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.