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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 $R1<R2$. 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?

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  • $\begingroup$ 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. $\endgroup$ – Chet Miller May 25 at 12:07

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