Definitions

# 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:

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

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

The formula can be rewritten as:

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

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

Thus,

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

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

Thus,

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