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:

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

where:

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

The formula can be rewritten as:

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

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.

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

Simple shear

For the case of simple shear:

$\nabla \mathbf {v} ={\begin{pmatrix}0&0&0\\{\dot {\gamma }}&0&0\\0&0&0\end{pmatrix}}$

Thus,

${\stackrel {\triangledown }{\mathbf {A} }}={\frac {D}{Dt}}\mathbf {A} -{\dot {\gamma }}{\begin{pmatrix}2A_{12}&A_{22}&A_{23}\\A_{22}&0&0\\A_{23}&0&0\end{pmatrix}}$

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:

$\nabla \mathbf {v} ={\begin{pmatrix}{\dot {\epsilon }}&0&0\\0&-{\frac {\dot {\epsilon }}{2}}&0\\0&0&-{\frac {\dot {\epsilon }}{2}}\end{pmatrix}}$

Thus,

${\stackrel {\triangledown }{\mathbf {A} }}={\frac {D}{Dt}}\mathbf {A} -{\frac {\dot {\epsilon }}{2}}{\begin{pmatrix}4A_{11}&A_{12}&A_{13}\\A_{12}&-2A_{22}&-2A_{23}\\A_{13}&-2A_{23}&-2A_{33}\end{pmatrix}}$

See also

• Upper-convected Maxwell model

References

• Macosko, Christopher (1993). Rheology. Principles, Measurements and Applications. VCH Publisher. ISBN 978-1-56081-579-2.

Notes