A Rivlin–Ericksen temporal evolution of the strain rate tensor such that the derivative translates and rotates with the flow field. The first-order Rivlin–Ericksen is given by

${\displaystyle \mathbf {A} _{ij(1)}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}}$

where

${\displaystyle v_{i}}$ is the fluid's velocity and

${\displaystyle A_{ij(n)}}$ is ${\displaystyle n}$-th order Rivlin–Ericksen tensor.

Higher-order tensor may be found iteratively by the expression

${\displaystyle A_{ij(n+1)}={\frac {\mathcal {D}}{{\mathcal {D}}t}}A_{ij(n)}.}$

The derivative chosen for this expression depends on convention. The upper-convected time derivative, lower-convected time derivative, and Jaumann derivative are often used.

References

• Truesdell, Clifford & Noll, Walter (2004). The Non-Linear Field Theories of Mechanics. Springer. ISBN 978-3-662-10388-3.