What is the shear stress of a fluid?

Question

One book defines the shear stress $\tau$ of a (Newtonian) fluid as

$$\tau = \eta \frac{\partial v}{\partial r} $$

where $\eta$ is the viscosity. There is not much context, so I've made some guesses. Are my following assumptions correct?

• $v$ is the velocity of the flow line, parallel to the wall.
• $r$ is the distance of the flow line from the wall.
• the flow must be laminar for the above to hold. (Otherwise, what would $v$ mean?)
• the "wall" must be a tube for the above to hold. (Otherwise, what would $r$ mean?)

Answer 1

Your assumptions are correct (but $r$ is often defined as the distance from the pipe centerline). However, this is a very specific case: laminar pipe flow.

In general, the stress will be a tensiorial quantity, defined as

$$ \tau_{ij}= \eta \frac{\partial u_i}{\partial x_j}$$

which is true for turbulent flow, in arbitrary geometries. Where $i,j$ are in the range ${1,2,3}$ for the $x,y,z$ components.

For your case, you only have velocities in the streamwise direction, and variations in the radial direction, which makes all other components zero.