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:
where:
The formula can be rewritten as:
By definition the upper-convected time derivative of the Finger tensor is always zero.
It can be shown that the upper-convected time derivative of a spacelike vector field is just its Lie derivative by the velocity field of the continuum.
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:
Thus,

Uniaxial extension of incompressible fluid

In this case a material is stretched in the direction X and compresses in the directions Y and Z, so to keep volume constant.
The gradients of velocity are:
Thus,