Discussion
Conservation of mass for a steady axisymmetric flow field in the cylindrical \((r, z)\) coordinates is:
\[ \begin{aligned} &\frac{1}{r}\frac{\partial(rV_{r})}{\partial r} \\ &+ \frac{\partial V_{z}}{\partial z} = 0 \end{aligned} \]
Here, \(V_{r}\) and \(V_{z}\) are radial and axial components of velocity, respectively.
Which one of the following options is correct if \(\psi\) is the stream function?
Answer: B
The stream function is defined by \(V_{r} = \frac{1}{r}\frac{\partial \psi}{\partial z}\) and \(V_{z} = -\frac{1}{r}\frac{\partial \psi}{\partial r}\).
A stream function must satisfy the continuity equation identically. Substitute these definitions:
\[ \begin{aligned} \frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \psi}{\partial z}\right) + \frac{\partial}{\partial z}\left(-\frac{1}{r}\frac{\partial \psi}{\partial r}\right) \\ = \frac{1}{r}\psi_{rz} - \frac{1}{r}\psi_{zr} = 0 \end{aligned} \]
The mixed derivatives cancel, as required. Without the \(1/r\) in \(V_{r}\) (option A) the first term becomes \(\frac{1}{r}(\psi_{z} + r\psi_{rz})\), which does not cancel. With a plus sign in \(V_{z}\) the terms add instead of cancelling, and without the \(1/r\) in \(V_{z}\) the two terms differ by a factor \(r\).
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