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Help with similarity solution

Hello Math Help Forum! This is my first time posting.

I am currently working to understand a derivation that involves a similarity solution and there is one step that I am stuck on. In this step, one of the governing equations is transformed into a new equation based on the new variables. However, when I insert the new variables into the governing equation, I am unable to produce the transformed equation.

I have attached a .pdf file called "ProblemStatement" in which I explain the step that has me stuck.

I have also attached a .pdf file called "ProgressSoFar" in which I outline the attempt that I have made so far. I am hoping that I am close--that there is just something that I am not seeing--and that it will not be too much work for someone to pick up from where I have left off. However, I have made two separate files because I don't want to get anyone started off on the wrong foot if I have gone about this all wrong.

Thanks for your help!

Re: Help with similarity solution

I think it would benefit everyone if you wrote down the entire system of equations and let everyone know what are the independent and dependent variables. For example is $\rho$ constant or a function of ???. I'm guessing you have 3 unknowns, $\rho$, $u$ and $v$. You have a stream function which I'm guessing came from a continuity equation? and I'm sure there's another equation floating around (again, you have 3 unknown's).

I think by giving more details on what you've done will help those wanting to help you.

2 Attachment(s)

Re: Help with similarity solution

Thank you for your reply. I apologize, I did not realize that my problem was unclear--probably because I have spent so much time looking at it. I have tried to clarify in the attached file "UpdatedProblemStatement.pdf".

I am trying to obtain the new form of the conservation of momentum equation, which was originally written in terms of $x$, $r$, and $\rho$, in terms of $x$ and $\eta$ (where $\eta$ is a function of $r$ and $\rho$, so all of the expressions are still dependent upon $\rho$). My independent variables are $x$ and $r$. The values $\rho_0$ and $\mu_0$ are constants.

I have also attached a document, called "StreamFunctionsNewVariables.pdf," showing the steps that I used to obtain the new expressions for $u$ and $v$, in case that is helpful. Again, my best attempt at obtaining the conservation of momentum equation in terms of $x$ and $\eta$ is attached to my original post as "ProgressSoFar.pdf". Let me know if this is still not clear.

Thanks again!

Re: Help with similarity solution

You should have a third equation. Unknowns $u$, $v$ and $\rho$. You've given two equations, the continuity and momentum. Do you have a third equation to complete the system?

Re: Help with similarity solution

Thank you for your quick reply. I do not have any other equations. I understand that there are three unknowns, $u$, $v$, and $\rho$, and that I need a third equation to solve for $u$, $v$, and $\rho$ in terms of $r$ and $x$. However, at this stage, I do not need expressions that are solely dependent on $r$ and $x$. The expression for $u$ that I am trying to arrive at is dependent upon $C$ and $\eta$ (and therefore dependent upon $\rho$).

Thanks again for your help.

Re: Help with similarity solution

I have consulted with a friend who understands the problem but was also unable to produce the given expression for the conservation of momentum equation. He suggested that the other equation that you are looking for might be buried in the definition of $\eta$ and $\tilde{r}^2$. This was a point that I was trying to make in my last two replies:

$\tilde{r}^2 = 2\int_{0}^{r} \frac{\rho}{\rho_0}r\,dr$

$\eta = \frac{\tilde{r}}{x}$

Re: Help with similarity solution

Do you really want $r$ in the upper limit of integration, in the integral and the integration variable?

Re: Help with similarity solution

I've been able to work out your problem. I believe it's easiest to do it in steps.

Can you obtain

$\mu r u_r = C \dfrac{\mu_0^2}{\rho_0} \dfrac{\eta}{x}\, \left( \dfrac{f'}{\eta}\right)'$?