Flow inside a Circular Duct

Here is a simple and yet interesting situation. Most of us at one point or another have seen the flow of a fluid through a circular duct. This can be for example the flow of water through a hose. This very common situation has some interesting features. Suppose we have a flow of air coming at 0.1m/s towards a hose that is 1cm in diameter. As the flow enters the pipe, the velocity profiles starts to develop. Due to boundary conditions the velocity of the flow at the wall must be zero. 
From an experimental study using CFD software it can be observed from the picture above that the velocity is always a maximum at the center and decreases to zero at the wall. It can also be observed that for this case it took about ¼ of the length of the pipe for the velocity profile to develop. If one probes the velocity at different cross-sections along the pipe, it can then be observed how the velocity profiles changes from a slug flow velocity profile towards a parabolic fully developed velocity profile with a maximum right at the center of the pipe (see picture below). 
The distance along the duct that it takes for the velocity profile to develop is called the hydrodynamic entrance length. This effect becomes of great importance in micro/nano ducts where the entry length is relevant compared to the total length of the duct affecting the pressure drop and heat transfer characteristics significantly.

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