Rivlin–Ericksen tensor (original) (raw)

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Concept in physics

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

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

where

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

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

Higher-order tensor may be found iteratively by the expression

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