Upper-convected time derivative

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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.

Examples for the symmetric tensor A[edit]

Simple shear[edit]

For the case of simple shear:


Uniaxial extension of uncompressible fluid[edit]

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:


See also[edit]


  • Macosko, Christopher (1993). Rheology. Principles, Measurements and Applications. VCH Publisher. ISBN 1-56081-579-5.