Rotating Liquid Paraboloid (3D)

Why the surface curves

In the frame that rotates with the liquid (angular speed w) every fluid element feels gravity and a centrifugal force. The pressure gradient must balance both, and the free surface is a surface of constant pressure, so it is perpendicular to the total effective force. The surface slope is therefore dz/dr = w^2 r / g and integrating gives the paraboloid z(r) = z0 + w^2 r^2 / (2 g).

Volume conservation

The liquid cannot be created or lost, so with container radius R and rest depth h0 the centre sinks to z0 = h0 - w^2 R^2 / (4 g) while the wall rises to h0 + w^2 R^2 / (4 g). When w^2 R^2 > 4 g h0 the vertex reaches the floor, a dry circle of radius r0 = sqrt(R^2 - 2 R sqrt(g h0) / w) appears, and the depth becomes w^2 (r^2 - r0^2) / (2 g). If the liquid would climb over the rim it spills and the lost volume stays lost when the spin slows.

Newton's bucket and spin-up

When the bucket is suddenly spun, the water is at first at rest and flat even though the bucket moves; the surface curves only as viscosity drags the liquid up to speed. Newton took this to show that the curvature follows rotation relative to absolute space, not to the bucket. In the model the liquid speed relaxes toward the container speed with the spin-up time tau = h / sqrt(nu W) (Greenspan), where nu is the kinematic viscosity and W the rotation rate. Thick liquids spin up in seconds; water takes about a minute.

The liquid mirror

A paraboloid focuses parallel rays perfectly: z = r^2 / (4 f) with f = g / (2 w^2). A rotating pool of mercury is used as the mirror of real liquid mirror telescopes; the model traces rays off the surface to confirm the focus.

Floating and sinking objects

The pressure gradient supplies exactly the inward force needed to keep the surrounding liquid on its circle, rho_l w^2 r V for a volume V. A body denser than the liquid needs more than that, so relative to the liquid it moves outward; a lighter one needs less and drifts toward the axis. The net radial force is (rho_b - rho_l) w^2 r V. The test object moves at the terminal speed of quadratic drag, v^2 = (8/3) a ((rho_b - rho_l) / rho_l) w^2 r / C_d, with C_d = 0.47 and a 0.25 s response time.

Simplifications

The liquid is in solid-body rotation once spun up (no waves or Ekman layers in the shape), spin-up is a single exponential, the surface tension and the meniscus at the wall are ignored, and the test object does not disturb the surface.

1. Pick Newton's bucket and watch the first graph: the container reaches its speed in a few seconds, the liquid takes tens of seconds. Raise the time speed to hurry it along.
2. When the liquid has caught up, compare the green surface with the graph of slope: it follows tan(slope) = w^2 r / g at every radius.
3. Use Dry bottom: above the critical speed shown in the model check, the bottom of the container is bare in the middle.
4. Use Spill over the rim, then stop the motor: the liquid does not climb back into the container.
5. Change the test-object density below 1 (floats to the axis), above 1 (flung to the wall) and exactly 1 (stays where it is).
6. Pick Liquid mirror telescope and read the focal length; halving the rotation speed quadruples it.