Q: I have noticed that when I try to dissolve a granular compound by stirring it in a beaker of liquid, the compound collects in the centre of the beaker. The faster I stir the liquid the more rapidly the compound heads towards the centre of the beaker. Can anyone explain why this happens?
A: This problem has vexed me for many years, as applied to the similar question of why tea leaves collect in the centre of a cup of tea when it is stirred and why rubbish collects in the centre of urban whirlwinds. Intuition suggests that heavy objects should be flung to the outside of the rotating fluid, not collect in the middle.
The standard approach is to examine the pressure-momentum balance in the rotating fluid. If the rotating liquid is to remain within the beaker, then the inertial force created by the fluid rotation must be balanced by a pressure force or, more accurately, pressure gradient, within the liquid. It is like a mass being swung round on a piece of string: to prevent the mass flying off, a tension is set up in the string. This tension is the mechanical equivalent of the fluid pressure gradient.
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By considering the direction of the pressure gradient, it is quite easy to show that a particle that is less dense than the surrounding fluid will tend to move inwards, whereas a particle of higher density will move outwards. This is in tune with intuition. So why do heavy things move inwards in practice?
The answer lies in the behaviour of the fluid close to the bottom of the beaker. Fluid viscosity causes fluid in this region to rotate more slowly than the main body of fluid. This means the local pressure gradient is more than that required to balance the inertia forces, resulting in a net force towards the centre of rotation. The fluid, and any heavy particles that have settled on the bottom are swept towards the centre of the beaker. The fluid obviously cannot accumulate, so carries on moving, flowing up the axis of rotation. The heavy particles, meanwhile, are deposited in a central pile at the bottom of the beaker.
This theory can be tested by stirring fluid in a tall measuring cylinder. Heavy particles dropped into the fluid move outwards on the way down until they reach the bottom, at which point they move inwards.
A: As the liquid rotates, the 鈥渃entrifugal force鈥� pushes it outwards while the outer wall of the beaker restrains the liquid. Because the bottom of the beaker will always be slightly rough, liquid near the bottom rotates more slowly than liquid at the top (where there is less friction). The centrifugal force is lower at slow rotation speeds, and so a circulation current is set up, with liquid slowly moving outwards across the top surface, down near the outer wall, inwards along the bottom, and back up at the centre of the beaker.
Granules are carried along across the bottom by this circulation current until they are left behind at the centre when the current moves upwards, because gravity is able to hold them there. The result is a heap of granules in the centre at the bottom of the beaker.
Physicists call this a boundary layer effect. It wouldn鈥檛 occur if there was no rough surface along the bottom of the beaker and is much less noticeable when surfaces are smooth.