Re: Do fluids in a rotating sphere have viscous energy losses as t goes to inf?

Christopher,

Assuming your spherical shell is not filled with super fluid Helium, you are correct that eventually the shell and the fluid would have the same angular velocity. In the absence of external forces acting on the shell this angular velocity would continue indefinitely. Angular momentum is indeed conserved.

If it were filled with super fluid Helium, this question would be better answered by a quantum physicist. Most, if not all, of the Helium would not rotate.

Viscous forces are what would bring the shell and fluid to the same angular velocity. The more viscous the fluid the faster the final velocity would be reached.

The linear velocity of a point in either the fluid or the shell would depend on the angular velocity and the radius from the axis of rotation. This would range from zero on the axis to a maximum on the “equator” of the shell. Adjacent points would not be moving relative to each other so there would be no further viscous force.

The earth differs from your shell in that it does have external forces acting on it. Both the sun and Moon contribute to tides in the atmosphere, ocean and solid earth. The Moon contributes the most to the tides. For an introduction to the long term secular variation see Ocean Tides and the Earth’s Rotation. The term tidal friction covers a number of poorly understood effects. Currently the earth is slowing down at 2.3 milliseconds per day per century.

The Moon doesn’t have an atmosphere or an ocean. Earth caused tides in the solid Moon have already slowed the Moon’s rotation to the point where one side always faces the earth. Eventually the earth’s rotation will slow to the point that the earth and Moon will be rotationally locked. This will take billions of years.

David