Schröder–Hipparchus number

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;
• $\lambda _{1}$ is the relaxation time;
• $\lambda _{2}$ 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;
• $\eta _{0}$ is the total viscosity composed of solvent and polymer components, $\eta _{0}=\eta _{s}+\eta _{p}$;
• $\mathbf {D}$ is the deformation rate tensor or rate of strain tensor, $\mathbf {D} ={\frac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {v} +({\boldsymbol {\nabla }}\mathbf {v} )^{T}\right]$.

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

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