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} )\cdot \mathbf {A} -\mathbf {A} \cdot (\nabla \mathbf {v} )^{T}

where:

${\stackrel {\triangledown }{\mathbf {A} }}$ is the upper-convected time derivative of a tensor field $\mathbf {A}$
${\frac {D}{Dt}}$ is the substantive derivative
$\nabla \mathbf {v} ={\frac {\partial v_{i}}{\partial x_{j}}}$ 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.^[1]

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:

\nabla \mathbf {v} ={\begin{pmatrix}0&{\dot {\gamma }}&0\\0&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_{21}&A_{31}\\A_{12}&-2A_{22}&-2A_{23}\\A_{13}&-2A_{23}&-2A_{33}\end{pmatrix}}

References

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

Notes

↑ Matolcsi, Tamás; Ván, Péter (2008). "On the Objectivity of Time Derivatives". Atti della Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali (1): 1–13. doi:10.1478/C1S0801015.

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[1] Matolcsi, Tamás; Ván, Péter (2008). "On the Objectivity of Time Derivatives". Atti della Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali (1): 1–13. doi:10.1478/C1S0801015.

[1]

Examples for the symmetric tensor A

Simple shear

Uniaxial extension of incompressible fluid

See also

References