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?