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:

In laminar Hagen–Poiseuille flow, $\Delta P \propto V$. In turbulent flows, the exponent $n$ is typically higher (close to 2).

Answer 1

The Correct Option is A

Step 1: Recall Hagen–Poiseuille law.
For fully developed laminar flow in a circular pipe: \[ Q = \frac{\pi R^4}{8\mu L} \Delta P, \] where $Q$ = volumetric flow rate, $R$ = radius, $\mu$ = dynamic viscosity, $L$ = length.

Step 2: Express in terms of average velocity.
Average velocity: \[ V = \frac{Q}{\pi R^2} = \frac{R^2}{8\mu L}\Delta P. \]

Step 3: Rearrange relation.
\[ \Delta P = \frac{8\mu L}{R^2} V. \] This shows that $\Delta P \propto V^1$.

Step 4: Identify $n$.
Comparing with $\Delta P = K V^n$, we get $n=1$. Final Answer:
\[ \boxed{n=1} \]