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:

$\mathbf{A}^{\nabla} = \frac{D}{Dt} \mathbf{A} - (\nabla \mathbf{v})^T \cdot \mathbf{A} - \mathbf{A} \cdot (\nabla \mathbf{v})$

where:

$\mathbf{A}^{\nabla}$ is the Upper convected time derivative of a tensor field $\mathbf{A}$
$\frac{D}{Dt}$ is the Substantive derivative
$\nabla \mathbf{v}=\frac {\partial v_j}{\partial x_i}$ is the tensor of velocity derivatives for the fluid.

The formula can be rewritten as:

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

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.

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1 Examples for the symmetric tensor A
- 1.1 Simple shear
- 1.2 Uniaxial extension of uncompressible fluid
2 See also
3 References

Examples for the symmetric tensor A

Simple shear

For the case of simple shear:

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

Thus,

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

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:

$\nabla \mathbf{v} = \begin{pmatrix} \dot \epsilon & 0 & 0 \\ 0 & -\frac {\dot \epsilon} {2} & 0 \\ 0 & 0 & -\frac{\dot \epsilon} 2 \end{pmatrix}$

Thus,

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