Then what would be the right way of telling Mathematica what I want? I tried to input my equations e.g. like this

eqn={D[u[x,y],x]+D[v[x,y],y]==0,eta[x,y]==Sqrt[ue/(nu*x)]*y,psi==Sqrt[ue*nu*x]*f
[eta[x,y]],u[x,y]==D[psi[x,y],y],v[x,y]==-D[psi[x,y],x]}
FullSimplify[eqn]

Thus specifying which functions I have, and what their parameters are (u is a function of x and y, f is a function of eta, eta is a function of x and y again, ...) and then using FullSimplify, or Eliminiate[eqn, {x,y}] to get rid of the variables I don't want, but nothing happened, so I guess this is the wrong approach.