Discussion
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?
Answer: A
The velocity is a two-dimensional function of \(r\) and \(z\).
In the converging section the flow area shrinks along the axis, so the axial velocity grows with \(z\) to keep the flow rate constant, and the walls turn the stream inward, giving a radial component that also varies with \(r\). Because the section is axisymmetric, nothing changes with the angle \(\theta\).
A function of \(r\) only describes fully developed flow in the straight part of the pipe, not the converging end. A function of \(z\) only would ignore the no-slip variation across the section. A dependence on \(\theta\) contradicts the axial symmetry of the geometry.
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