only for the ideal case of irrotational and steady flow of an incompressible and non-viscous liquid. We need to include some correction terms in a non-ideal case for greater accuracy. Anyway, I just used it to highlight why differential pressure arises due to the difference in speed of flow in different regions. I don't feel the outcomes of this non-ideal case will vary hugely from that of the ideal case. You may consider this to be a kind of approximation. Mar 27 '20 at 13:35
• @Krishna: As for the second part of your comment, imagining this system to be that of a glass of water placed on a rotating disc is a good idea but it differs from this one. In order to witness similar effects, you might need to observe your system from a rotating frame of reference due to which you need to consider some inertial (pseudo) forces like centrifugal and Coriolis forces. But, the main difference arises from the fact there will be no vertical pressure difference due to difference in flow speeds (if you assume a cylindrical container). Mar 27 '20 at 14:02