Laminar and turbulent flow in ducts

In laminar flow, all liquid fluid elements are moving in parallel paths without crossing each other. 

The flow in a tube is an appropriate example to demonstrate the ditFerences of laminar and turbulent flow. According to Fig. 3.1-3 the force balance for a cyhndrical element with the radius r and the length dz can be written as 

According to the assumption that the velocity is zero at the pipe wall, 

The volumetric flow V can be expressed as a result of the integration of the local velocity w(r)  

By this the velocity profile in a pipe vahd for laminar flow is obtained  

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