Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 2 / b / ii

Impermeability gives $\hat{w}(\pm 1)=0$. At $\zeta =\pm h$, the base velocity is continuous but its derivative jumps. The normal velocity and pressure must remain continuous. From incompressibility and the streamwise linearized momentum equation,Therefore, with derivatives now taken in $\zeta$, the vorticity-jump matching for an inviscid shear flow isAt the upper jump $[U^{\prime }]=-1/h$ and at the lower jump $[U^{\prime }]=1/h$, where brackets mean the limit from larger $\zeta$ minus that from smaller $\zeta$. Equivalently $(U-c)[{\hat{w}}^{\prime }]=[U^{\prime }]\hat{w}$ away from $U=c$. Setting $[{\hat{w}}^{\prime }]=0$ would discard the vorticity jumps and give the wrong eigenproblem.