Singular behaviour of a rarefied gas on a planar boundary is clarified on the basis of the Boltzmann equation. The thermal transpiration between two parallel plates is taken as a specific example. First, the flow velocity is shown to behave like $x\ln x$ in the vicinity of the boundary, where $x$ is a distance from the boundary. This implies a logarithmic divergence of the flow velocity gradient as $x\rightarrow 0$ . Then, such a spatial singularity is shown to induce a similar singularity of the velocity distribution function (VDF) with respect to ${\zeta }_{n} $ on the boundary, where ${\zeta }_{n} $ is a normal component of the molecular velocity to the boundary. Moreover, the spatial singularity is shown to be quantitatively related to the discontinuity of the VDF on the boundary at ${\zeta }_{n} = 0$ . These macroscopic and microscopic singularities should be observed generally in a rarefied gas on a planar boundary.

Poiseuille and thermal transpiration flows of a highly rarefied gas are investigated on the basis of the linearized Boltzmann equation, with a special interest in the over-concentration of molecules on velocities parallel to the walls. An iterative approximation procedure with an explicit error estimate is presented, by which the structure of the over-concentration is clarified. A numerical computation on the basis of the procedure is performed for a hard-sphere molecular gas to construct a database that promptly gives the induced net mass flow for an arbitrary value of large Knudsen numbers. An asymptotic formula of the net mass flow is also presented for molecular models belonging to Grad's hard potential. Finally, the resemblance of the profiles between the heat flow of the Poiseuille flow and the flow velocity of the thermal transpiration is pointed out. The reason is also given.