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} = rac{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 $mathbf\{A\}$

★ is the Substantive derivative

★ $$
abla mathbf{v}=rac {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} = 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

Contents

Uniaxial extension of uncompressible fluid

See also

References

=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} = rac{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 & -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 egin{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

★ Rheology. Principles, Measurements and Applications, Macosko, Christopher, , , VCH Publisher, 1993, ISBN 1-56081-579-5