We investigate the flow physics of non-equilibrium gases in interaction with solid particles in a microscale shock tube and the collection efficiency in the jet impingement on a permeable surface. One interesting application of flows in shock tubes at low pressures or micro-shock tubes is needle-free injection technology where drug particles are delivered by shock waves. To investigate such problems, a new two-fluid model system coupled with second-order Boltzmann–Curtiss-based constitutive relationships for modeling a non-equilibrium gas was developed. We were specifically interested in how rarefaction affects the complex wave patterns observed in dusty gas flows and the role of bulk viscosity in diatomic and polyatomic gases exposed to moving shocks. Simulation results demonstrated how significantly the bulk viscosity can affect the topology of the solution in the Sod shock tube problem. Counter-intuitive flow features were noted, resulting from bulk viscosity effects and the incapability of the first-order theory, even when Stokes' hypothesis was abandoned (i.e., the Navier–Fourier model). After detailed analyses in one-, two-, and three-dimensional space for simplified flow problems, a case was designed to represent a needle-free injection device. In addition, a new concept of “collection efficiency” was introduced that quantifies the efficiency of drug delivery in the two-phase jet impingement on the skin. We also derived a new “vorticity transport equation” that takes the bulk viscosity and multiphase effects into account. Based on the new equation, the time evolution of vorticity growth rates was analyzed for all the contributing terms in the equation.