To accurately model the inhaled particle motion, equations governing particle trajectories in carrier flow are solved together with the Navier-Stokes equations. Under the relatively dilute particle condition in the mixture, equations for two phases are coupled through the interface drag shown in the solid-phase momentum equations. The present study investigates bifurcation flow in the human central airway using the finite element method. In the gas phase, we employ the biquadratic streamline upwind Petrov-Galerkin finite element model to simulate the incompressible air flow. To solve the equations of motion for the inhaled particles, we apply another biquadratic streamline upwind finite element model. A feature common to two models applied to each phase of equations is that both of them provide nodally exact solutions to the convection-diffusion and the convection-reaction equations, which are prototype equations for the gasphase and the solid-phase equations, respectively. In two dimensions, both models have ability to introduce physically meaningful artificial damping terms solely in the streamline direction. With these terms added to the formulation, the discrete system is enhanced without compromising the numerical diffusion error. Tests on inspiratory problem were conducted, and the results are presented, with an emphasis on the discussion of particle motion.