Question

Fluid at a constant flow rate passes through a long, straight, cylindrical pipe that has an axisymmetric convergent section at the end. Which one of the following options correctly represents the velocity field in the converging section in cylindrical $(r,\theta,z)$ coordinates?

• A. Two-dimensional function of $r$ and $z$
• B. One-dimensional function of $r$
• C. Two-dimensional function of $r$ and $\theta$
• D. One-dimensional function of $z$

Hint:

For axisymmetric flows in pipes, velocity fields depend only on $r$ and $z$. The azimuthal variation ($\theta$) vanishes due to symmetry.

Answer 1

The Correct Option is A

Step 1: Understand the problem.
We have fluid flow in a cylindrical pipe with an axisymmetric converging section. Flow is incompressible and axisymmetric about the pipe axis.

Step 2: Coordinate system.
In cylindrical coordinates $(r,\theta,z)$: - $r$ = radial direction, - $\theta$ = azimuthal direction, - $z$ = axial direction. Because the pipe is axisymmetric, the velocity components do not depend on $\theta$.

Step 3: Dependency of velocity.
In the convergent section, the velocity varies along the axis ($z$) as well as radially ($r$). Thus velocity is a function of both $r$ and $z$. Final Answer:
\[ \boxed{\text{Two-dimensional function of $r$ and $z$}} \]