How is mass flow rate within an annular region of a pipe taken to be an inexact differential?


I read it in Fluid Mechanics textbook by Yunus A. Cengel and John M. Cimbala.

The mass flow rate through the annulus is given to be inexact differential. Why is mass flow through the annulus not equal to $$(m2) ̇-(m1) ̇$$ Given any 2 radius r2 and r1?

Wont the mass flow rate in the annulus be equal to (mass flow in the area with radius r2)-(mass flow in the area with radius r1) ie m2-m1? Also then it goes on to say that the mass flow rate is exact.


  • $\begingroup$ I'm wondering if it is not related to the possibility of non-zero curl to the field: physics.stackexchange.com/a/75486/520 $\endgroup$ Nov 13, 2017 at 0:03
  • $\begingroup$ @GRANZER, Could you provide more context? $\endgroup$
    – stafusa
    Nov 13, 2017 at 0:52
  • $\begingroup$ @stafusa I read it in Fluid mechanics textbook by Cenge & Cimbala. This is the only book where I could find any resource on it. $\endgroup$
    Nov 13, 2017 at 2:00
  • 1
    $\begingroup$ Came across the same note on rates dm v. δm The way I had considered it was there is more than one way to get from 10 kg/hr to 20 kg/hr .. running a constant speed pump vs variable speed $\endgroup$
    – ARinLA
    Aug 11, 2018 at 19:08
  • 1
    $\begingroup$ Thus .. a path function where mass rate can vary at constant rate say 10 rpm/min vs . variable rate of 5 rpm/.5min then 15 rpm/.5min. Both paths yield the same mass flow rate $\endgroup$
    – ARinLA
    Aug 11, 2018 at 19:12


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