fluid dynamics - Why is oil a better lubricant than water?

How come mineral oil is a better lubricant than water, even though water has a lower viscosity?

When two surfaces slide over each other with a gap filled with a fluid, the different layers of the fluid are dragged at different speeds. The very top layer touching the top metal surface will have the same speed as the surface itself, while the bottommost layer is stationary. The speed in the layers between is distributed linearly and there exist friction forces between those layers that slow the movement. Those frictional forces should be reduces however, if a fluid with a lower viscosity is chosen.

How come this is not so?

Does it have to do with water's polarity, so that it sticks to surfaces in a different way than oil?

Answer

Your derivation is composed of correct statements and indeed, if something is known to act as a lubricant, we want the viscosity to be as low as possible because the friction will be reduced in this way. For example, honey is a bad lubricant because it's too viscous.

However, your derivation isn't the whole story. The second condition is that the two surfaces must stay apart. If you use a lubricant with too low a viscosity, the surfaces will come in contact and the original friction will reappear.

So the optimum lubricant is the least viscous liquid that is viscous enough to keep the surfaces apart. Which of them is the optimal one depends on the detailed surfaces and other conditions. For example, there exist situations in which water is a better lubricant than oil – for example when ice slides on ice. Some of the ice melts and the water is why the ice slides so nicely.