Algebra of Communicating Processes

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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:

A=DDtA(v)TAA(v)

where:

The formula can be rewritten as:

Ai,j=Ai,jt+vkAi,jxkvixkAk,jvjxkAi,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:

v=(000γ˙00000)

Thus,

A=DDtAγ˙(2A12A22A23A2200A2300)

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:

v=(ϵ˙000ϵ˙2000ϵ˙2)

Thus,

A=DDtAϵ˙2(4A11A12A13A122A222A23A132A232A33)

See also

References

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