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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 3

Choose the polar axis along $\mathbf d$ and write $d=|\mathbf d|$. Since
$$
\mathbf d=d(\cos\theta\,\mathbf e_r-\sin\theta\,\mathbf e_\theta),
$$
the velocity components are
$$
u_r=\frac{d\cos\theta}{r^2},
\qquad
u_\theta=\frac{d\sin\theta}{r^2}.
$$
Their <divergence> is
$$
\nabla\cdot\mathbf u
=\frac1r\frac{\partial}{\partial r}(ru_r)
 +\frac1r\frac{\partial u_\theta}{\partial\theta}
=-\frac{d\cos\theta}{r^3}
 +\frac{d\cos\theta}{r^3}=0.
$$
\b[Thus the flow is <incompressible flow>[incompressible] for $r\ne0$.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 7C / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 7C / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 7C / a / Solution

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