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For laminar flow in a pipe or annulus of diameter $D$, there is a region from the circle of entry where velocity profile slowly develops until it is fully developed and no longer changes with the axial direction of the pipe.

The length from the start of the pipe till the end of varying velocity distribution is the entrance length $L_e$. Several empirical correlations show that $L_e = f(Re,D)$, as given by most fluid mechanics textbooks. However, none cover why this is a function of Reynold’s number and Diameter from a dimensional analysis and mass balance perspective. How can this expression be proven mathematically?

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  • $\begingroup$ I suggest that most relationships like this are proven empirically, not mathematically $\endgroup$
    – Tiger Guy
    Apr 4 at 4:38

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We had an experiment with a long tube and an axial fan where the flow profile was measured with a pitot tube across the diameter at various points along the length.

As the developing profile can be plotted and the fully developed profile as well, then the results can be compared with other fluids with dimensional analysis - something you @TheTenthBox can check out (Rayleigh and Buckingham also).

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  • $\begingroup$ My main question revolves around why entrance length is a function of $Re,L$ and how it can be proven. $\endgroup$
    – TheTenthBox
    Apr 4 at 10:50
  • $\begingroup$ @TheTenthBox because the Reynolds number defines the characteristic of the flow, as in Laminar, Critical or Turbulent. The velocity profile develops for each. The expressions come from work by Reynolds, Colebrooke White and many others. there is SO much to read from the names I have given you which will expand the knowledge. $\endgroup$
    – Solar Mike
    Apr 4 at 10:58
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It may have to do something with random nature of turbulence, which is connected to Reynolds number. You basically need to "destroy" the information about the entrance for the flow to be developed and turbulence may help you with that.

In case of laminar flow, the main mechanism may be different. You could take element of the pipe with the fluid far from the entrance and determine equilibrium velocity profile. Any different profile should gravitate to that one and rate of the change will be resisted by inertia and helped by viscous forces, the ratio of which is the Reynolds number.

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