The laminar flow in the entrance region of pipes having a circular cross
section is investigated by the use of finite element solutions to the full
Navier-Stokes equations in cylindrical coordinates. Because these solutions
cover the complete three-dimensional geometry the usual development length is
no longer a very interesting parameter as explained herein. The velocities and
pressures are determined throughout the pipe and presented in graphic form.
Schiller, L. (1922) Die Entwicklung der laminaren
Geschwindigkeitsverteilung und ihre Bedeutung für Zahigkeitsmessungen, (Mit
einem Anhang über den Druckverlust turbulenter Stromung beim Eintritt in ein
Rohr.). Zeitschrift für angewandte Mathematik und Mechanik, 2, 96-106. http://dx.doi.org/10.1002/zamm.19220020203
Gerrad, J.H. and Taylor, L.A. (1977) Mathematical Model Representing Blood Flow Inarteries. Medical and Biological Engineering and Computing, 19, 611-617. http://dx.doi.org/10.1007/BF02457918
Hornbeck, R.W. (1964) Laminar Flow in the Entrance Region of a Pipe. Applied Scientific Research, Section A, 13, 224-236. http://dx.doi.org/10.1007/BF00382049
Durst, F., Ray, S., U’nsal, B. and Bayoumi, O.A. (2005) The Development Lengths of
Laminar Pipe
and Channel Flows. Transactions
of the ASME, 127, 1154-1160. http://dx.doi.org/10.1115/1.2063088
T'oppel, M.,
Rada, M. and Pasche, E. (2006) Hydrodynamic Flow Simulation in Pipe Networks Including Heat Transfer. Dresdener Wasserbauliche Mitteilungen, 32, 461.