Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 334 / 1 / b / Solution

Expand $\psi =ϵ{\psi }_1+{ϵ}^2{\psi }_2+\cdots$ and $U=ϵU_1+{ϵ}^2{U}_2+\cdots$. A Taylor expansion of the sheet boundary conditions about $y=0$ givesThe first-order Fourier mode is ${\psi }_1=f(y)\sin \theta$, with $({\partial }_y^2-1{)}^{2}f=0$, $f(0)=1$, and $f^{\prime }(0)=0$. Decay excludes the growing exponentials, leaving $f=(a+by)e^{-y}$. The two boundary values give $a=b=1$:There is no first-order mean boundary motion. Boundedness excludes mean shear, so $\boxed{U_1=0}$.