Desperately seeking a fitting function

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Clausius2

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I am seeking a function [tex]r=r(\eta)[/tex] 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:

[tex] r(\eta=0)=0[/tex]
[tex] r'(\eta=0)\sim 0[/tex]
[tex] r(\eta=\eta_o)=1[/tex]
[tex] r'(\eta=\eta_o)\sim 0[/tex]
[tex] r''(\eta=\eta_o)\sim 0[/tex]
[tex] r(\eta=\eta_{max})=r_{max}[/tex]

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

Please Help!
Thanks!
 

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