Transforming Boundary Conditions in Similarity Transform

[member 428835]

Hi PF!

I am wondering how to bring boundary conditions through a similarity transform. The transform is as follows $$h(z,\tau) = \tau^a F(\eta)\\ \eta = C_2 z \tau^b\\L=\eta_{tip} C_2^{-1}\tau^{-b}$$ Before I continue, I have a pdf of a tex doc I made for this, but since I don't have the same syntax as these forums it would be easier if I could pm someone with the problem.

If not, please let me know, as I am totally fine with posting all of the work. I just want someone to check and see if my work is okay.

Thanks so much!

Actually, for ease I'll post my work for one here now so you get a better idea of what I'm talking about. The first boundary condition is $F^+(\eta^+)=0$ where $\eta^+=1$ so essentially $F^+(1)=0$. I should say $F^+\lambda^2=F$ and $\eta^+ \lambda = \eta$. Now we have $F^+(1)=0 \implies \lambda^{-2} F(1)=0 \implies F(1)=0$.

From here $\eta=1\implies z=C_2^{-1} \tau^{-b}$ and thus $h(C_2^{-1} \tau^{-b},\tau) = \tau^a F(1) = 0$.

 

[jvicens]

Hi there!

To bring boundary conditions through a similarity transform, you can use the following steps:

1. Start with the original boundary condition in terms of the transformed variables, in this case, $F^+(\eta^+)=0$.

2. Substitute the transformation equations $\eta = C_2 z \tau^b$ and $L=\eta_{tip} C_2^{-1}\tau^{-b}$ into the boundary condition.

3. Simplify the resulting expression using the properties of the transformation.

4. Use the fact that $F^+\lambda^2=F$ and $\eta^+ \lambda = \eta$ to rewrite the boundary condition in terms of the original variables.

5. Use the boundary condition for the original variables, in this case, $F(1)=0$, to obtain the corresponding boundary condition for the transformed variables.

6. Substitute the values of $\eta$ and $\tau$ at the boundary, in this case, $\eta=1$ and $\tau=\eta_{tip} C_2^{-1} L^{-b}$, into the transformed boundary condition to obtain the final form of the boundary condition.

I hope this helps and please feel free to post your work for further clarification. Good luck!