All Problems Hydrodynamics
Problem 1.333
A fluid with viscosity \(\eta\) fills the space between two long co-axial cylinders of radii \(R_{1}\) and \(R_{2},\) with \(R_{1}< R_{2} .\) The inner cylinder is stationary while the outer one is rotated with a constant angular velocity \(\omega_{2} .\) The fluid flow is laminar. Taking into account that the friction force acting on a unit area of a cylindrical surface of radius \(r\) is defined by the formula \(\sigma=\eta r(\partial \omega / \partial r),\) find: (a) the angular velocity of the rotating fluid as a function of radius \(r\) (b) the moment of the friction forces acting on a unit length of the outer' cylinder.
(a) \(\omega=\omega_{2} \frac{R_{1}^{2} R_{2}^{2}}{R_{2}^{2}-R_{1}^{2}}\left(\frac{1}{R_{1}^{2}}-\frac{1}{r^{2}}\right)\) (b) \(N=4 \pi \eta \omega_{2} \frac{R_{1}^{2} R_{2}^{2}}{R_{2}^{2}-R_{1}^{2}}\)