Let $a=\lambda\dot\gamma$. For simple shear, the steady conformation equation is
$$
F\mathbf C-\mathbf I
=\lambda[(\nabla\mathbf u)\mathbf C
+\mathbf C(\nabla\mathbf u)^T].
$$
Its nonzero components are
$$
C_{yy}=C_{zz}=\frac1F,
\qquad
C_{xy}=\frac a{F^2},
\qquad
C_{xx}=\frac1F+\frac{2a^2}{F^3}.
$$
Since $\boldsymbol\tau=\eta_s\dot{\boldsymbol\gamma}+(\eta_p/\lambda)(F\mathbf C-\mathbf I)$,
$$
\boxed{\tau_{xy}=\left(\eta_s+\frac{\eta_p}{F}\right)\dot\gamma},
$$
$$
\boxed{\tau_{xx}=\frac{2\eta_p\lambda\dot\gamma^2}{F^2}},
\qquad
\tau_{yy}=\tau_{zz}=0.
$$
Thus the <FENE-P model> has
$$
\boxed{N_1=\frac{2\eta_p\lambda\dot\gamma^2}{F^2}>0,
\qquad N_2=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 2 / b / i / Solution

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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.


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