I am trying to work through a paper *Instability in Parallel Flows Revisited* by Friedlander and Howard, and there are a couple steps in the beginning that I do not understand. I apologize in advance for being long-winded, but I want to be candid.

Consider a steady two-dimensional parallel-shear incompressible flow $$\nabla\cdot u=0,$$ $$u\cdot\nabla u=-\nabla p+\frac{1}{Re}\Delta u+\frac{1}{Re}f$$ for $u=(U(y),0)$, pressure $p$ and (steady) force density $f=(h(y),0)$. If we consider a (incompressible) perturbation of this base flow $\tilde{u}=u+w$ (and $\tilde{p}=p+q$), then when plugging $\tilde{u},\tilde{p}$ into the incompressible NSE and dropping the quadratic term, we have the linearized equations for the perturbed flow $w=(w_1,w_2)$ as $$\nabla\cdot w=0,$$ $$\partial_t w+U\partial_x w+(U'(y)w_2,0)=\nabla q+\frac{1}{Re}\Delta w.$$ Next, taking the curl ($\nabla^\perp\cdot(v_1,v_2)=\partial_y v_1-\partial_x v_2$) of this equation, along with sufficient smoothness so derivatives commute, we have $$\partial_t(\nabla^\perp\cdot w)+U(y)\partial_x (\nabla^\perp\cdot w)+\partial_y(U'(y)w_2)=\frac{1}{Re}\Delta (\nabla^\perp\cdot w).$$ At this point, by incompressibility we can switch to using the stream function for $w$, calling it $\phi$, writing $\nabla^\perp\cdot w=\Delta \phi$, and renaming in the last equation $w_2=-\partial_x\phi$, we should achieve the equation for the perturbed stream function as $$\partial_t\Delta\phi+U(y)\partial_x\Delta\phi-\left[U'(y)\partial^2_{xy}\phi+U''(y)\partial_x\phi\right]=\frac{1}{Re}\Delta^2\phi.$$

My

first questioninvolves the bracketed terms: in the paper, they present the linearized stream function equation as $$\partial_t\Delta\phi+U(y)\partial_x\Delta\phi-U''(y)\partial_x\phi=\frac{1}{Re}\Delta^2\phi,$$ where the $-U'(y)\partial^2_{xy}\phi$ term seems to be omitted. Have I missed something completely obvious, or have they made an assumption about $\phi$ (common enough to not be mentioned in the paper)?

I have worked through the derivation a couple times now, and always seem to be getting the "extra term" from the product rule...

For my

second question, more of a soft question, the authors proceed to provide motivation for deriving the Orr-Sommerfeld equation by suggesting $\phi$ be of the form $$\phi=\Psi(y)e^{i\alpha(x-ct)}$$ Is there an intuitive reason for the motivation behind this ansatz? This assumption seems common in linear stability analyses, but I haven't come across any reasoning behind it. Obviously it works, but is there a physical interpretation of this "travelling phase" perturbation?