The gravitational settling of a small asymmetric particle in a two-dimensional periodic flow given by Stuart solution of the inviscid Euler equations, the so-called Stuart vortex flow field, is investigated with various buoyancy parameters. The numerical sheme used in this study couples the quasi-static Stokes equations for the fluid with the equilibrium equations for the particles. The force and torque expressions are used in the equations of motion of the particle, to predict the trajectory and orientation histories for a variety of initial orientations and length-to-diameter ratios. The equations of these dynamical systems are solved by using the fourth-order Runge-Kutta scheme.