Question

A fully developed laminar viscous flow through a circular tube has the ratio of maximum velocity to average velocity as

• A. \( 3.0 \)
• B. \( 2.5 \)
• C. \( 2.0 \)
• D. \( 1.5 \)

Hint:

For fully developed laminar flow in a circular pipe, the velocity profile is parabolic. A key characteristic is that the average velocity is half of the maximum velocity at the center of the pipe. Therefore, the ratio of maximum to average velocity is always 2.0.

Answer 1

The Correct Option is C

Step 1: Understand the velocity profile for fully developed laminar flow in a circular tube.
For fully developed laminar flow through a circular tube, the velocity profile is parabolic. The velocity is maximum at the center of the tube and zero at the tube walls due to the no-slip condition. The velocity profile \( u(r) \) at a radial distance \( r \) from the center of the pipe is given by: \[ u(r) = -\frac{1}{4\mu} \left(\frac{dP}{dx}\right) (R^2 - r^2) \] where:
\( \mu \) = dynamic viscosity of the fluid
\( \frac{dP}{dx} \) = pressure gradient along the pipe
\( R \) = radius of the pipe
\( r \) = radial distance from the center
Step 2: Determine the maximum velocity.
The maximum velocity, \( u_{max} \), occurs at the center of the pipe where \( r=0 \). \[ u_{max} = -\frac{1}{4\mu} \left(\frac{dP}{dx}\right) R^2 \]
Step 3: Determine the average velocity.
The average velocity, \( u_{avg} \), for fully developed laminar flow in a circular tube can be found by integrating the velocity profile over the cross-sectional area and dividing by the area, or by using the relationship from Hagen-Poiseuille equation.
It is a standard result that for a parabolic velocity profile, the average velocity is exactly half of the maximum velocity. \[ u_{avg} = \frac{1}{2} u_{max} \] To verify this, the average velocity can be calculated as: \[ u_{avg} = \frac{\int_0^R u(r) (2\pi r) dr}{\pi R^2} \] Substituting \( u(r) \) and performing the integration, we indeed find \( u_{avg} = \frac{1}{8\mu} \left(-\frac{dP}{dx}\right) R^2 \).
Comparing this to \( u_{max} = \frac{1}{4\mu} \left(-\frac{dP}{dx}\right) R^2 \), we get:
\( u_{avg} = \frac{1}{2} u_{max} \).
Step 4: Calculate the ratio of maximum velocity to average velocity.
\[ \frac{u_{max}}{u_{avg}} = \frac{u_{max}}{\frac{1}{2} u_{max}} = 2 \] So, the ratio of maximum velocity to average velocity for fully developed laminar viscous flow through a circular tube is 2.0. The final answer is $\boxed{\text{3}}$.