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 question** involves 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?
> 
> 
> 



Linearized stream function
First compare your derivation to the following. Consider the Navier-Stokes equation with the plane symmetric ansatz, which reduces to $v = (v\_1,v\_2)$ solving 
$$ \partial\_t v + (v\cdot \nabla)v + \nabla p = \epsilon \triangle v $$
(We ignore the external force for now, since we are only concerned with the linearized equation.) Taking the two-dimensional curl of $v$ we have the vorticity equation 
$$\tag{V} \partial\_t \omega + (v \cdot\nabla)\omega = \epsilon \triangle \omega $$
using, in its derivation, the incompressibility assumption. The stream function satisfies 
$$\tag{1} \triangle \phi = \omega $$
and
$$\tag{2} v = w + (\partial\_2 \phi, - \partial\_1 \phi) $$
where $w$ is both divergence free and curl free. This $w$ corresponds to the freedom of replacing $\phi\mapsto \phi + \psi$ where $\psi$ is a harmonic function so doesn't change (1). 


Plugging into (V) we arrive at the equation for stream function
$$ \tag{S} \partial\_t \triangle\phi + (w\cdot \nabla) \triangle\phi + (\nabla\phi)\wedge(\nabla\triangle \phi) = \epsilon \triangle^2 \phi $$


Now write $\phi = \phi\_0 + \tilde{\phi}$, where $\phi\_0$ satisfies that $\triangle \phi\_0 = U'$ (the stream function corresponding to the background solution) and that $w$ is defined so that $(U,0) = w + (\partial\_2 \phi\_0, -\partial\_1\phi\_0)$, we see that the linearized equation is exactly as you quoted:


$$\tag{SL} \partial\_t \triangle \tilde{\phi} + U \partial\_x \triangle \tilde{\phi} - U'' \partial\_x \tilde{\phi} = \epsilon \triangle^2\tilde{\phi} $$




---


So what's the difference? The key difference is in the phrase "linearized vorticity equation" on page 2 of the article you mentioned. Converting from velocity to vorticity you are losing a bit of information! The vorticity only captures the curl part of the velocity, and the curl-free part is left un-prescribed! So while you considered "perturbations of velocity" from which you derived the "perturbations of vorticity", the article starts from considering "perturbations of vorticity"; and this additional degree of freedom allows you to essentially assume that the quantity "$\partial\_y w\_2$" in your computation vanishes. 


