Upper convected time derivative 3507217 128246777 2007-05-04T17:15:47Z Cydebot 1215485 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]]