Let's perform the substitutions step by step in the given expression. The substitutions that need to be applied are:

1. $u = u' \cdot \frac{U}{\delta^{1/2}}$
2. $v = v' \cdot U$
3. $w = w' \cdot U$
4. $\kappa(s) = \frac{\delta(s)}{a}$

We will apply these substitutions to the original equation:

$\omega \cdot \frac{\partial u(s,r,\theta)}{\partial s} + \frac{\partial v(s,r,\theta)}{\partial r} + \frac{1}{r} \cdot \frac{\partial w(s,r,\theta)}{\partial \theta} + \frac{v(s,r,\theta)}{r} + \kappa(s) \cdot \omega \cdot (v(s,r,\theta) \cdot \sin(\theta) + w(s,r,\theta) \cdot \cos(\theta)) = 0$

Now, let's substitute:

1. Substitute for $u(s, r, \theta)$: $u(s,r,\theta) = u'(s,r,\theta) \cdot \frac{U}{\delta^{1/2}}$

   So, $\frac{\partial u(s,r,\theta)}{\partial s} = \frac{\partial}{\partial s} \left( u'(s,r,\theta) \cdot \frac{U}{\delta^{1/2}} \right)$ Using the product rule: $\frac{\partial u(s,r,\theta)}{\partial s} = \frac{U}{\delta^{1/2}} \cdot \frac{\partial u'(s,r,\theta)}{\partial s} + u'(s,r,\theta) \cdot \frac{\partial}{\partial s} \left( \frac{U}{\delta^{1/2}} \right)$

2. Substitute for $v(s, r, \theta)$: $v(s,r,\theta) = v'(s,r,\theta) \cdot U$

   So, $\frac{\partial v(s,r,\theta)}{\partial r} = U \cdot \frac{\partial v'(s,r,\theta)}{\partial r}$

3. Substitute for $w(s, r, \theta)$: $w(s,r,\theta) = w'(s,r,\theta) \cdot U$

   So, $\frac{\partial w(s,r,\theta)}{\partial \theta} = U \cdot \frac{\partial w'(s,r,\theta)}{\partial \theta}$

4. Substitute for $\kappa(s)$: $\kappa(s) = \frac{\delta(s)}{a}$

Finally, substitute these into the original equation:

$\omega \cdot \left( \frac{U}{\delta^{1/2}} \cdot \frac{\partial u'(s,r,\theta)}{\partial s} + u'(s,r,\theta) \cdot \frac{\partial}{\partial s} \left( \frac{U}{\delta^{1/2}} \right) \right) + U \cdot \frac{\partial v'(s,r,\theta)}{\partial r} + \frac{1}{r} \cdot U \cdot \frac{\partial w'(s,r,\theta)}{\partial \theta} + \frac{U \cdot v'(s,r,\theta)}{r}$ $+ \frac{\delta(s)}{a} \cdot \omega \cdot \left( U \cdot v'(s,r,\theta) \cdot \sin(\theta) + U \cdot w'(s,r,\theta) \cdot \cos(\theta) \right) = 0$

We can factor out the common terms of $U$:

U \cdot \left[ \omega \cdot \left( \frac{1}{\delta^{1/2}} \cdot \frac{\partial u'(s,r,\theta)}{\partial s} + u'(s,r,\theta) \cdot \frac{\partial}{\partial s} \left( \frac{1}{\delta^{1/2}} \right) \right) + \frac{\partial v'(s,r,\theta)}{\partial r} + \frac{1}{r} \cdot \frac{\partial w'(s,r,\theta)}{\partial \theta} + \frac{v'(s,r,\theta)}{r} + \frac{\delta(s)}{a} \cdot \omega \cdot \left( v'(s,r,\theta) \cdot \sin(\theta) + w'(s,r,\theta) \cdot \cos(\theta) \right) \right] = 0

This is the final expression after the substitution.

Would you like further details or a breakdown of specific parts?

Here are some follow-up questions you might explore:

1. How would the equation change if $\omega$ were not constant?
2. What happens to the equation in the case where $\delta(s)$ is constant?
3. How does the presence of $\frac{\delta(s)}{a}$ affect the physical interpretation of the problem?
4. Can this equation be simplified further under any specific conditions?
5. What role does each variable (such as $\omega$, $U$, and $a$) play in the overall system?

Tip: When substituting variables, keep track of dimensionless quantities. It helps in checking the consistency of units in complex systems.