Influence of collisions in a fluid model for the warm-ion sheath around a cylindrical Langmuir probe

A long cylindrical Langmuir probe immersed in a collisional plasma is modelled by using fluid equations in which the collision term is introduced assuming a constant ion mean free path. The physical magnitudes of the system are expanded in a perturbation series in the collision parameter, that is, the ratio between the Debye length and the ion mean free path, and the system is thus separated into zeroth-order and first-order perturbation systems. The ion temperature is arbitrary and a continuous solution from the plasma to the probe is obtained for the perturbation systems. Using this methodology, a continuous solution for the ion sheath around a cylindrical Langmuir probe is obtained for long ion mean free path, for any ion temperature with no need of two-scale study or asymptotic matching. It is found that the collisions and the ion temperature have opposite effects, and these effects are quantified. The solution for cold ions cannot be obtained directly using this method, but it can be recovered from the warm-ion solution. The methodology is usable for long cylindrical probes in which the scale of the probe is much higher than the scale of the sheath, so that the geometry is essentially cylindrical. The model is numerically solved to obtain the potential profile, the ion temperature profile, the ion density profile and the ion velocity profile, and the dependence of the sheath on the ion mean free path and the ion temperature is studied. The ion-current to probe-potential profiles for different probe radius are obtained. Finally, the Sonin plot is calculated, in order to use the solution in plasma diagnosis.