Algebra of Communicating Processes

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:

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

where:

• $\stackrel{▽}{𝐀}$ is the upper-convected time derivative of a tensor field $𝐀$
• $\frac{D}{Dt}$ is the substantive derivative
• $\mathrm{\nabla }𝐯=\frac{\partial v_j}{\partial x_i}$ is the tensor of velocity derivatives for the fluid.

The formula can be rewritten as:

${\stackrel{▽}{A}}_{i,j}=\frac{\partial A_{i,j}}{\partial t}+v_k\frac{\partial A_{i,j}}{\partial x_k}-\frac{\partial v_i}{\partial x_k}{A}_{k,j}-\frac{\partial v_j}{\partial x_k}{A}_{i,k}$

By definition the upper-convected time derivative of the Finger tensor is always zero.

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:

$\mathrm{\nabla }𝐯=\left(\begin{array}{ccc}0& 0& 0\\ \dot{\gamma }& 0& 0\\ 0& 0& 0\end{array}\right)$

Thus,

$\stackrel{▽}{𝐀}=\frac{D}{Dt}𝐀-\dot{\gamma }\left(\begin{array}{ccc}2A_{12}& A_{22}& A_{23}\\ A_{22}& 0& 0\\ A_{23}& 0& 0\end{array}\right)$

Uniaxial extension of uncompressible fluid

In this case a material is stretched in the direction X and compresses in the direction s Y and Z, so to keep volume constant. The gradients of velocity are:

$\mathrm{\nabla }𝐯=\left(\begin{array}{ccc}\dot{ϵ}& 0& 0\\ 0& -\frac{\dot{ϵ}}{2}& 0\\ 0& 0& -\frac{\dot{ϵ}}{2}\end{array}\right)$

Thus,

$\stackrel{▽}{𝐀}=\frac{D}{Dt}𝐀-\frac{\dot{ϵ}}{2}\left(\begin{array}{ccc}4A_{11}& A_{12}& A_{13}\\ A_{12}& -2A_{22}& -2A_{23}\\ A_{13}& -2A_{23}& -2A_{33}\end{array}\right)$

See also

• Upper-convected Maxwell model

References

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