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:
The formula can be rewritten as:
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.
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.