note that $$\nabla^4 \psi_1 - R_e \frac{\partial(\psi_1, \delsq \psi_1)}{\partial(x, y)} \equiv \lambda^2 R_e^{4\beta} \psi_1$$ Next suppose that we consider a more general stream function $\psi$ as $$\psi = \psi_0 + \psi_1$$ Then a trivial calculation tells us that $$\nabla^4 \psi - R_e \frac{\partial(\psi, \delsq \psi)}{\partial(x, y)} = \lambda^2 R_e^{4\beta} \psi_1 - \lambda R_e^{2\beta+1} \frac{\partial(\psi_0, \psi_1)}{\partial(x, y)}   \label{expandedns}$$ Our approach is therefore to combine solutions for potential flow, $\psi_0$, with a solution $\psi_1$ of the Helmholtz equation Eq. (23) that also satisfies the fourth order equation given by Eq. (24) without the identically zero non-linear term: $$\nabla^4 \psi_1 = \lambda^2 R_e^{4\beta} \psi_1$$ When $R_e \rightarrow 0$ this PDE of course approaches the biharmonic equation provided $\beta >0$. Then Eq. (\[expandedns\]) offers the possibility that we can solve a PDE of the form: $$\nabla^4 \psi - R_e \frac{\partial(\psi, \delsq \psi)}{\partial(x, y)} = O(R_e^k) f(x,y)$$ where $k$ and $f$ can be calculated and analyzed. The introduction of the term of order $R_e^{4\beta}$ in the right side of Eq. (\[expandedns\]) may seem like an artificial device, but given that (a) we are seeking solutions of the problem as $R_e \rightarrow 0$; (b) we are not modifying higher derivatives in the equations; (c) we have 