= Solution
{c}
Use the same <stream function> convention and denote differentiation with respect to the <dimensionless variable> $\eta=y/\delta$ by a prime. The proposed <similarity solution> gives
$$
u=Ax f'(\eta),\qquad v=-A\delta f(\eta).
$$
Its <continuity equation> holds identically. Substitution in the tangential <Prandtl boundary-layer equation> gives
$$
A^2x\bigl((f')^2-ff''\bigr)
=\frac{\nu Ax}{\delta^2}f'''+A^2x.
$$
Choose the constant scale $\delta=\sqrt{\nu/A}$. The <Hiemenz flow> equation and its <boundary conditions> are
$$
\boxed{f'''+ff''-(f')^2=-1,\qquad
f(0)=0,\quad f'(0)=0,\quad f'(\infty)=1.}
$$
The first two conditions impose the <no-penetration boundary condition> and <no-slip boundary condition>. The last matches the tangential outer <velocity>. One also has $f''\to0$ and $f(\eta)/\eta\to1$. It is too strong to impose $f-\eta\to0$: the <displacement thickness> generally permits a constant offset, $f\sim\eta-B$.
The normal <Navier-Stokes equation> makes the approximation in part (c) explicit. With this <similarity solution>, it gives
$$
p_y=-\rho g-\rho A^2\delta\bigl(ff'+f''\bigr).
$$
At the wall this differs from $(p_\infty)_y=-\rho g$ by $-\rho A^2\delta f''(0)$. This is a concrete next-order counterexample to interpreting the printed normal <pressure gradient> equality exactly; the tangential <Hiemenz flow> equation remains valid at its stated order.
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