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 radius and length of the pipe, respectively.

• A. Delta P
• B. 1 / R squared
• C. 1 / mu
• D. 1 / L

Hint:

Remember the Hagen–Poiseuille law: velocity is maximum at the pipe centerline and depends on pressure difference, radius squared, viscosity, and pipe length.

Answer 1

The Correct Option is A, C, D

Step 1: Governing law.
For laminar flow of a Newtonian liquid in a circular pipe, the velocity profile is given by the Hagen–Poiseuille equation. The maximum velocity at the pipe centerline is: v max = (Delta P * R squared) / (4 * mu * L) Step 2: Identify proportionalities.
From the above formula, v max is directly proportional to Delta P.
It is directly proportional to R squared.
It is inversely proportional to viscosity mu.
It is inversely proportional to length L. Step 3: Check the given options.
(A) Delta P – correct, velocity increases with pressure difference.
(B) 1 / R squared – incorrect, velocity increases with R squared, not decreases.
(C) 1 / mu – correct, velocity decreases with viscosity.
(D) 1 / L – correct, velocity decreases with length of pipe. Step 4: Conclusion.
Therefore, the maximum velocity is proportional to (A), (C), and (D). \[ \boxed{\text{Correct: (A), (C), and (D)}} \]