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.
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.
[1] The upper-convected derivative is widely used in polymer rheology for the description of the behavior of a viscoelastic fluid under large deformations.
The form the equation is written in is not entirely clear due to different definitions for
The notation used here is picked to be consistent with the literature using the upper-convected derivative.