Question

A fluid is in streamline flow across a horizontal pipe of variable area of cross section. For this which of the following statements is correct?

• A. The velocity is maximum at the narrowest part of the pipe and pressure is maximum at the widest part of the pipe.
• B. Velocity and pressure both are maximum at the narrowest part of the pipe.
• C. Velocity and pressure both are maximum at the widest part of the pipe.
• D. The velocity is minimum at the narrowest part of the pipe and the pressure is minimum at the widest part of the pipe.

Answer 1

The Correct Option is A

The question relates to the flow of fluid through a pipe with varying cross-sectional areas, described as a streamline or laminar flow. We need to understand the relationship between velocity and pressure at different sections of the pipe.

According to Bernoulli's principle, which applies to steady, incompressible, and non-viscous fluid flow, the total mechanical energy of the fluid remains constant. This is expressed as:

P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}

Where:

• P is the pressure energy per unit volume.
• \frac{1}{2} \rho v^2 is the kinetic energy per unit volume.
• \rho gh is the gravitational potential energy per unit volume.

In this problem, since the pipe is horizontal, the gravitational potential energy change can be ignored. Therefore, the equation simplifies to:

P + \frac{1}{2} \rho v^2 = \text{constant}

From this relationship, we see that if the velocity of the fluid increases (e.g., at a narrow section of the pipe), the pressure decreases so that the sum remains constant. Conversely, if the velocity decreases (e.g., at a wider section of the pipe), the pressure increases.

Thus, the correct explanation for the fluid flow across the horizontal pipe is:

• The velocity is maximum at the narrowest part of the pipe due to the conservation of mass and the requirement for flow rate to remain constant. The narrower section forces the fluid to flow faster.
• Pressure is maximum at the widest part of the pipe because the fluid slows down in areas of larger cross-section, causing the pressure to increase.

Therefore, the correct answer is:

The velocity is maximum at the narrowest part of the pipe and pressure is maximum at the widest part of the pipe.