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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 3

Set
$$
A=\left(\frac{\rho\nu^2}{S}\right)^{1/3},
\qquad U_0=\frac{\nu}{A^2},
\qquad \eta=\frac{y}{A\sqrt x},
$$
and choose stream <function>
$$
\psi=U_0A\sqrt x,f(\eta).
$$
Then
$$
u=U_0f'(\eta),
\qquad
v=-\frac{U_0A}{2\sqrt x}\{f-\eta f'\}.
$$
Substitution gives
$$
f'''+\frac12ff''=0,
$$
with
$$
f(0)=0,
\qquad f''(0)=1,
\qquad f'(\infty)=0.
$$
Treat $f'(0)$ as a shooting parameter, integrate the initial-value problem numerically, and adjust it until the far-field condition holds. The adjacent-fluid speed is then $u(x,0)=U_0f'(0)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 38C / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 38C / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 38C / c / Solution

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