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:
$$\lambda = -\frac{2}{3}\mu$$