This paper provides an approach for the deconfliction problem for a pair of constant altitude, constant speed unmanned aerial vehicles (UAVs) modeled as unicycles in the presence of static constraints. We use a combination of navigation and swirling functions to direct the unicycle vehicles along the planned trajectories while avoiding intervehicle collisions. The main feature of our contribution is proposing means of designing a deconfliction algorithm for unicycle vehicles that more closely capture the dynamics of constant speed UAVs as opposed to double integrator models. Specifically, we consider the issue of UAV turn-rate constraints and proceed to explore the selection of key algorithmic parameters in order to minimize undesirable trajectories and overshoots induced by the avoidance algorithm. The avoidance and convergence analysis of the proposed algorithm is then performed for two cooperative UAVs and simulation results are provided to support the viability of the proposed framework for more general mission scenarios.