Question

In the entrance region of a pipe, the boundary layer grows and the inviscid core accelerates. This is accompanied by the fluid's pressure drop per unit length is

• A. lower compared to the fully developed region
• B. same compared to the fully developed region
• C. Lower or same compared to the fully developed region
• D. higher compared to the fully developed region

Hint:

The pressure gradient in a pipe is highest at the entry point and decreases gradually as the boundary layer develops, eventually reaching a constant, lower value in the fully developed region.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks about the characteristics of the pressure drop per unit length (pressure gradient) in the entrance (developing) region of a pipe flow in comparison with the fully developed region.

Step 2: Key Formula or Approach: This is a fluid mechanics question concerning boundary layer development. At the entrance, the boundary layer is thin, which results in high velocity gradients and high shear stress at the wall. Additionally, the inviscid central core accelerates to maintain mass conservation.

Step 3: Detailed Explanation:

• When a fluid enters a pipe, a viscous boundary layer begins to grow along the inner walls due to the no-slip condition.

• Near the entrance, the boundary layer is extremely thin, which creates a very steep velocity gradient near the wall (\(\frac{\partial u}{\partial y}\)).

• According to Newton's Law of Viscosity, a higher velocity gradient produces a significantly larger wall shear stress (\(\tau_{\text{w}}\)) compared to the fully developed region.

• Furthermore, as the boundary layer grows, it restricts the flow near the walls, forcing the central "inviscid core" of the fluid to accelerate to maintain a constant volumetric flow rate.

• This acceleration of the core requires an additional pressure drop to increase the fluid's kinetic energy.

• In contrast, in the fully developed region, the boundary layer thickness has stabilized, the velocity profile is constant, and there is no further core acceleration.

• Therefore, the total pressure drop per unit length (\(\frac{dP}{dx}\)) in the entrance region is much higher because it must overcome both the elevated frictional wall shear stress and provide the force needed to accelerate the flow core.

Step 4: Final Answer:
In the entrance region, the fluid's pressure drop per unit length is higher compared to the fully developed region.