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.^[1]

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}}

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". doi:10.1478/C1S0801015.

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[1] Matolcsi, Tamás; Ván, Péter (2008). "On the Objectivity of Time Derivatives". doi:10.1478/C1S0801015.

Upper-convected time derivative

Examples for the symmetric tensor A

Simple shear

Uniaxial extension of incompressible fluid

See also

References