Oldroyd-B model

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 dumbbells. The model is named after its creator James G. Oldroyd.

The model can be written as:

 \mathbf{T} + \lambda_1 \stackrel{\nabla}{\mathbf{T}} = 2\eta_0 (\mathbf{D} + \lambda_2 \stackrel{\nabla}{\mathbf{D}})

where:

 \stackrel{\nabla}{\mathbf{T}} = \frac{\partial}{\partial t} \mathbf{T} + \mathbf{v} \cdot \nabla \mathbf{T} -( (\nabla \mathbf{v})^T \cdot \mathbf{T} + \mathbf{T} \cdot (\nabla \mathbf{v}))
  • \mathbf{v} is the fluid velocity
  • η0 is the total viscosity composed of solvent and polymer components, η0 = ηs + ηp;;
  • \mathbf {D} is the deformation rate tensor.

The model can also be written split into polymeric (viscoelastic) part separately from the solvent part:  \mathbf{T} = 2\eta_s \mathbf{D} + \mathbf{\tau} .

where

 \mathbf{\tau} + \lambda_1 \stackrel{\nabla}{\mathbf{\tau}} = 2\eta_p \mathbf{D}



Whilst the model gives good approximations of viscoelastic fluids in shear flow, it has an unphysical singularity in extensional flow, where the dumbbells are infinitely stretched; If the solvent viscosity is zero then the Oldroyd-B becomes the Upper Convected Maxwell model.


References

  • Owens, R. G.,Phillips, T. N. (2002). Computational Rheology.. Imperial College Press. ISBN 978-1-86094-186-3. 

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