 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2020

Paper 3

Let
$$
\epsilon=\sqrt{\frac\nu Q},
\qquad
\eta=\frac{y}{\epsilon x},
\qquad
\psi=\sqrt{\nu Q}\,f(\eta)=Q\epsilon f(\eta).
$$
Using the <Cartesian streamfunction> convention $u=\psi_y$, $v=-\psi_x$ gives
$$
u=\frac Qx f'(\eta),
\qquad
v=\frac{Q\epsilon}{x}\eta f'(\eta).
$$
At fixed $y$,
$$
u_x=-\frac Q{x^2}[f'+\eta f''],
\qquad
u_y=\frac Q{\epsilon x^2}f'',
\qquad
u_{yy}=\frac Q{\epsilon^2x^3}f'''.
$$
The terms proportional to $\eta f'f''$ cancel, leaving
$$
u u_x+v u_y=-\frac{Q^2}{x^3}(f')^2.
$$
Since $U=-Q/x$, $UU'=-Q^2/x^3$, while $\nu u_{yy}=Q^2f'''/x^3$. The boundary-layer equation therefore becomes
$$
\boxed{f'''=1-(f')^2}.
$$
Choosing the wall value of the streamfunction as zero, imposing no slip, and matching to $U=-Q/x$ give
$$
\boxed{f(0)=0,
\qquad f'(0)=0,
\qquad f'(\eta)\to-1\quad(\eta\to\infty)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 38B / b / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 38B / b / iii / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 38B / b / iii / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)