Suppose there is a hollow ball of radius 1 meter. It is filled some viscous liquid. Then the ball is rotated at 1 radian per second. So the speed of any point on the ball's surface is 1 meter per second.
Also let us assume there is no gravity.
My understanding of viscosity is not too deep. So the next argument I am going to make may be faulty. From what I understand viscous forces make adjacent layers of liquid to move at the same speed. Suppose we keep applying external torque until every point in the liquid is moving at 1 meter per second. Now there will be no viscous force in the liquid and if we stop applying external torque this ball will go on rotating indefinitely.
But if the speed is same for every point in the liquid that must mean the angular speed is inversely proportional to distance from center. This means that water nearer to the center will have very high angular speed. This seems to be unrealistic.
So it seems that we cannot really achieve zero viscous force. There must be some limit on how high the angular speed can go. I think for some fixed torque we may be able to find a relationship between angular speed and distance from center.
Since there will always be a viscous force the ball will eventually stop rotating. My question is about how long it will be before the ball stops.
There seems to be two step to solve this problem. First find out out how the angular speed varies with distance at a steady state condition for some fixed torque and second from this find out how the angular speed of the ball will change with time when the external torque is removed.
I would love to analyze this problem further but I lack the mathematical background. My grasp of viscosity is elementary and most of the material I have found in the internet use complicated vector calculus to describe viscosity.
I would love it if you give an answer without using vector calculus. I would also love it if you just give some pointer to relevant reading material that are easy to follow.