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

σ i j = − p δ i j + η 0 A i j ( 1 ) + α 1 A i k ( 1 ) A k j ( 1 ) + α 2 A i j ( 2 ) ,

where

− p δ i j is the indeterminate spherical stress due to the constraint of incompressibility, A i j ( n ) is the n -th Rivlin–Ericksen tensor, η 0 is the zero-shear viscosity, α 1 and α 2 are constants related to the zero shear normal stress coefficients.