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 ) , {\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

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

• 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