= Taylor-expanded no-slip boundary condition
{c}
For a displaced wall $y=\epsilon h_1(x,t)$, write fluid velocity as $\mathbf u=\epsilon\mathbf u_1+\epsilon^2\mathbf u_2+\cdots$ and prescribed wall velocity as $\epsilon\mathbf v_1+\epsilon^2\mathbf v_2+\cdots$. 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.
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