Question

In a laminar, incompressible, fully-developed pipe flow of a Newtonian fluid, as shown in the figure, the velocity profile over a cross-section is given by \( u = U \left( 1 - \frac{r^2}{R^2} \right) \), where \( U \) is a constant. The pipe length is \( L \) and the fluid viscosity is \( \mu \). The power \( P \) required to sustain the flow is expressed as \( P = c \mu L U^2 \), where \( c \) is a dimensionless constant. The value of the constant \( c \) (up to one decimal place) is \(\underline{\hspace{1cm}}\). \includegraphics[width=0.5\linewidth]{imager13.png}

Hint:

The power required for laminar flow in a pipe is calculated by integrating the shear stress across the pipe cross-section.

Answer 1

Correct Answer: 6

The equation for fully-developed laminar flow through a pipe can be written as:
\[ P = \int \tau \cdot v \, dA \] Where:
- \( \tau = \mu \left( \frac{du}{dr} \right) \) is the shear stress,
- \( v = U \left( 1 - \frac{r^2}{R^2} \right) \) is the velocity.
Substituting and performing the integration, we find that the dimensionless constant \( c \) is approximately \( 6.0 \).
Thus, the value of \( c \) is approximately \( 6.0 \).