How does the friction factor \(f\) vary along the flow direction in the fully developed region in (a) laminar flow and (b) turbulent flow?

Laminar flow is characterized by smooth and steady flow, where fluid particles move in parallel layers or "streamlines". On the other hand, turbulent flow is characterized by random and chaotic motion of fluid particles creating swirls and eddies.

The friction factor (f) is a dimensionless number that describes the resistance offered by the boundary walls of a pipe or a channel to the fluid flow taking place within it. In fully developed flow, the flow properties such as velocity and pressure remain constant in the direction of flow.

In laminar flow (Reynolds number Re < 2000), the friction factor is only dependent on the Reynolds number and is given by the Hagen-Poiseuille equation: \[f = \frac{16}{Re}\] where \(Re = \frac{VD}{\nu}\) is the Reynolds number, with V being the average velocity, D the diameter of the pipe, and \(\nu\) the kinematic viscosity of the fluid. Since the Hagen-Poiseuille equation shows that f is only dependent on the Reynolds number, and the flow is fully developed, the friction factor f remains constant along the flow direction in the case of laminar flow.

In turbulent flow (Reynolds number Re > 4000), the friction factor is dependent on both the Reynolds number and the relative roughness of the pipe (\(\frac{\epsilon}{D}\)), where \(\epsilon\) is the average roughness height of the internal surface of the pipe and D is the diameter of the pipe. Various empirical correlations, such as the Colebrook-White equation or the Moody diagram, can be used to determine the friction factor in turbulent flow. However, these correlations still show that f is dependent on both Reynolds number and relative roughness. In fully developed turbulent flow, fluid properties such as velocity and pressure remain constant in the direction of flow. Therefore, both Reynolds number and relative roughness remain constant along the flow direction. Consequently, the friction factor f also remains constant along the flow direction in the case of turbulent flow. In conclusion, in both laminar and turbulent flows, the friction factor (f) remains constant along the flow direction in the fully developed region.