  We study a two phase flow with interactions of liquid and rarefied gas inside
the gas phase. The gas phase is modeled by the BGK model of the Boltzmann
equation. The liquid phase is modeled by the incompressible Navier-Stokes
equations. In this paper we have excluded the heat transfer between two phases.
The interface boundary conditions for the liquid and gas phases is presented.
The BGK model is solved by a semi-Lagrangian scheme with a meshfree
reconstruction since the droplet moves inside the gas phase and its interface
deforms with respect to time and space. Therefore, the similar meshfree
particle method is also suitable to solve the incompressible Navier-Stokes
equations for liquid phase. To validate the coupled solutions of the BGK model
and the incompressible Navier-Stokes equations, we have compared the results of
the BGK model and the incompressible Navier-Stokes equations, with those of the
Boltzmann and the incompressible Navier-Stokes equations, where the Boltzmann
equation is solved by a DSMC method. Results in $1D$ and $2D$ physical spaces
are presented.
