Question

The velocity profile for a Bingham plastic fluid flowing (under laminar conditions) in a pipe.

• A. Parabolic
• B. Flat
• C. Flat near the wall and parabolic in the middle
• D. Parabolic near the wall and flat in the middle

Hint:

Bingham Plastic Pipe Flow. Exhibits yield stress (\(\tau_y\)). Below \(\tau_y\), acts solid; above \(\tau_y\), flows. In pipe flow: Sheared (parabolic-like) region near wall (\(\tau>\tau_y\)), unsheared plug flow (flat profile) in center (\(\tau<\tau_y\)).

Answer 1

The Correct Option is D

A Bingham plastic is a type of non-Newtonian fluid that exhibits a yield stress (\(\tau_y\)).
It behaves like a rigid solid below the yield stress and flows like a viscous fluid once the shear stress exceeds the yield stress.
When a Bingham plastic flows under laminar conditions in a circular pipe due to a pressure gradient: - Near the pipe wall, the shear stress is highest (maximum at the wall, zero at the center).
If the wall shear stress exceeds the yield stress, the fluid near the wall will shear and flow.
The velocity profile in this sheared region is typically parabolic (similar to Newtonian flow but modified).
- In the central region of the pipe, the shear stress may fall below the yield stress (\(\tau<\tau_y\)).
In this region, the fluid does not shear internally but moves together as a solid plug (plug flow).
The velocity across this central plug is constant (flat velocity profile).
Therefore, the overall velocity profile consists of a parabolic (or near-parabolic) region near the wall where shearing occurs, and a flat, constant velocity region (plug flow) in the center where the stress is below the yield stress.
Option (4) correctly describes this profile.