What is the difference between strain rate and velocity gradient?

[v_pino]

What is the difference between strain rate and velocity gradient of a Newtonian fluid?

 

[minger]

For a Newtonian fluid, they are the same, which is proportional to the the shear stress. They are proportional by the second viscosity coefficient. For example, the normal shear stress in the x-direction is given by:

\tau_{xx} = \lambda (\vec{\nabla}\cdot\vec{V})+2\mu\frac{\partial u}{\partial x}
You can see the velocity gradient term in there, with the leading coefficient being the proportional part. Do note that \lambda is hard to measure, and this is where Stoke's Hypothesis (see number fudge so the equations can be solved) comes into play, where we just assume that:
\lambda = -\frac{2}{3}\mu