The plasma–sheath matching problem

The plasma–sheath matching problem has attracted new interest during the last few years. It is complicated by the singular structure of the asymptotic (λD/L → 0) plasma and sheath solutions and by a coupling with the eigenvalue problem originating from the plasma balance of bounded plasmas. Due to these difficulties the existence of a matched asymptotic expression uniformly valid from the plasma core to the wall is widely questioned. The issue is clarified both analytically and numerically by the explicit construction of a matched asymptotic expression and comparison with exact solutions for the hydrodynamic plane Tonks–Langmuir problem. The approximations obtained by consistent matching show excellent agreement with numerical potential curves. The singularities of the asymptotic components are reflected by small discontinuities in the derivatives that vanish in the limit λD/L → 0.