= Similarity solution for a sink-flow boundary layer
Near a radial wall, let $x$ measure distance from the sink along the wall and $y$ point into the fluid. With
$$
\delta=x\sqrt{\frac\nu Q},
\qquad
\eta=\frac y\delta,
\qquad
\psi=\sqrt{\nu Q}\,f(\eta),
$$
the <Prandtl boundary-layer equation> reduces to
$$
f'''=1-(f')^2,
\qquad
f(0)=f'(0)=0,
\qquad
f'(\infty)=-1.
$$
One physical solution has
$$
f'(\eta)=\frac{5-\cosh(\sqrt2\eta+c)}{1+\cosh(\sqrt2\eta+c)},
\qquad
c=\operatorname{arcosh}5.
$$
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