Definitions

Material time derivative

Upper convected time derivative

In continuum mechanics, including fluid dynamics upper convected time derivative or Oldroyd derivative 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:

mathbf{A}^{nabla} = frac{D}{Dt} mathbf{A} - (nabla mathbf{v})^T cdot mathbf{A} - mathbf{A} cdot (nabla mathbf{v})
where:

The formula can be rewritten as:

{A}^{nabla}_{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 derivatives is widely use in polymer rheology for the description of behavior of a visco-elastic 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,

mathbf{A}^{nabla} = frac{D}{Dt} mathbf{A}-dot gamma begin{pmatrix} 2 A_{12} & A_{22} & A_{23} A_{22} & 0 & 0 A_{23} & 0 & 0 end{pmatrix}

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

Thus,

mathbf{A}^{nabla} = 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

References

  • Macosko, Christopher (1993). Rheology. Principles, Measurements and Applications. VCH Publisher. ISBN 1-56081-579-5.

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