Assuming a magnetic field perpendicular to ion extraction axis x, a selfconsistent model of the plasma and of the ion beam, including all the structures in-between (presheath, sheath and meniscus), is written and numerically solved in two dimension x, y. A series expansion of the one dimensional solution is used to represent most of the plasma analytically and to precise boundary conditions. Five field variables are used to treat the positive ion case, namely the three fluid velocities V, the ion density profile n and the adimensional potential u. Collisional effect and their implication on equation kind and stability are discussed, as well as correction to fluid approximation.