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): Viscious Stress General Equation (Tensor)

So the magnitude of the shear rate is: So the magnitude of the shear rate is:

Is this shear rate magnitude formula correct?


  • $\begingroup$ Shouldn't there be a factor of 2 out front? $\endgroup$ May 5 '18 at 12:31
  • $\begingroup$ @ChesterMiller the 2 factor is referring to the Linear Stress Constitutive Equation (en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations) $\endgroup$ May 5 '18 at 16:17
  • $\begingroup$ The equation for the shear rate is kinematic, and is independent of any constitutive equation. $\endgroup$ May 5 '18 at 16:34
  • $\begingroup$ So, in case I'd like to get the shear rate magnitude expression for Compressible flow, what is the right formula? $\endgroup$ May 5 '18 at 16:40
  • 1
    $\begingroup$ In my judgment, what you have is correct, except for the factor of 2. $\endgroup$ May 5 '18 at 16:56

I didn't catch this the first time I read this thread, but your equation for the rate of deformation tensor D is incorrect; it should not have the dilatation terms along the diagonal. The definition of the rate of deformation tensor is "the symmetric part of the velocity gradient tensor":

$$\mathbf{D}=\frac{(\nabla \mathbf{v})+(\nabla \mathbf{v})^T}{2}$$

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.


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