Second-order Fluid

A second-order fluid is a fluid where the stress tensor is the sum of all tensors that can be formed from the velocity field with up to two derivatives, much as a Newtonian fluid is formed from derivatives up to first order. This model may be obtained from a retarded motion expansion truncated at the second-order. For an isotropic, incompressible second-order fluid, the total stress tensor is given by
:$\sigma_ = -p \delta_ + \eta_0 A_ + \alpha_1 A_A_ + \alpha_2 A_,$
where
:$-p \delta_$ is the indeterminate spherical stress due to the constraint of incompressibility,
:$A_$ is the $n$-th Rivlin–Ericksen tensor,
:$\eta_0$ is the zero-shear viscosity,
:$\alpha_1$ and $\alpha_2$ are constants related to the zero shear normal stress coefficients.

References

*Bird, RB., Armstrong, RC., Hassager, O., Dynamics of Polymeric Liquids: Second Edition, Volume 1: Fluid Mechanics. John Wiley and Sons 1987 {{ISBN, 047180245X(v.1)
*Bird R.B, Stewart W.E, Light Foot E.N.: Transport phenomena, John Wiley and Sons, Inc. New York, U.S.A., 1960

Non-Newtonian fluids