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Fluid Mechanics: Fully Developed Flow

Fully developed flow is when the viscous effects due to the shear stress between the fluid particles and pipe wall create a fully developed velocity profile for a fluid as it travels through the length of a straight pipe. The velocity of the fluid for a fully developed flow will be at its fastest at the center line of the pipe (equation 1 laminar flow), and the velocity of the fluid at the walls of the pipe will be at its slowest. Due to the change of velocity across the velocity profile it is common to describe the fluid velocity as an average velocity.

Max velocity equation for fully developed flow (1)

Vc = Maximum Velocity

Q = Flow Rate

R = Pipe Radius

Fully Developed Flow

As mentioned earlier the viscous effects are caused by the shear stress between a fluid and the wall of a pipe. This shear stress is always present despite how smooth the pipe is. Also, the shear stress between the fluid particles moving past one another is a product of the wall shear stress and the distance from the wall. Refer to equation 2 to calculate the shear stress between fluid particles for laminar flow.

shear stress equation between fluid particles during laminar flow (2)

τ = Shear Stress

τw = Shear Stress at the Wall

r = radial distance from the center of the pipe to point of interest

D = Pipe Diameter

Due to the shear stress on the fluid particles as the fluid moves past the pipe wall a pressure drop will occur as can be seen in equation 3.

Pressure drop equation between fluid particles due to shear stress between fluid particles (3)

The viscous effects, pressure drop, and pipe length will affect the flow rate. To calculate the average flow rate, taking these into account, equation 4 would be used to calculate the flow rate for laminar flow.

Flow rate equation for fully developed laminar flow (4)

L = Pipe Length

μ = Dynamic Viscosity

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