title: Second-order fluid
text: # 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
  
      $\sigma _{ij}=-p\delta _{ij}+\eta _{0}A_{ij(1)}+\alpha _{1}A_{ik(1)}A_{kj(1)}+\alpha _{2}A_{ij(2)},$
  
  where
  
      $-p\delta _{ij}$ is the indeterminate spherical stress due to the constraint of incompressibility,
      $A_{ij(n)}$ 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.
  