For laminar incompressible flow through a smooth circular pipe of diameter $D$ and length $L$, which of the following statements is/are TRUE according to the Hagen-Poiseuille equation?

(A) The velocity profile across the pipe cross-section is parabolic

(B) The maximum centerline velocity is exactly twice the average flow velocity

(C) The pressure drop is inversely proportional to the fourth power of pipe diameter ($\Delta P \propto 1/D^4$)

(D) The Darcy friction factor $f$ is independent of Reynolds number

Answer Key:

A,B,C

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Conservation Equations of Mass, Momentum & Energy** under *Basic Fluid Mechanics* in GATE AE. 2. **Stage 2: Mathematical Derivation:** * Hagen-Poiseuille flow: $u(r) = u_{max}(1 - (r/R)^2)$ is parabolic (A is true). $u_{max} = 2 \bar{u}$ (B is true). $\Delta P = \frac{32 \mu v L}{D^2} = \frac{128 \mu Q L}{\pi D^4}$ (C is true). For laminar flow, $f = 64/Re$, so D is false. 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Assuming the Darcy friction factor is constant in laminar flow. *Reference: John D. Anderson Jr., Fundamentals of Aerodynamics (McGraw-Hill).*

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