Yes, you can use this approach to verify that the stream function $\psi$ satisfies the continuity equation for the given steady, incompressible, axisymmetric flow.

Steps to Verify

1. Continuity Equation for Axisymmetric Flow: The continuity equation for steady, incompressible, axisymmetric flow can be expressed as: $\frac{1}{r} \frac{\partial (r u_r)}{\partial r} + \frac{\partial u_z}{\partial z} = 0$

2. Substituting the Stream Function: Given: $u_r = -\frac{1}{r} \frac{\partial \psi}{\partial z}, \quad u_z = \frac{1}{r} \frac{\partial \psi}{\partial r}$

   We first compute $r u_r$: $r u_r = -\frac{\partial \psi}{\partial z}$ Then differentiate with respect to $r$: $\frac{\partial (r u_r)}{\partial r} = -\frac{\partial^2 \psi}{\partial z \partial r}$

   Next, we compute $\frac{\partial u_z}{\partial z}$: $\frac{\partial u_z}{\partial z} = \frac{1}{r} \frac{\partial^2 \psi}{\partial r \partial z}$

3. Substituting Back into the Continuity Equation: Plugging these into the continuity equation gives: $\frac{1}{r} \left(-\frac{\partial^2 \psi}{\partial z \partial r}\right) + \frac{1}{r} \frac{\partial^2 \psi}{\partial r \partial z} = 0$

   Since mixed partial derivatives are equal (assuming $\psi$ is sufficiently smooth): $-\frac{\partial^2 \psi}{\partial z \partial r} + \frac{\partial^2 \psi}{\partial r \partial z} = 0$

   This confirms that the continuity equation is satisfied.

Conditions/Restrictions on $\psi$

To ensure the validity of this approach, $\psi$ must satisfy the following conditions:

1. Smoothness: $\psi$ should be continuously differentiable in the domain of interest, allowing for the interchange of partial derivatives.
2. Boundary Conditions: Appropriate boundary conditions should be defined for the physical problem to ensure uniqueness of the solution.
3. Non-singularity: The stream function should not be singular in the flow domain, particularly as $r$ approaches zero to avoid undefined behavior.

Would you like more details or clarification on any part of this? Here are some related questions you might consider:

1. How do boundary conditions affect the solution for $\psi$?
2. Can you provide a specific example of a flow that uses this stream function?
3. What are the implications of axisymmetry on the flow field?
4. How does the incompressibility assumption simplify the analysis?
5. What techniques can be used to solve for $\psi$ in practice?

Tip: Always check the physical significance of your derived equations in the context of fluid dynamics.