Oldroyd-B model

The Oldroyd-B model combines a Newtonian solvent with infinitely extensible Hookean polymer dumbbells. It predicts constant shear viscosity, a positive first normal-stress difference, and an extensional catastrophe at a finite extension rate.

In an inertialess cone-and-plate rheometer, reverse a steady torque from T to -T at time zero. The polymer shear stress remains continuous while the positive solvent viscosity allows the rate to jump. The linear stress equation gives ${\lambda }_2={\mu }_0\tau /({\mu }_0+G_0\tau )$ and the displayed velocity for t>0, where ${\Omega }_{+}=3\alpha T/[2\pi a^3({\mu }_0+G_0\tau )]$. The transient backward speed exceeds the final backward speed because stored polymer stress initially retains its former sign.

For circular Couette flow with $v=Ar+B/r$, the rotating-frame reduction of circular viscoelastic shear gives shear rate $g=-2B/r^2$. The total shear stress is $\mu g$, and the azimuthal minus radial normal-stress difference is $2(\mu -{\mu }_r)\tau g^2$ in the stress convention where ${\sigma }_{rr}=-p$. Radial force balance without inertia is $p^{\prime }=-2(\mu -{\mu }_r)\tau g^2/r$, which integrates to the displayed pressure. The pressure constant is undetermined by these equations.

The FENE-P model replaces infinitely extensible Hookean polymers by finitely extensible nonlinear elastic springs with a mean-field closure. Finite extensibility regularizes the Oldroyd-B extensional catastrophe and produces shear thinning.