Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 77 / 2 / d

Expand $\psi =ϵ{\psi }_1+{ϵ}^2{\psi }_2+\cdots$ and $\mathcal{U}={ϵ}^2{\mathcal{U}}_2+\cdots$. The screened first-order transverse sheet flow has ${\psi }_1=f(y)\sin \xi$. Set $s=\sqrt{1+A^2}$. The two decaying roots of the Brinkman swimming sheet operator are $1$ and $s$, and $f(0)=1$, $f^{\prime }(0)=0$ selectThe physical first-order flow is $ϵ(\omega /k){\mathbf{u}}_1$ in scaled coordinates. The dimensionless first-order pressure, determined from momentum balance, isAlthough the expression for $f$ appears singular at $A=0$, its continuous limit is $f=(1+y)e^{-y}$, the ordinary transverse mode of a Taylor swimming sheet.