= Fourier traction map for a viscous half-space
{c}
{title2=$\sigma_{xy}=-2\mu|k|U,\quad\sigma_{yy}=-2\mu|k|V$}
For a two-dimensional decaying <Stokes flow> in $y>0$, prescribe boundary <Fourier mode> $(U,V)e^{ikx}$. The boundary <shear stress> and normal stress are the displayed diagonal map, with the common exponential understood. For $k>0$ and $V=0$, the <Unscaled Papkovich–Neuber representation> gives $u=U(1-ky)e^{ikx-ky}$, $v=-ikUy e^{ikx-ky}$ and $p=-2i\mu kUe^{ikx-ky}$. Differentiation proves the tangential traction and vanishing normal stress. The normal-velocity calculation supplies the second diagonal entry. The map converts a viscous exterior into a nonlocal boundary resistance proportional to $|k|$.
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