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:
- 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.
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.
Examples for the symmetric tensor A
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:
- Macosko, Christopher (1993). Rheology. Principles, Measurements and Applications. VCH Publisher. ISBN 978-1-56081-579-2.