Friday, 16 October 2020

What is the shear stress of a fluid?


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



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.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...