Upper convected time derivative

In continuum mechanics, including fluid dynamics upper convected time derivative or Oldroyd derivative is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate system rotating and stretching with the fluid.

The operator is specified by the following formula:: $mathbf{A}^{
abla} = frac{D}{Dt} mathbf{A} - (
abla mathbf{v})^T cdot mathbf{A} - mathbf{A} cdot (
abla mathbf{v})$ where:
* $mathbf{A}^{
abla}$ is the Upper convected time derivative of a tensor field $mathbf{A}$
* $frac{D}{Dt}$ is the Substantive derivative
* $abla mathbf{v}=frac {partial v_j}{partial x_i}$ is the tensor of velocity derivatives for the fluid.

The formula can be rewritten as:

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

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:: $abla mathbf{v} = egin{pmatrix} 0 & 0 & 0 \ {dot gamma} & 0 & 0 \ 0 & 0 & 0 end{pmatrix}$

Thus,: $mathbf{A}^{
abla} = frac{D}{Dt} mathbf{A}-dot gamma egin{pmatrix} 2 A_{12} & A_{22} & A_{23} \ A_{22} & 0 & 0 \ A_{23} & 0 & 0 end{pmatrix}$

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:: $abla mathbf{v} = egin{pmatrix} dot epsilon & 0 & 0 \ 0 & -frac {dot epsilon} {2} & 0 \ 0 & 0 & -frac{dot epsilon} 2 end{pmatrix}$

Thus,: $mathbf{A}^{
abla} = frac{D}{Dt} mathbf{A}-frac {dot epsilon} 2 egin{pmatrix} 4A_{11} & A_{12} & A_{13} \ A_{12} & -2A_{22} & -2A_{23} \ A_{13} & -2A_{23} & -2A_{33} end{pmatrix}$

ee also

*Upper Convected Maxwell

References

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