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 ${\displaystyle n}$-th Rivlin–Ericksen tensor,

${\displaystyle \eta _{0}}$ is the zero-shear viscosity,

${\displaystyle \alpha _{1}}$ and ${\displaystyle \alpha _{2}}$ are constants related to the zero shear normal stress coefficients.