  The inability of the Navier-Stokes-Fourier equations to capture rarefaction
effects motivates us to adopt the extended hydrodynamic equations. In the
present work, a hydrodynamic model, which consists of the conservation laws
closed with the recently propounded coupled constitutive relations (CCR), is
utilized. This model is referred to as the CCR model and is adequate for
describing moderately rarefied gas flows. A numerical framework based on the
method of fundamental solutions is developed to solve the CCR model for
rarefied gas flow problems in quasi two dimensions. To this end, the
fundamental solutions of the linearized CCR model are derived in two
dimensions. The significance of deriving the two-dimensional fundamental
solutions is that they cannot be deduced from their three-dimensional
counterparts that do exist in literature. As applications, the developed
numerical framework based on the derived fundamental solutions is used to
simulate (i) a rarefied gas flow between two coaxial cylinders with evaporating
walls and (ii) a temperature-driven rarefied gas flow between two non-coaxial
cylinders. The results for both problems have been validated against those
obtained with the other classical approaches. Through this, it is shown that
the method of fundamental solutions is an efficient tool for addressing quasi
two-dimensional multiphase microscale gas flow problems at a low computational
cost. Moreover, the findings also show that the CCR model solved with the
method of fundamental solutions is able to describe rarefaction effects, like
transpiration flows and thermal stress, generally well.
