text: For laminar (smooth) flows, it is a consequence of Poiseuille's law (which stems from an exact classical solution for the fluid flow) that
  where  is the Reynolds number
  and where  is the viscosity of the fluid and 
  is known as the kinematic viscosity. In this expression for Reynolds number, the characteristic length  is taken to be the hydraulic diameter of the pipe, which, for a cylindrical pipe flowing full, equals the inside diameter. In Figures 1 and 2 of friction factor versus Reynolds number, the regime  demonstrates laminar flow; the friction factor is well represented by the above equation.
  In effect, the friction loss in the laminar regime is more accurately characterized as being proportional to flow velocity, rather than proportional to the square of that velocity: one could regard the Darcy–Weisbach equation as not truly applicable in the laminar flow regime.
  In laminar flow, friction loss arises from the transfer of momentum from the fluid in the center of the flow to the pipe wall via the viscosity of the fluid; no vortices are present in the flow. Note that the friction loss is insensitive to the pipe roughness height : the flow velocity in the neighborhood of the pipe wall is zero.