A Linearized Continuity (Fluids)

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The discussion revolves around the continuity equation for incompressible flow in cylindrical coordinates, specifically addressing a discrepancy between two authors regarding the linearized conservation of mass. One author presents the equation as ##\partial_ru_r + u_r/r + u_z = 0##, while the other uses ##\partial_ru_r + u_r/r + \partial_z u_z = 0##. The confusion arises from the use of ##u_z## instead of ##\partial_z u_z##, leading to questions about the meaning of "linearized" in this context. Participants note that dimensions must be consistent, and there is skepticism about the authors' interpretations. Ultimately, the discussion highlights the complexities and potential inconsistencies in physics textbook explanations.
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!
 
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Authors of physics textbooks ramble sometimes. You understand the equation by itself do not you? Then skip this comment and go on.
 
zwierz said:
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##.
 
O! now I see what the point is. This is strange indeed. Dimensions must be incomparable
 
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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).
 
perhaps in some books ##u_z=\frac{\partial u}{\partial z}##
 
zwierz said:
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.
 
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