Children in rural areas inevitably do household chores. Of course, my home didn’t have any particularly heavy chores; they were usually simple tasks like sweeping the floor, cooking, and washing vegetables. Speaking of washing vegetables, after finishing, I always liked to stir the water while it was draining, which would form a rather interesting vortex on the water surface. Now, let us analyze this vortex from a mathematical and physical perspective.
Before discussing the vortex in the washbasin, let’s look at a similar and more ancient problem—Newton’s rotating liquid surface problem. Newton assumed there is a bucket (assume it is cylindrical, though this is not critical) rotating around its central axis at a uniform angular velocity until the water inside also rotates at the same uniform angular velocity (i.e., the water is relatively stationary with respect to the bucket). At this point, the shape of the water surface is concave. Let’s examine the shape of this surface.
Newton’s Bucket
To analyze the shape, we need to review the principle of hydrostatic equilibrium mentioned previously: https://kexue.fm/archives/1964
This law tells us that when a fluid reaches an equilibrium state, the resultant external force must be perpendicular to the liquid surface. With this, the problem becomes quite manageable. Take a small droplet of water on the surface with mass m. It is subjected to two forces: the Earth’s gravity and the centrifugal force generated by rotation. As shown in the figure:
Centrifugal force: m\omega^2 x
Gravity: mg
Tangent vector: (dx, dy)
Perpendicularity implies: \frac{dy}{dx} = \tan\theta = \frac{m\omega^2 x}{mg} = \frac{\omega^2 x}{g}
Solving this gives: y = \frac{\omega^2}{2g}x^2
This is a paraboloid of revolution!
What does a paraboloid remind you of? Astronomy enthusiasts might easily think of the "Newtonian reflector"! Yes, the primary mirror of a Newtonian reflecting telescope is also a paraboloid of revolution, though it is generally made of solid material. In fact, when Newton first discovered that this was a paraboloid of revolution, he also thought about using rotating mercury to create a telescope mirror. However, for a long time, various disturbances were difficult to overcome (it would deform with the slightest vibration), so it remained only on paper. Nevertheless, with the development of technology, such liquid-mirror telescopes have now been realized, and their prospects are very bright.
The following information is sourced from the internet:
The world’s first liquid telescope was made in the early 1950s by the Soviet physicist Ud using a basin of mercury. After continuous improvement and exploration, Canadian scientist Ermanno Borra built the first liquid telescope suitable for astronomical observations in the early 1980s, with a lens diameter of 45 cm. Later, Borra used 250 kg of mercury to build two telescopes with a diameter of 1 meter and one with a diameter of 1.6 meters. He added a special layer of transparent resin on the mercury surface, which solved the problem of external interference and avoided the evaporation of mercury, which is hazardous to human health.
In 1995, NASA built a 3-meter diameter mercury telescope installed in New Mexico, specifically for monitoring space debris and near-Earth asteroids. Its total cost was less than $500,000, whereas the production cost for a telescope of this size is usually at least $10 to $20 million.
Back to the Washbasin
Friends with life experience will find that the vortex formed by water in a washbasin is clearly not a paraboloid of revolution. The reason is simple: after the water in the washbasin is stirred, it has not yet formed a collective uniform angular velocity rotation. Careful observation reveals that it is more like a collective uniform linear velocity rotation. That is to say, the magnitude of the velocity is fixed; the angular velocity is higher near the center and lower further away. Thus, we only need to replace the centrifugal force term with \frac{m v^2}{x}, obtaining:
\frac{dy}{dx} = \tan\theta = \frac{\frac{m v^2}{x}}{mg} = \frac{v^2}{gx}
Solving this gives: y = \frac{v^2}{g} \ln x
This is a logarithmic curve. When it is rotated around the y-axis, we get the approximate shape, as drawn below:
Although it is not entirely perfect (as there is no intersection point with the y-axis), it is clearly closer to what we observe.
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