Question

Conservation of mass for a steady axisymmetric flow field in the cylindrical \( (r, z) \) coordinates is:
\[ \frac{1}{r} \frac{\partial}{\partial r} (r V_r) + \frac{\partial V_z}{\partial z} = 0 \] 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?

• A. \( V_r = \frac{\partial \psi}{\partial z} \) and \( V_z = -\frac{1}{r} \frac{\partial \psi}{\partial r} \)
• B. \( V_r = \frac{1}{r} \frac{\partial \psi}{\partial z} \) and \( V_z = -\frac{1}{r} \frac{\partial \psi}{\partial r} \)
• C. \( V_r = \frac{\partial \psi}{\partial z} \) and \( V_z = \frac{1}{r} \frac{\partial \psi}{\partial r} \)
• D. \( V_r = \frac{\partial \psi}{\partial z} \) and \( V_z = -\frac{\partial \psi}{\partial r} \)

Hint:

In axisymmetric flows, the stream function is used to relate the velocity components in cylindrical coordinates, ensuring conservation of mass.

Answer 1

The Correct Option is B

In a steady, axisymmetric flow, the velocity components \( V_r \) and \( V_z \) can be related to the stream function \( \psi \). The stream function satisfies the continuity equation for incompressible flow, and it can be used to express the velocity components in cylindrical coordinates.
The conservation of mass equation in cylindrical coordinates for steady, axisymmetric flow is given as: \[ \frac{1}{r} \frac{\partial}{\partial r} (r V_r) + \frac{\partial V_z}{\partial z} = 0 \] The stream function \( \psi \) for axisymmetric flow is related to the velocity components by the following relationships: \[ V_r = \frac{1}{r} \frac{\partial \psi}{\partial z}, V_z = -\frac{1}{r} \frac{\partial \psi}{\partial r} \] This ensures that the flow satisfies the conservation of mass. Therefore, the correct answer is (B). Thus, the correct answer is (B) \( V_r = \frac{1}{r} \frac{\partial \psi}{\partial z} \) and \( V_z = -\frac{1}{r} \frac{\partial \psi}{\partial r} \).