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Upper convected time derivative

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In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, 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.

Contents

The operator is specified by the following formula:

A ▽ = D D t A − ( ∇ v ) T ⋅ A − A ⋅ ( ∇ v )

where:

  • A ▽ is the upper-convected time derivative of a tensor field A
  • D D t is the substantive derivative
  • ∇ v = ∂ v j ∂ x i is the tensor of velocity derivatives for the fluid.

    • The formula can be rewritten as:

    A ▽ i , j = ∂ A i , j ∂ t + v k ∂ A i , j ∂ x k − ∂ v i ∂ x k A k , j − ∂ v j ∂ x k A i , k

    By definition the upper-convected time derivative of the Finger tensor is always zero.

    The upper-convected derivative is widely use in polymer rheology for the description of behavior of a viscoelastic fluid under large deformations.

    Simple shear

    For the case of simple shear:

    ∇ v = ( 0 0 0 γ ˙ 0 0 0 0 0 )

    Thus,

    A ▽ = D D t A − γ ˙ ( 2 A 12 A 22 A 23 A 22 0 0 A 23 0 0 )

    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:

    ∇ v = ( ϵ ˙ 0 0 0 − ϵ ˙ 2 0 0 0 − ϵ ˙ 2 )

    Thus,

    A ▽ = D D t A − ϵ ˙ 2 ( 4 A 11 A 12 A 13 A 12 − 2 A 22 − 2 A 23 A 13 − 2 A 23 − 2 A 33 )

    References

    Upper-convected time derivative Wikipedia
    (Text) CC BY-SA


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