Question

Consider the steady, incompressible, and fully developed laminar flow of a fluid through a circular pipe. Here, \( \Delta P \) is the pressure drop in the direction of the flow and \( V \) is the average axial velocity of the fluid at any cross-section. The relation between \( \Delta P \) and \( V \) is:
\[ \Delta P = K V^n \] Here, \( K \) and \( n \) are constants.
Which one of the following options is the correct value of \( n \)?

• A. 1
• B. 2
• C. 1.75
• D. 0.5

Hint:

For laminar flow in pipes, the pressure drop is directly proportional to the velocity, hence \( n = 1 \), making the equation \( \Delta P = K V^1 \) valid.

Answer 1

The Correct Option is A

The relation \( \Delta P = K V^n \) represents the pressure drop in a fully developed laminar flow through a circular pipe. In laminar flow, the relationship between the pressure drop \( \Delta P \) and the fluid velocity \( V \) follows a well-known pattern. Specifically, for fully developed laminar flow in pipes, the pressure drop is directly proportional to the velocity raised to the first power, meaning \( n = 1 \).
This relationship is derived from the Darcy-Weisbach equation for laminar flow, where the pressure drop depends linearly on the velocity. For laminar flow, the value of \( n \) is 1, indicating that the pressure drop increases directly as the velocity increases. This simple relationship holds because the resistance to flow in laminar flow is a linear function of the velocity.
Thus, the correct answer is (A) 1, which represents the case for fully developed laminar flow in a circular pipe.