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  Velocity distribution in smooth pipes  In the vicinity of a smooth boundary there exists a laminar sublayer. The flow in the laminar sublayer being laminar has a parabolic velocity distribution. Since the thickness of laminar sublayer ( δ ') is generally very small, the parabolic velocity distribution in the region may be approximated by a straight line without appreciable error.  So, in the zone of laminar sublayer, since the flow is laminar, the viscous stress predominates and the turbulent stresses tend to become zero. Therefore, in the laminar sublayer, the shear is  For linear velocity distribution within the laminar sublayer, 'du/dy' becomes 'u/y'. Ad: Fig: Velocity distribution for turbulent flow near a smooth boundary Furthermore, if it is assumed that in the laminar sublayer, i.e. up to y= δ ', ' τ ' remains constant and equal to ' τ o'. The shear stress at the pipe boundary is Which is the velocity distribution laminar number sublayer

 Hydrodynamically smooth and rough boundaries Fig: Definition of smooth and rough boundaries In general, a boundary with irregularities of large average height 'K' on its surface is considered to  be a rough boundary and one with smaller 'K' value is considered as a smooth boundary. ✓ However, for a proper classification of smooth and rough boundaries, the flow and fluid characteristics are required to be considered in addition to the boundary characteristics. ✓ As the flow outside the laminar sub-layer is turbulent, eddies of various sizes are present which try  to penetrate through the laminar sublayer. But, due to greater thickness of laminar sub-layer ,   eddies can't reach the surface irregularities and thus the boundary acts as a smooth boundary. Such  a boundary is termed as " Hydro-dynamically Smooth Boundary" ✓ With the increase in Reynold's no (Re), the thickness of the laminar sub-layer  decreases and it's  can even become much smaller