# Viscous Force On A Rotating Cylinder

In this question asked in Irodov, it is taken that the friction force acting on a unit area of a cylindrical surface of radius $$r$$ is given by $$σ = ηr(∂ω/∂r)$$.

A fluid with viscosity $$η$$ fills the space between two long co-axial cylinders of radii $$R1$$ and $$R2$$, with $$R1. The inner cylinder is stationary while the outer one is rotated with a constant angular speed $$ω2$$. The fluid flow is laminar. Take that the friction force acting on a unit area of a cylindrical surface of radius $$r$$ is given by $$σ = ηr(∂ω/∂r)$$.

Let a cylindrical surface of water with radius $$r$$ be spinning at an angular velocity $$ω$$ about its axis. From Newton's law of viscosity, $$σ = η(dv/dr).$$ Now $$v = ωr$$. So, $$σ = ηr(dω/dr) + ηω.$$ Why is the second term not taken in the question?

• This is the result of generalizing Newton's law of viscosity to 3D using tensor representation, and then using that to determine the $r-\theta$ component of the rate of deformation tensor in cylindrical coordinates. See Transport Phenomena by Bird, Stewart, and Lightfoot for details. – Chet Miller May 25 at 12:07