Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 37A (Fluid Dynamics II)

The proposed stream function gives $u=Cx{H}^{-2}{\varphi }^{\prime }$ and $v=-C{H}^{-1}\varphi$. Since $\eta =y/H$, ${\partial }_t\eta =-\eta H^{\prime }/H$, the vorticity equation terms areFor $H=\sqrt{2Ct}$, $H{H}^{\prime }=C$. Divide by $C^{2}x/H^5$ (or compare coefficients at $x=0$ by continuity) to obtainThe planes move normally with velocities $v(\pm H)=\pm H^{\prime }=\pm C/H$ and have zero tangential velocity. No-slip boundary conditions therefore requireThese ensure both normal impermeability relative to the moving planes and tangential no slip.

Put $K=k\pi$, $s_k=(-1{)}^k$ and $\varphi =s_k\sin (K\eta )/K-\eta$. Its derivatives are ${\varphi }^{\prime }=s_k\cos (K\eta )-1$, ${\varphi }^{\prime \prime }=-s_{k}K\sin (K\eta )$, ${\varphi }^{\prime \prime \prime }=-s_k{K}^2\cos (K\eta )$ and ${\varphi }^{\prime \prime \prime \prime }=s_k{K}^3\sin (K\eta )$. 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 $(\eta +\varphi ){\varphi }^{\prime \prime \prime }$ and $-{\varphi }^{\prime }{\varphi }^{\prime \prime }$ cancel, leavingThus each displayed profile is an exact solution when $R=k^2{\pi }^2/4$.

For large $k$, $\varphi$ stays within $1/(k\pi )$ of $-\eta$, but the tangential velocity factor ${\varphi }^{\prime }$ 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.