ω*Partial[u\(40)s\(44)r\(44)θ\(41),s]+ Partial[v\(40)s\(44)r\(44)θ\(41),r] + Divide[1,r]*Partial[w\(40)s\(44)r\(44)θ\(41),θ]+ Divide[v\(40)s\(44)r\(44)θ\(41),r] + κ(s)*ω*(v(s,r,θ)*Sin[θ]+ w(s,r,θ)*Cos[θ])=0,u=u'*Divide[U,Power[δ,Divide[1,2]]],v=v'*U,w=w'*U,κ(s)=Divide[δ\(40)s\(41),a]kindly do the substitution

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.

ω*Partial[u(s,r,θ),s] + Partial[v(s,r,θ),r] + Divide[1,r]*Partial[w(s,r,θ),θ] + Divide[v(s,r,θ),r] + κ(s)*ω*(v(s,r,θ)*Sin[θ] + w(s,r,θ)*Cos[θ]) = 0, u = u'*Divide[U,δ^(1/2)], v = v'*U, w = w'*U, κ(s) = δ(s)/a. Kindly do the substitution.

This problem involves substituting variables into a partial differential equation containing trigonometric functions, focusing on physics-based mathematical modeling. Step-by-step breakdowns include key substitutions for u, v, w, and κ(s), applicable to undergraduate or graduate students studying advanced mathematics or physics.

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