A gas fills up the space between two long coaxial cylinders of radii `R_1` and `R_2`, with `R_1 lt R_2`. The outer cylinder rotates with a fairly low angular velocity `omega` about the stationary inner cylinder. The moment of friction forces acting on a unit length of the inner cylinder is equal to `N_1`. Find the viscosity coefficient `eta` of the gas taking into account that the friction force acting on a unit area of the cylindrical surface of radius `r` is determined by the formule `sigma = eta r(del omega//del r)`.

Explanation

We neglect the moment of inertia of the gas in a shell. Then the moment of friction forces on a unit length of the cylinder must be a constant as a function of `r`.
So, `2 pi r^3 eta (d omega)/(dr) = N_1` or `omega (r) = (N_1)/(4 pi eta) ((1)/(r_1^2) -(1)/(r^2))`
and `omega = (N_1)/(4 pi eta) ((1)/(r_1^2) -(1)/(r_2^2))` or `eta = (N_1)/(4 pi omega) ((1)/(r_1^2) -(1)/(r_2^2))`.