In the final term in your list, system, you are differentiating functions that do not depend on the variable r since the evaluation is done before the differentiation is carried out. You can fix this like this:

system = {2 \[CapitalOmega] sin[\[Phi]] \!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\(\[Omega][r, \[Phi]]\)\) == -(
      v[r, \[Phi]]/r) cos[\[Phi]], 1/r^2 \!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\((
\*SuperscriptBox[\(r\), \(2\)] \[Omega][r, \[Phi]])\)\) + 1/r  \!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Phi]\)]\(v[r, \[Phi]]\)\) == 0,
   \[Omega][R, \[Phi]] == \[Omega][R + \[CapitalDelta]R, \[Phi]] == 
    0, \[Omega][r, 0] == \[Omega][r, 2 \[Pi]], 
   v[r, 0] == v[r, 2 \[Pi]], (\!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\(v[r, \[Phi]]\)\) /. (r -> 
        R)) == 0, (\!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\(v[r, \[Phi]]\)\) /. (r -> 
        R + \[CapitalDelta]R)) == 0};

However, when this is done you get the following error message concerning boundary conditions and initial conditions which you will then need to address (I am not going to address that here since I am not sure of the details of your problem):

NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable. >>