# A taylor expansion for fluids

A while back I asked the question In fluid dynamics does $u_x-v_y$ have a name?. The reason for my question was due to it's presence in the following Taylor expansion about the orgin (see (1)): $$\newcommand{\p}{\frac{\partial #1}{\partial #2}} \newcommand{\f}{\frac{ #1}{ #2}} \newcommand{\l}{\left(} \newcommand{\r}{\right)} \newcommand{\mean}{\langle #1 \rangle}\newcommand{\e}{\varepsilon} \newcommand{\ket}{\left|#1\right>} u(x,y)\approx u_0+\f{1}{2}x(u_x+v_y)+\f{1}{2}x(u_x-v_y)+\f{1}{2}y(v_x+u_y)-\f{1}{2}y(v_x-u_y)$$ $$v(x,y)\approx v_0+\f{1}{2}y(u_x+v_y)-\f{1}{2}y(u_x-v_y)+\f{1}{2}x(v_x+u_y)+\f{1}{2}x(v_x-u_y)$$ where all derivatives are evaluated at $x=y=0$ and each coefficient has a specific meaning as explained in (1). Now I know1 that these equations have something to do with 2d turbulence but can find no information on the relation. Thus my question is this does the above expansions have a name (so I can look it up) and what is their relation to 2d turbulence?

Reference

(1) Mak, M. 2011. Atmospheric Dynamics. Cambridge: Cambridge University Press (page 50 Link to Google Books)

1From a source which is not publicly available and that goes into very little detail except mentioning the existence of such a relation.

• What source is not publicly available? Is it behind a firewall or is it like a handwritten note on a book or something? In 2D, the $u_x + v_y$ term is just the dilation term, and for incompressible flows it will be zero -- but then $u_x - v_y$ would just be $2 u_x$ and so I assume this is related to compressible flows only. But I don't know why expanding things the way it is done there would be terribly useful. – tpg2114 Jan 4 '17 at 19:13
• @tpg2114 they are old notes on a server only accessible thru my uni. (I have provided as much information if not more then in those notes), although I think it's relation to turbulence is to do with the follows in the following paper onlinelibrary.wiley.com/doi/10.1111/j.2153-3490.1955.tb01147.x/… (pdf page 7) – Quantum spaghettification Jan 4 '17 at 19:22
• The only other relation I can think of that may matter: if it is 2D, you have a stream function $\psi$ such that $u = \psi_y$ and $v = \psi_x$. In that case, $u_x - v_y = 2 \psi_{xy}$. In that situation, $u \approx u_0 + x \psi_{xy} + ...$ and $v \approx v_0 - y \psi_{xy}$. What that gets you that you didn't already know... not sure I have any idea. I haven't found this quantity in any other place, and the link to the Atmospheric Dynamics book says that page is blocked for me. – tpg2114 Jan 4 '17 at 19:47