### Abstract:

A method was sought to predict the flight paths and collisions for closely spaced ink droplets of various sizes as a design aid for ink-jet printing development. Computational fluid dynamics models of two rigid aligned spheres, as a proxy for ink droplets, were initiated in atmospheric pressure air at constant velocities of 6 m/s to 14 m/s, and drag coefficients were calculated from the forces which developed on the spheres. Simulations were run for the Reynolds number range of 5 to 25 for equal sized 20 μm and 26 μm sphere pairs at center-to-center separation distances of 1.5 to 19 sphere diameters. For separations of 5 diameters and less, where relative sphere size was found to influence the drag, simulations were also conducted using smaller trailing spheres of 0.6 and 0.8 times the diameter of the leading sphere. Empirical equations were found for the drag coefficients as a function of separation distance, which also incorporated a factor to account for unequal sized spheres. From these equations a calculator was developed to estimate the sphere trajectories, and the collision time and distance, for two spheres given the initial diameters, velocity and separation distance. The calculator predicts that in a 2 mm distance between an inkjet cartridge and the paper, equal sized spheres will collide when separated by no more than 3 diameters for the 14 m/s, 26 μm diameter case and up to 5 diameters for the 6 m/s, 20 μm diameter case. The collision distance, in meters, can be estimated for equal sized spheres from 1 x 10⁻⁵δ⁰ ̇²³d1 for separations of 5 diameters and less, where δ is the separation in diameters and d1 is the leading sphere diameter. Similarly, the collision time in seconds for equal sized spheres at δ = 5 or less, can be estimated from 1 x 10⁻⁵δ⁰ ̇¹⁸d₁/U, where U is the start velocity of the spheres. For unequal sized sphere pairs, the decreased drag on the smaller trailing sphere is counteracted by faster loss of momentum compared to the leading sphere. For cases where the ratio of drag coefficients for the trailing sphere divided by the leading sphere is greater than the ratio of their diameters, there is less drag improvement from the separation distance than there is relative deceleration increase from the differences in mass, and the spheres will not collide. At separations less than six diameters, drag reduction was found to be significant for both the leading and trailing spheres compared to a single sphere at the same Reynolds number. At the closest separation of 1.5 diameters, over 10 % drag reduction was found for the leading sphere and up to 50 % drag reduction was found for the trailing sphere. The primary contributor to the drag reduction on the trailing sphere was found to be the reduced velocity field created in front of the trailing sphere. The leading sphere has drag reduction caused by modification to its wake region. The average drag reduction for both spheres was also found to be within ± 3 % of the creeping flow analytical two sphere solution.