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^[1] truncated at the second-order. For an isotropic, incompressible second-order fluid, the total stress tensor is given by

${\displaystyle \sigma _{ij}=-p\delta _{ij}+\eta _{0}A_{ij(1)}+\alpha _{1}A_{ik(1)}A_{kj(1)}+\alpha _{2}A_{ij(2)},}$

where

${\displaystyle -p\delta _{ij}}$ is the indeterminate spherical stress due to the constraint of incompressibility,

${\displaystyle A_{ij(n)}}$ 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

1. Rivlin, R. S. & Ericksen, J. L (1955). "Stress-deformation relations for isotropic materials". J. Ration. Mech. Anal. 4. Hoboken. pp. 523–532.

• 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