My watch list
my.chemeurope.com  
Login  

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:

  • \mathbf{A}^{\nabla} 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:

{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.

Additional recommended knowledge

Contents

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. 
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Upper_convected_time_derivative". A list of authors is available in Wikipedia.
Your browser is not current. Microsoft Internet Explorer 6.0 does not support some functions on Chemie.DE