Linearized Continuity (Fluids)

[member 428835]

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"?

Thanks for your help!

 

[zwierz]

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

 

[member 428835]

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

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

 

[zwierz]

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

 

[member 428835]

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).

 

[zwierz]

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

 

[member 428835]

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

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.