  A system of fluid-dynamic-type equations and their boundary conditions
derived from a system of the Boltzmann equation is of great importance in
kinetic theory when we are concerned with the motion of a slightly rarefied
gas. It offers an efficient alternative to solving the Boltzmann equation
directly and, more importantly, provides a clear picture of the flow structure
in the near-continuum regime. However, the applicability of the existing
slip-flow theory is limited to the case where both the boundary shape and the
kinetic boundary condition are smooth functions of the boundary coordinates,
which precludes, for example, the case where the kinetic boundary condition has
a jump discontinuity. In this paper, we discuss the motion of a slightly
rarefied gas caused by a discontinuous wall temperature in a simple two-surface
problem and illustrate how the existing theory can be extended. The discussion
is based on our recent paper [Taguchi and Tsuji, J. Fluid Mech. 897, A16
(2020)] supported by some preliminary numerical results for the newly
introduced kinetic boundary layer (the Knudsen zone), from which a source-sink
condition for the flow velocity is derived.
