= Solution
The proposed <stream function> gives $u=CxH^{-2}\phi'$ and $v=-CH^{-1}\phi$. Since $\eta=y/H$, $\partial_t\eta=-\eta H'/H$, the <vorticity equation> terms are
$$
\begin{aligned}
\partial_t\omega&=CxH'H^{-4}(3\phi''+\eta\phi'''),\\
u\partial_x\omega&=-C^2xH^{-5}\phi'\phi'',\\
v\partial_y\omega&=C^2xH^{-5}\phi\phi''',\\
\nu\Delta\omega&=-\nu CxH^{-5}\phi''''.
\end{aligned}
$$
For $H=\sqrt{2Ct}$, $HH'=C$. Divide by $C^2x/H^5$ (or compare coefficients at $x=0$ by <continuity>) to obtain
$$
\boxed{3\phi''+\eta\phi'''-\phi'\phi''+\phi\phi'''+R^{-1}\phi''''=0,
\qquad R=C/\nu.}
$$
The <planes> move normally with velocities $v(\pm H)=\pm H'=\pm C/H$ and have zero tangential velocity. <No-slip boundary conditions> therefore require
$$
\boxed{\phi(1)=-1,\quad\phi(-1)=1,\quad\phi'(1)=\phi'(-1)=0.}
$$
These ensure both normal impermeability relative to the moving <planes> and tangential no slip.
Put $K=k\pi$, $s_k=(-1)^k$ and $\phi=s_k\sin(K\eta)/K-\eta$. Its <derivatives> are $\phi'=s_k\cos(K\eta)-1$, $\phi''=-s_kK\sin(K\eta)$, $\phi'''=-s_kK^2\cos(K\eta)$ and $\phi''''=s_kK^3\sin(K\eta)$. The sine vanishes at both walls and $s_k\cos(K)=1$, verifying all four <boundary conditions>. In the ODE, the sine-cosine products from $(\eta+\phi)\phi'''$ and $-\phi'\phi''$ cancel, leaving
$$
s_kK\sin(K\eta)(K^2/R-4).
$$
Thus \b[each displayed profile is an exact solution when $R=k^2\pi^2/4$].
For large $k$, $\phi$ stays within $1/(k\pi)$ of $-\eta$, but the tangential velocity factor $\phi'$ continues to oscillate between $-2$ and $0$, and its shear grows like $k$. The sketch shows both effects; a nearly straight stream-function profile does not mean nearly uniform velocity. Such fine oscillatory branches occur at special discrete large <Reynolds numbers> and require specially prepared flow. Their many inflection points and strong shear permit instability, so generic disturbances are unlikely to preserve them. This is a physical plausibility discussion, not a proved stability classification.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-3-expanding-channel.png]
{title=Large-k expanding-channel stream-function profile and its oscillatory tangential-velocity factor}
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