The interplay of gas flow and depletion by plasma dissociation determines the spatial distribution of species and the deposition uniformity in a plasma source. Many plasma reactors use a gas showerhead and the design of the flow dynamics is a critical aspect of the reactor performance. In this paper, plasma deposition is considered as chemically reacting gas flow in an ideal showerhead reactor. The gas fluid flow is described by finite-gap stagnation-point creeping flow. The distribution of neutral species across the electrode gap is determined by diffusion equations, whereas their lateral transport is purely convective. Parameters relevant to large-area radio-frequency plasma deposition are particularly suitable for a complete analytical solution of the multi-component transport. A representative reaction scheme for hydrogen/silane plasma deposition is used for an analytical example from first principles which shows good agreement with numerical simulation. For a laterally uniform plasma, apart from edge effects, the deposition uniformity is limited only by the lateral uniformity of the pressure: if the electrode gap is very small in a large-area reactor, the pressure and deposition rate will be non-uniform even for a uniform showerhead. The deposition mass flux is self-consistently accounted for by the Stefan velocity for arbitrary levels of gas concentration and depletion, and its influence on streamlines and fluid velocity is shown.