Viscous Flow in Pipes.pdf

• Hence the velocity profile can be written as,( )u r( )u r22 2⎡ ⎤ ⎡ ⎤ΔpD ⎛2r ⎞ ⎛2r⎞= ⎢1− ⎜ ⎟ ⎥ = Vc⎢1−⎜ ⎟ ⎥16μl⎢⎣ ⎝ D ⎠ ⎥⎦ ⎢⎣ ⎝ D ⎠ ⎥⎦2ΔpDwhere Vc= is the centerline velocity16μlIn terms of wall shear stress,2τ D ⎡ ⎛1 r ⎞ ⎤= w ⎢ − ⎜ ⎟ ⎥ where R=D is the pipe radius.4μ⎢⎣⎝ R ⎠ ⎥⎦2• Volume flow rate2r=RR⎡⎛ r ⎞ ⎤Q = ∫udA = ∫ u ( r)2πrdr = 2πV 1r 0c ⎢ − ⎥rdr=∫0⎜ ⎟⎢⎣⎝ R ⎠ ⎥⎦π=2RVc2