Upper convected 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}^{ abla} = frac{D}{Dt} mathbf{A} - ( abla mathbf{v})^T cdot mathbf{A} - mathbf{A} cdot ( abla mathbf{v}) where:
* mathbf{A}^{ abla} is the Upper convected time derivative of a tensor field mathbf{A}
*frac{D}{Dt} is the Substantive derivative
* abla 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}^{ abla}_{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:: abla mathbf{v} = egin{pmatrix} 0 & 0 & 0 \ {dot gamma} & 0 & 0 \ 0 & 0 & 0 end{pmatrix}

Thus,: mathbf{A}^{ abla} = frac{D}{Dt} mathbf{A}-dot gamma egin{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:: abla mathbf{v} = egin{pmatrix} dot epsilon & 0 & 0 \ 0 & -frac {dot epsilon} {2} & 0 \ 0 & 0 & -frac{dot epsilon} 2 end{pmatrix}

Thus,: mathbf{A}^{ abla} = frac{D}{Dt} mathbf{A}-frac {dot epsilon} 2 egin{pmatrix} 4A_{11} & A_{12} & A_{13} \ A_{12} & -2A_{22} & -2A_{23} \ A_{13} & -2A_{23} & -2A_{33} end{pmatrix}

ee also

*Upper Convected Maxwell

References

*


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Upper Convected Maxwell model — The Upper Convected Maxwell model (or UCM model) is a generalisation of the Maxwell material for the case of large deformations using the Upper convected time derivative. The model was proposed by J. G. Oldroyd.The model can be written as::… …   Wikipedia

  • Material derivative — The material derivative[1][2] is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and classical mechanics. It describes the time rate of change of some quantity (such as heat or momentum) by following… …   Wikipedia

  • Oldroyd-B model — The Oldroyd B model is a constitutive model used to describe the flow of viscoelastic fluids. This model can be regarded as an extension of the Upper Convected Maxwell model and is equivalent to a fluid filled with elastic bead and spring… …   Wikipedia

  • List of mathematics articles (U) — NOTOC U U duality U quadratic distribution U statistic UCT Mathematics Competition Ugly duckling theorem Ulam numbers Ulam spiral Ultraconnected space Ultrafilter Ultrafinitism Ultrahyperbolic wave equation Ultralimit Ultrametric space… …   Wikipedia

  • Dirac equation — Quantum field theory (Feynman diagram) …   Wikipedia

  • Maxwell material — A Maxwell material is a viscoelastic material having the properties both of elasticity and viscosity. It is named for James Clerk Maxwell who proposed the model in 1867. It is also known as a Maxwell fluid. Contents 1 Definition 2 Effect of a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”