Strain rate, velocity gradient

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

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minger
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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:

[tex]\tau_{xx} = \lambda (\vec{\nabla}\cdot\vec{V})+2\mu\frac{\partial u}{\partial x}[/tex]
You can see the velocity gradient term in there, with the leading coefficient being the proportional part. Do note that [tex]\lambda[/tex] 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:
[tex]\lambda = -\frac{2}{3}\mu[/tex]
 

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