\psi_2)}{\partial(x, y)}$$ and we write the remaining parts of the Navier-Stokes equation as $$\begin{split}
0 = & \nabla^4 \psi_2  +\gamma_{NL}(\epsilon, r)\sin\theta \cos\theta +\gamma_{HA}(\epsilon,r)  \sin\theta \\
& - R_e \frac{\partial(\psi_2, \delsq \psi)}{\partial(x, y)}- R_e \frac{\partial(\psi, \delsq \psi_2)}{\partial(x, y)}
- R_e \frac{\partial(\psi_2, \delsq \psi_2)}{\partial(x, y)}
\end{split}$$ We do not know how to solve this full non-linear system, and propose instead to treat a linearized form. The question now is whether to work with the raw system $$0 =  \nabla^4 \psi_2  +\gamma_{NL}(\epsilon, r)\sin\theta \cos\theta +\gamma_{HA}(\epsilon,r)  \sin\theta \\$$ or to propose a Helmholtz-damped variation along the lines of our original approach. We also need to understand which non-linear term to treat first. [*Provided*]{} $\epsilon$ is of the form $\epsilon = \sqrt{\lambda} R_e$, for example, we know that $\Gamma_{NL}$ is the lowest order in $R_e$: $\Gamma_{NL} = O(R_e^2/\log^2 R_e)$, so we will give priority to killing this lowest order correction.

Naive analysis of the raw forms
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To develop the solution we need to identify the Greens’ s function for the biharmonic operator. In fact we do not need the full form as the angular structure of the right side is simple - we need only look for a pair of appropriate radial Green’s functions 