Upper Convected Time Derivative

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Upper Convected Time Derivative

The operator is specified by the following formula:
:

\mathbf{A}^{
abla} = rac{D}{Dt} \mathbf{A} - (
abla \mathbf{v})^T \cdot \mathbf{A} - \mathbf{A} \cdot (
abla \mathbf{v})

where:

$\mathbf{A}^{$

\mathbf{A}

$rac{D}{Dt}$ is the Substantive Derivative

The formula can be rewritten as:

:

{A}^{
abla}_{i,j} = rac {\partial A_{i,j}} {\partial t} + v_k rac {\partial A_{i,j}} {\partial x_k} - rac {\partial v_i} {\partial x_k} A_{k,j} - rac {\partial v_j} {\partial x_k} A_{i,k}

By definition the upper convected time derivative of the Finger Tensor is always zero.

The upper convected derivatives is widely use in Polymer Rheology for the description of behavior of a Visco-elastic fluid under large deformations.

EXAMPLES FOR THE SYMMETRIC TENSOR A

Simple Shear

For the case of Simple Shear :
:

abla \mathbf{v} = \begin{pmatrix} 0 & 0 & 0 \ {\dot \gamma} & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}

Thus,
:

\mathbf{A}^{
abla} = rac{D}{Dt} \mathbf{A}-\dot \gamma \begin{pmatrix} 2 A_{12} & A_{22} & A_{23} \ A_{22} & 0 & 0 \ A_{23} & 0 & 0 \end{pmatrix}

Uniaxial extension of uncompressible fluid

In this case a material is stretched in the direction X and compresses in the direction s Y and Z, so to keep volume constant.
The gradients of velocity are:
:

abla \mathbf{v} = \begin{pmatrix} \dot \epsilon & 0 & 0 \ 0 & -rac {\dot \epsilon} {2} & 0 \ 0 & 0 & -rac{\dot \epsilon} 2 \end{pmatrix}

Thus,
:

\mathbf{A}^{
abla} = rac{D}{Dt} \mathbf{A}-rac {\dot \epsilon} 2 \begin{pmatrix} 4A_{11} & A_{12} & A_{13} \ A_{12} & -2A_{22} & -2A_{23} \ A_{13} & -2A_{23} & -2A_{33} \end{pmatrix}

SEE ALSO

Upper Convected Maxwell

REFERENCES

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