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})

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}


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}


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


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

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