How long will it take for rotating spherical container full of viscous liquid to stop rotating when there is no gravity? 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.
 A: 
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.

No, it isn't faulty. Given enough time all the layers will eventually move at the same angular velocity. It takes time because the viscous forces between the layers provide accelerating torque but even with acceleration taking place, it takes time for all layers to move at the same speed.

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.

Here you're conflating things: angular velocity and tangential speed.
The angular velocity $\omega$ (in $\mathrm{rad/s}$) is independent of the distance from the center, at least when we've achieved equilibrium as described above.
But tangential speed $v$ does depend on distance from the center $r$
$$v=\omega r$$

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.

This is in contradiction with what you correctly wrote higher up:

Now there will be no viscous force in the liquid and if we stop applying external torque this ball will go on rotating indefinitely.

Without viscous forces in the liquid (at equilibrium) the sphere will keep rotating at constant $\omega$ indefinitely. The only way to change that, by Newton's second law, is to apply a braking torque to the shell. The layers of fluid will then continue to rotate but will experience angular deceleration due to viscous shear force. First the layer next to the shell will slow down, this will then affect the 'next' layer and so on.
The principle is the basis of a simple party trick in which participants are asked to try and spin a hard boild egg and a raw one (without knowing which is which). The raw egg is much harder to get to spin, due to viscous forces between the fluid's layers.
A: The moment of inertia of a fluid body is difficult to calculate. Non-rigid mechanics means that we have to go to the integration of fluids from Navier Stokes terms. Look that up. The idea is to use the continuity equation ddt rho. The rho is the density. That density also appears in the force equation which is how you get a characteristic frequency for the rotation of the bulk fluid as a function of time. It should be inversely proportional to the mass of the object, the radius of the pointlike particle if in the Brownian regime, and it should have some dependence on the viscosity. A good way to go is to find the force from the viscosity term proportional to the velocity squared (for obvious reasons). That term is related to the collision time between particles, but in a bulk fluid, we approximate collisions from kinetic theory. Don’t worry about rarefied rotating fluids, those are tough to work out, use a particle model instead. If this is unclear, then use the divergence and time derivative in the continuity equation, plus all the forces acting on a test spherical section. The answer should show that inverse mass dependence is my bet.
