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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 352

For uniaxial extension,
$$
\mathbf u=\left(-\frac{\dot\epsilon x}{2},
-\frac{\dot\epsilon y}{2},\dot\epsilon z\right),
\qquad a=\lambda\dot\epsilon.
$$
The diagonal steady conformation tensor is
$$
\boxed{C_{xx}=C_{yy}=\frac1{F+a},
\qquad C_{zz}=\frac1{F-2a}},
$$
where physical solutions require $F>2a$. The <extensional viscosity> is
$$
\boxed{\eta_{\rm ext}
=\frac{\tau_{zz}-\tau_{xx}}{\dot\epsilon}
=3\eta_s+\eta_p\left[
\frac2{F-2a}+\frac1{F+a}
\right]}.
$$
The implicit closure is
$$
\boxed{T=\frac2{F+a}+\frac1{F-2a},
\qquad
F=\frac{L^2-3}{L^2-T}}.
$$

Solution

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

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

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

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