Question

Consider a fully developed, steady, one-dimensional, laminar flow of a Newtonian liquid through a pipe. The maximum velocity in the pipe is proportional to which of the following quantities? 
Given: \( \Delta P \) is the difference between the outlet and inlet pressure, \( \mu \) is the dynamic viscosity of the liquid, and \( R \) and \( L \) are the radius and length of the pipe, respectively.

• A. \( \Delta P \)
• B. \( \frac{1}{R^2} \)
• C. \( \frac{1}{\mu} \)
• D. \( \frac{1}{L} \)

Hint:

In laminar flow, the maximum velocity is proportional to the pressure difference \( \Delta P \), inversely proportional to viscosity \( \mu \), and inversely proportional to the pipe length \( L \). Additionally, the velocity is proportional to the square of the pipe radius \( R^2 \).

Answer 1

The Correct Option is A, C, D

In a fully developed, steady, laminar flow of a Newtonian fluid through a pipe, the maximum velocity (\(V_{{max}}\)) is governed by the following equation based on the Hagen-Poiseuille equation for laminar flow: \[ V_{{max}} = \frac{R^2}{4 \mu} \frac{\Delta P}{L} \] Where: - \( R \) is the radius of the pipe, - \( \mu \) is the dynamic viscosity of the fluid, - \( \Delta P \) is the pressure difference between the inlet and outlet of the pipe, - \( L \) is the length of the pipe. From this equation, we can observe the dependencies of the maximum velocity on the given quantities: 
Step 1: Analyzing each option 
- Option (A): \( \Delta P \) - Correct: The maximum velocity is directly proportional to the pressure difference \( \Delta P \). An increase in \( \Delta P \) will increase the maximum velocity, as indicated by the equation. 
- Option (B): \( \frac{1}{R^2} \) - Incorrect: The maximum velocity is proportional to \( R^2 \), not \( \frac{1}{R^2} \). A larger radius results in a higher maximum velocity, as seen from the equation. Thus, Option B is incorrect. 
- Option (C): \( \frac{1}{\mu} \) - Correct: The maximum velocity is inversely proportional to the dynamic viscosity \( \mu \). A lower viscosity results in a higher maximum velocity, as per the equation. 
- Option (D): \( \frac{1}{L} \) - Correct: The maximum velocity is inversely proportional to the length of the pipe \( L \). A longer pipe reduces the maximum velocity, as indicated in the equation. 
Step 2: Conclusion The correct answers are Option A, Option C, and Option D. These quantities are all directly involved in determining the maximum velocity in laminar flow through a pipe.