ANS:C - 1/A

For laminar flow of fluid through a long pipe, the pressure drop per unit length (ΔP/L) is proportional to 1/A2, where A is the cross-sectional area of the pipe. Explanation:

1. Hagen-Poiseuille Equation: For laminar flow through a long, straight pipe, the pressure drop per unit length (ΔP/L) can be described by the Hagen-Poiseuille equation:

• ΔP/L is the pressure drop per unit length,
• μ is the dynamic viscosity of the fluid,
• Q is the volumetric flow rate,
• D is the diameter of the pipe.

2. Cross-Sectional Area: The cross-sectional area A of the pipe is related to the diameter D by A=πD2/4.
3. Substitution: Substituting the expression for the cross-sectional area into the Hagen-Poiseuille equation, we get:

4. Proportionality: From the derived equation, we can see that the pressure drop per unit length (ΔP/L) is inversely proportional to the fourth power of the diameter (D4). This implies that as the diameter increases, the pressure drop per unit length decreases. Therefore, the pressure drop per unit length is proportional to 1/D4.
5. Conclusion: Hence, the pressure drop per unit length for laminar flow of fluid through a long pipe is proportional to 1/A2, where A is the cross-sectional area of the pipe. Since A=πD2/4, this can also be expressed as proportional to 1/D4.