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:
- is the upper-convected time derivative of a tensor field
- is the substantive derivative
- is the tensor of velocity derivatives for the fluid.
The formula can be rewritten as:
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.
For the case of simple shear:
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:
- Macosko, Christopher (1993). Rheology. Principles, Measurements and Applications. VCH Publisher. ISBN 1-56081-579-5.