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:

• $\mathbf{T}$ is the stress tensor;
• λ₁ is the relaxation time;
• λ₂ is the retardation time = $\frac{\eta_s}{\eta_0}\lambda_1$;
• $\stackrel{\nabla}{\mathbf{T}}$ is the Upper convected time derivative of stress tensor:

$\stackrel{\nabla}{\mathbf{T}} = \frac{\partial}{\partial t} \mathbf{T} + \mathbf{v} \cdot \nabla \mathbf{T} -( (\nabla \mathbf{v})^T \cdot \mathbf{T} + \mathbf{T} \cdot (\nabla \mathbf{v}))$

• $\mathbf{v}$ is the fluid velocity
• η₀ is the total viscosity composed of solvent and polymer components, η₀ = η_s + η_p;;
• $\mathbf {D}$ 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.