A Linearized Continuity (Fluids)

1. May 11, 2017

joshmccraney

Hi PF!

Continuity for incompressible flow independent of $\theta$ is $\nabla\cdot u = \partial_ru_r+u_r/r+\partial_z u_z=0$. However, I'm following a problem in cylindrical coordinates, same assumptions as above, and the author states the linearized conservation of mass is $\partial_ru_r+u_r/r+u_z$ (two different authors wrote this so I doubt it's a typo). How are they arriving at this? Also, continuity is already linear in $u$, so what do they mean by "linearized"?

2. May 11, 2017

zwierz

Authors of physics textbooks ramble sometimes. You understand the equation by itself do not you? Then skip this comment and go on.

3. May 11, 2017

joshmccraney

I do, but that still doesn't explain how they get $u_z$ rather than $\partial_z u_z$.

4. May 11, 2017

zwierz

O! now I see what the point is. This is strange indeed. Dimensions must be incomparable

5. May 11, 2017

joshmccraney

Good call on the dimensions. I don't know what they're doing but I'll just move on and forget it (it was the Plateau-Rayleigh instability if you're curious).

6. May 11, 2017

zwierz

perhaps in some books $u_z=\frac{\partial u}{\partial z}$

7. May 11, 2017

joshmccraney

I don't think so because this does not match continuity as they wrote it, since they're using subscripts to denote a particular velocity component.