Upper convected time derivative
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2007-05-04T17:15:47Z
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Robot - Moving category Multivariate calculus to Multivariable calculus per [[WP:CFD|CFD]] at [[Wikipedia:Categories for discussion/Log/2007 April 27]].
In [[continuum mechanics]], including [[fluid dynamics]] '''upper convected time derivative''' or '''Oldroyd derivative''' is the [[derivative|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:
:<math> \mathbf{A}^{\nabla} = \frac{D}{Dt} \mathbf{A} - (\nabla \mathbf{v})^T \cdot \mathbf{A} - \mathbf{A} \cdot (\nabla \mathbf{v}) </math>
where:
*<math> \mathbf{A}^{\nabla} </math> is the Upper convected time derivative of a tensor [[Field (physics)|field]] <math> \mathbf{A} </math>
*<math>\frac{D}{Dt}</math> is the [[Substantive derivative]]
*<math>\nabla \mathbf{v}=\frac {\partial v_j}{\partial x_i} </math> is the tensor of [[velocity]] [[derivative]]s for the fluid.
The formula can be rewritten as:
:<math> {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} </math>
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]]:
:<math> \nabla \mathbf{v} = \begin{pmatrix} 0 & 0 & 0 \\ {\dot \gamma} & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} </math>
Thus,
:<math> \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} </math>
===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:
:<math> \nabla \mathbf{v} = \begin{pmatrix} \dot \epsilon & 0 & 0 \\ 0 & -\frac {\dot \epsilon} {2} & 0 \\ 0 & 0 & -\frac{\dot \epsilon} 2 \end{pmatrix} </math>
Thus,
:<math> \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} </math>
==See also==
*[[Upper Convected Maxwell]]
==References==
* {{cite book | author=Macosko, Christopher| title=Rheology. Principles, Measurements and Applications | publisher=VCH Publisher | year=1993 | id=ISBN 1-56081-579-5}}
[[Category:Multivariable calculus]]
[[Category:Fluid dynamics]]
[[Category:Non-Newtonian fluids]]