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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 1

The shear stress is $\tau_{rz}=\mu u'(r)$, where
$$
u'(r)=-\frac{\rho g}{2\mu}r+\frac Ar.
$$
Thus the axial forces exerted by the fluid per unit length on the inner and outer cylinders are
$$
\boxed{F_{\rm in}=2\pi\alpha R\mu u'(\alpha R)
=2\pi\mu A-\pi\rho g\alpha^2R^2},
$$
$$
\boxed{F_{\rm out}=-2\pi R\mu u'(R)
=\pi\rho gR^2-2\pi\mu A}.
$$
Their sum is
$$
F_{\rm in}+F_{\rm out}=\pi\rho gR^2(1-\alpha^2),
$$
the weight per unit length of the fluid in the annulus. This is the expected overall vertical force balance.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 39C / e / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 39C / e / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 39C / e / Solution

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