# How to calculate Shear Rate magnitude for Compressible Flow?

I'd like to calculate the shear rate formula for CFD (Non-newtonian Fluid) and want to know if the following formula is the good one:

Viscious Stress General Equation (Tensor):

So the magnitude of the shear rate is:

Is this shear rate magnitude formula correct?

Thanks

• Shouldn't there be a factor of 2 out front? May 5 '18 at 12:31
• @ChesterMiller the 2 factor is referring to the Linear Stress Constitutive Equation (en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations) May 5 '18 at 16:17
• The equation for the shear rate is kinematic, and is independent of any constitutive equation. May 5 '18 at 16:34
• So, in case I'd like to get the shear rate magnitude expression for Compressible flow, what is the right formula? May 5 '18 at 16:40
• In my judgment, what you have is correct, except for the factor of 2. May 5 '18 at 16:56

$$\mathbf{D}=\frac{(\nabla \mathbf{v})+(\nabla \mathbf{v})^T}{2}$$
Reiner and Rivlin derived a general constitutive equation for a non-viscoelastic non-linear fluid by expressing the stress tensor $\boldsymbol{\tau}$ as a Taylor series in D, and applying the Cauley Hamilton theorem to obtain: $$\boldsymbol{\tau}=a+b\mathbf{D}+c\mathbf{D^2}$$where the scalar material parameters a, b, and c are functions of the three invariants of D. The linearized version of this is a Newtonian fluid, with "a" being a function only of the dilatation (first invariant), b being a constant, and c being zero.