Optimal Function for Computational Mesh: Achieving Desired Shape and Flexibility

[Clausius2]

I am seeking a function $$r=r(\eta)$$ for a computational mesh. It has to have the same shape as the one shown in the figure attached. This figure has been achieved via polinomical approximation, but it doesn't give me many chances to change parameters of contractions. Maybe there is an analytical function which can behave as it is shown in the figure.

It has to check:

$$r(\eta=0)=0$$
$$r'(\eta=0)\sim 0$$
$$r(\eta=\eta_o)=1$$
$$r'(\eta=\eta_o)\sim 0$$
$$r''(\eta=\eta_o)\sim 0$$
$$r(\eta=\eta_{max})=r_{max}$$

Do you know some function (apart of a polinomical one) which behaves as the one shown in the figure?. The data is $$r_{max},\eta_{max}$$ and I must be able to vary successfully $$\eta_o$$.

Please Help!
Thanks!

 

[fresh_42]

Except from the start, where one could use a trig function, it looks as if $x^{2n+1}$ for some $n$ and appropriately shifted would do the job.