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.

For the case of simple shear:

Thus,

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,

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