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:

${\displaystyle {\stackrel {\triangledown }{\mathbf {A} }}={\frac {D}{Dt}}\mathbf {A} -(\nabla \mathbf {v} )^{T}\cdot \mathbf {A} -\mathbf {A} \cdot (\nabla \mathbf {v} )}$

where:

• ${\displaystyle {\stackrel {\triangledown }{\mathbf {A} }}}$ is the upper-convected time derivative of a tensor field ${\displaystyle \mathbf {A} }$
• ${\displaystyle {\frac {D}{Dt}}}$ is the substantive derivative
• ${\displaystyle \nabla \mathbf {v} ={\frac {\partial v_{j}}{\partial x_{i}}}}$ is the tensor of velocity derivatives for the fluid.

The formula can be rewritten as:

${\displaystyle {\stackrel {\triangledown }{A}}_{i,j}={\frac {\partial A_{i,j}}{\partial t}}+v_{k}{\frac {\partial A_{i,j}}{\partial x_{k}}}-{\frac {\partial v_{i}}{\partial x_{k}}}A_{k,j}-{\frac {\partial v_{j}}{\partial x_{k}}}A_{i,k}}$

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.^[1]

The upper-convected derivative is widely used in polymer rheology for the description of the behavior of a viscoelastic fluid under large deformations.