Upper-convected time derivative

In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, 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:

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

where:

${\stackrel{\triangledown}{\mathbf A}}$ 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:

{\stackrel{\triangledown}{A}}_{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 derivative is widely use in polymer rheology for the description of behavior of a viscoelastic fluid under large deformations.

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,

\stackrel{\triangledown}{\mathbf A} = \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}