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: 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. The upper-convected derivative is widely used in polymer rheology for the description of the behavior of a viscoelastic fluid under large deformations. Examples for the symmetric tensor A Simple shear For the case of simple shear: Thus, 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: Thus, See also Upper-convected Maxwell model References Notes Multivariable calculus Fluid dynamics Non-Newtonian fluids