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:

$\mathbf{T}+{\lambda }_1\stackrel{\mathrm{\nabla }}{\mathbf{T}}=2{\eta }_0(\mathbf{D}+{\lambda }_2\stackrel{\mathrm{\nabla }}{\mathbf{D}})$

where:

• $\mathbf{T}$ is the stress tensor;
• ${\lambda }_1$ is the relaxation time;
• ${\lambda }_2$ is the retardation time = $\frac{{\eta }_s}{{\eta }_0}{\lambda }_1$;
• $\stackrel{\mathrm{\nabla }}{\mathbf{T}}$ is the Upper convected time derivative of stress tensor:

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

• $\mathbf{v}$ 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, $\mathbf{D}=\frac{1}{2}[\mathrm{\nabla }\mathbf{v}+(\mathrm{\nabla }\mathbf{v}{)}^T]$.

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

where

$\mathit{\tau }+{\lambda }_1\stackrel{\mathrm{\nabla }}{\mathit{\tau }}=2{\eta }_p\mathbf{D}$

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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