Engr Help

Key Concepts: Flow of a fluid in a pipe may be laminar, transitional, or turbulent. A key

indicator is Reynolds Number which is the ratio of the inertial to viscous effects in the flow.

Laminar flow has generally a “smooth” parabolic velocity profile whereas turbulent flow is

“random” and has much more constant velocity profile except near the wall of the pipe

where there is a laminar boundary layer (viscous effects are important).

In a Nut Shell: For steady, fully-developed, laminar flow of a Newtonian fluid, the shear

stress, τ , is proportional to the velocity gradient, du//dr. The proportionality constant is the

dynamic viscosity, μ , of the fluid. Then the velocity profile is parabolic with zero on the

walls of the pipe (the no-slip condition) and maximum at the centerline of the pipe. The chart

below summarizes the key elements for steady, fully-developed, laminar pipe flow.

Steady, Laminar, Fully-Developed Flow

Linear Shear Stress Distribution

and

Parabolic Velocity Profile

Shear stress, τ = 2τ_w r/D

Velocity Profile, u( r ) = Vc [ 1 – (2r.D)² ]

Newtonian model for shear stress gives

τ = - μ du/dr

u( r ) = [τ_w D / (4μ)] [ 1 – (r/R)² ]
Q = [(πD⁴)/128 ] ∆P/μL