 Taylor-expanded no-slip boundary condition

For a displaced wall $y=ϵh_1(x,t)$, write fluid velocity as $\mathbf{u}=ϵ{\mathbf{u}}_1+{ϵ}^2{\mathbf{u}}_2+\cdots$ and prescribed wall velocity as $ϵ{\mathbf{v}}_1+{ϵ}^2{\mathbf{v}}_2+\cdots$. A Taylor expansion about $y=0$ gives ${\mathbf{u}}_1(0)={\mathbf{v}}_1$ and ${\mathbf{u}}_2(0)={\mathbf{v}}_2-h_1{\partial }_y{\mathbf{u}}_1(0)$. The second-order correction converts first-order velocity gradients into a mean boundary flow.