We study the dynamics of fragmentation for Newtonian and viscoelastic liquids in rotary atomization. In this common industrial process centripetal acceleration destabilizes the liquid torus that forms at the rim of a spinning cup or disk due to the Rayleigh-Taylor instability. The resulting ligaments leave the liquid torus with a remarkably repeatable spacing that scales inversely with the rotation rate. The fluid filaments then follow a well-defined geometrical path-line that is described by the involute of a circle. Knowing the geometry of this phenomenon we derive the detailed kinematics of this process and compare it with the experimental observations. We show that the ligaments elongate tangentially along the involute of the circle and thin radially as they separate from the cup. We use these kinematic conditions to develop an expression for the spatial variation of the filament deformation rate and show that it decays away from the spinning cup. Once the ligaments are sufficiently far from the cup, they are not stretched sufficiently fast to overcome the critical rate of capillary thinning and consequently undergo capillary-driven breakup forming droplets. We couple these kinematic considerations with the known properties of several Newtonian and viscoelastic test liquids to develop a quantitative understanding of this commercially important fragmentation process that can be compared in detail with experimental observations. We also investigate the resulting droplet size distributions and observe that the appearance of satellite droplets during the pinch-off process lead to the emergence of bidisperse droplet size distributions. These binary distributions are well described by the superposition of two separate $\mathrm{\Gamma }$ distributions that capture the physics of the disintegration process for the main and satellite droplets, respectively. Furthermore, as we consider more viscous Newtonian liquids or weakly viscoelastic test fluids, we show that changes in the liquid viscosity or viscoelasticity have a negligible effect on the average droplet size. However, incorporation of viscous/viscoelastic effects delays the thinning dynamics in the ligaments and thus results in broader droplet size distributions. The ratio of the primary to satellite droplet size increases monotonically with both viscosity and viscoelasticity. We develop a simple physical model that rationalizes the observed experimental trends and provides us a better understanding of the principal dynamical features of rotary fragmentation for both Newtonian and weakly viscoelastic liquids.

• Received 25 September 2019
• Accepted 14 January 2020

DOI:https://doi.org/10.1103/PhysRevFluids.5.033601