term is well behaved) and the time-independent NavierStokes equations reduce to $$\nabla^4 \psi = 0$$ which is the biharmonic limit, also known as Stokes flow. It is now well known (see for example, Chapter 8 of Van Dyke (1964)) that the neglect of the non-linear terms can lead to inconsistencies, as is evidenced by the lack of any solution for asymptotically uniform two-dimensional flow past a cylinder.

Here we introduce the scalar vorticity function $\omega = \nabla^2 \psi$ and write the NavierStokes as the pair $$\begin{split}
\omega & = \nabla^2 \psi\\
\nabla^2 \omega - R_e  \frac{\partial \omega}{\partial \tau}&= R_e \frac{\partial(\psi, \omega)}{\partial(x, y)}
\end{split}$$ If we had exponential growth in vorticity, $\omega = e^{\mu \tau}\hat{\omega}(x,y)$, then the linearized form would be $$\begin{split}
 e^{\mu \tau}\hat{\omega}(x,y) & = \nabla^2 \psi\\
\nabla^2\hat{\omega} - R_e \mu \hat{\omega}&= 0\end{split}$$ or in terms of a single condition: $$\nabla^4 \psi - R_e \mu \nabla^2 \psi=0\ ,$$ which is the Laplace-Helmholtz equation.

The instability of Stokes’ paradox
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Let us focus temporarily on the linearized time-dependent case based on the Laplace-Helmholtz equation. We let $\varepsilon^2 = R_e \mu$. It is an elementary exercise (and we shall give equivalent details in a different context later) to establish that for all 