Question

Conservation of mass for a steady axisymmetric flow field in the cylindrical $(r,z)$ coordinates is: \[ \frac{1}{r}\frac{\partial (rV_r)}{\partial r} + \frac{\partial V_z}{\partial z} = 0. \] Here, $V_r$ and $V_z$ are the radial and axial components of velocity, respectively. Which one of the following options is correct if $\psi$ is the stream function?

• A. $V_r = \dfrac{\partial \psi}{\partial z}, V_z = -\dfrac{1}{r}\dfrac{\partial \psi}{\partial r}$
• B. $V_r = \dfrac{1}{r}\dfrac{\partial \psi}{\partial z}, V_z = -\dfrac{1}{r}\dfrac{\partial \psi}{\partial r}$
• C. $V_r = \dfrac{1}{r}\dfrac{\partial \psi}{\partial z}, V_z = \dfrac{1}{r}\dfrac{\partial \psi}{\partial r}$
• D. $V_r = \dfrac{1}{r}\dfrac{\partial \psi}{\partial z}, V_z = -\dfrac{\partial \psi}{\partial r}$

Hint:

For axisymmetric flows, always remember the $\tfrac{1}{r}$ scaling when defining stream functions in cylindrical coordinates. This ensures continuity is automatically satisfied.

Answer 1

The Correct Option is B

Step 1: Recall the role of the stream function.
The stream function $\psi(r,z)$ is introduced so that the continuity equation is automatically satisfied. For axisymmetric flow in cylindrical coordinates (without swirl): \[ \frac{1}{r}\frac{\partial (rV_r)}{\partial r} + \frac{\partial V_z}{\partial z} = 0. \]

Step 2: Define velocity components in terms of $\psi$.
We define: \[ V_r = \frac{1}{r}\frac{\partial \psi}{\partial z}, V_z = -\frac{1}{r}\frac{\partial \psi}{\partial r}. \]

Step 3: Verify continuity equation.
Substitute into continuity: \[ \frac{1}{r}\frac{\partial (rV_r)}{\partial r} + \frac{\partial V_z}{\partial z} = \frac{1}{r}\frac{\partial}{\partial r}\left(r \cdot \frac{1}{r}\frac{\partial \psi}{\partial z}\right) + \frac{\partial}{\partial z}\left(-\frac{1}{r}\frac{\partial \psi}{\partial r}\right). \] Simplify: \[ = \frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \psi}{\partial z}\right) - \frac{1}{r}\frac{\partial^2 \psi}{\partial z \, \partial r}. \] Since mixed partial derivatives commute: \[ \frac{\partial}{\partial r}\left(\frac{\partial \psi}{\partial z}\right) = \frac{\partial}{\partial z}\left(\frac{\partial \psi}{\partial r}\right), \] the two terms cancel, giving: \[ = 0. \] Thus continuity is satisfied.

Step 4: Match with options.
- (A) misses the $\tfrac{1}{r}$ factor in $V_r$ — incorrect. - (B) matches perfectly. - (C) has the wrong sign for $V_z$. - (D) has $V_z$ without $\tfrac{1}{r}$ factor — incorrect. Final Answer:
\[ \boxed{V_r = \frac{1}{r}\frac{\partial \psi}{\partial z}, V_z = -\frac{1}{r}\frac{\partial \psi}{\partial r}} \]