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.
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.
Show Hint
- \( \Delta P \)
- \( \frac{1}{R^2} \)
- \( \frac{1}{\mu} \)
- \( \frac{1}{L} \)
The Correct Option is A, C, D
Solution and Explanation
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.
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