The laminar jet of a Newtonian liquid emanating from a long circular nozzle into a gaseous environment is studied. Provided that gravity effects can be neglected, the flow depends solely on the Reynolds number (Re) and the Capillary number (Ca) based on the nozzle bulk velocity (𝑈𝑏). At high Re, the jet contracts downstream of the nozzle, while at low Re, the jet expands, which is known as the Newtonian die-swell phenomenon. To study the influence of Reon the jet behavior, an integral momentum balance is theoretically derived for the entire flow, both in the nozzle and in the jet. A useful decomposition is made of the viscous force in the integral momentum balance into a contribution from the excess wall pressure drop (𝑃exc) and a contribution from the excess integral wall shear stress (𝑇exc) relative to, respectively, the pressure drop and integral wall shear stress for Poiseuille flow. The results of the numerical simulations for Re∈[4.2:47.4]show that 𝑃excis larger than zero for Re≲6, while negative for Re≳6. Furthermore, 𝑇excis always negative and is the dominant contribution to the overall viscous force for Re≲12. Normalized by the integral wall shear stress for Poiseuille flow, 𝑇excappears to scale with Re−1/3, implying that axial velocity perturbations near the nozzle wall scale with Re−1/3too when normalized by 𝑈𝑏. Self-similar behavior is also found for the axial velocity perturbation at the nozzle centerline, which scales with Re−2/3when normalized by 𝑈𝑏. The observed power-law scalings in Reare explained theoretically from the development of a boundary layer along the jet interface, originating from the stick-slip transition at the nozzle lip, and the constant integral mass flux along the jet. The boundary-layer flow is responsible for axial viscous stresses at the nozzle exit plane that result in a perturbation flow within the nozzle over an axial distance on the order of the nozzle radius. The die-swell phenomenon at low Reis caused by the radially outward flow at the nozzle exit and the accompanying compressive normal viscous stress that pushes the interface outwards till equilibrium is achieved. Self-similar behavior is also found for the perturbation flow far downstream of the nozzle with respect to the plug flow at equilibrium. With increasing Re, the exponential decay rate of the perturbation in the centerline velocity converges toward a prediction first made by Bohr for a related problem [N. Bohr, Philos. Trans. R. Soc. A 209, 281 (1909)], while the exponential decay rate of the perturbation in the jet radius is approximately twice as high. Finally, a discussion is given of the implications of our findings for applications in practice. ... Read more

Prilling is a spray solidification process of producing spherical beads, or prills, typically for fertilizers like urea and ammonium nitrate. The process involves molten material passing through perforated rotating containers, forming spiraling jets and breaking up into droplets due to a Rayleigh-Plateau instability by surface tension. These droplets undergo heat exchange with a cooling air stream and turn into solid prills. However, controlling prill size and shape remains challenging due to complex fluid dynamics, thermal effects, and non-Newtonian material properties. This research focuses on understanding jet breakup under the influence of thermal effects and body forces related to gravity and rotation (centrifugal and Coriolis forces). The overarching goal is to optimize the prilling process for a desired narrow prill size distribution by predicting and controlling the jet breakup process. A fundamental and holistic approach was taken based on a combination of experiments, stability analysis and various types of numerical simulations.
The study begins by exploring the dynamics of a spiraling liquid jet emitted from a rotating orifice. Using experiments, linear stability analysis and a nonlinear slender-jet model, the influence of the body forces from rotation and gravity on jet thinning and breakup is quantified. Imposed mechanical vibrations within the nozzle modulate the jet conditions at nozzle exit, enabling control over the jet breakup length and droplet size. The findings reveal that spiraling jets behave similarly to straight jets under gravity, where the body forces influence the thinning rate of the base jet and local base jet conditions control the growth rate of unstable jet perturbations. This insight opens pathways for more precise control mechanisms in various jet-based industrial applications.... ... Read more

We experimentally and numerically study the dynamics of a liquid jet issued from a rotating orifice, whose breakup is regulated by a vibrating piezo element. The helical trajectory of the spiralling jet yields fictitious forces varying along the jet whose longitudinal projections stretch and thin the jet, affecting the growth of perturbations. We show that by quantifying these fictitious forces, one can estimate the jet intact length and size distribution of drops formed at jet breakup. The presence of the locally varying fictitious forces may render high-frequency perturbations, that would otherwise be stable in the abscence of stretching, unstable, as observed similarly in the case of straight jets stretching under gravity. The perturbation amplitude then dictates how strong the perturbation is coupled to the jet compared with random noise that is inherently present in any experimental set-up. In the present study we exploit the slenderness of the jet to separate the calculation of the base flow and the growth of perturbations. The fictitious forces calculated from the base flow trajectory are then used in a nonlinear slender-jet model, which treats the spiralling jet as a quasi-straight jet with locally varying body forces. We show both experimentally and numerically that jet breakup characteristics (e.g. intact length and drop size distribution) can be controlled by finite-amplitude perturbations created by mechanically induced pressure modulations. Finally, we revisit the integrated net gain approach developed for straight jets under gravity and we provide simple analogous relations for spiralling jets. ... Read more

Using a slender-jet approach, we numerically investigate the control of jet breakup using temperature modulation at the nozzle with a specified frequency and amplitude. Our results show that temperature modulation does lead to instability through capillary and Marangoni stresses, providing control of the droplet formation in terms of intact length and resultant drop size distribution, which is otherwise irregular due to inevitable presence of background noise. For understanding the mechanisms underpinning the breakup of a thermally modulated jet in the presence of noise, it is useful to decompose the surface tension forces into a contribution from curvature-gradient forces and a contribution from surface tension-gradient forces, associated with axial variations in the jet curvature and the temperature-dependent surface tension coefficient, respectively. We show that in the limit of slow axial heat diffusion and slow cooling to the ambient, as considered here, the breakup of a thermally modulated jet is governed by the ratio of the surface tension-gradient force to the imposed random perturbation force at nozzle exit. This so-called “thermal modulation strength number” depends on the amplitude and frequency of the thermal modulation, the sensitivity of the surface tension coefficient to variations in temperature, the Weber number, and the strength of the Gaussian white noise added to the nozzle exit velocity. We show that the thermal modulation strength number governs the shift in breakup characteristics from forward to rear pinchoff for increasing modulation strength as well as the nature of the instability. When thermal modulation is weak, the surface tension-gradient forces act only as a trigger, and curvature-gradient forces soon take over and grow exponentially downstream from the jet due to inertio-capillary growth. When thermal modulation is strong, the surface tension-gradient forces not only act as a trigger, but remain significant until breakup. The thermal modulation strength number is thus useful to the design of thermal modulation in practical applications as a possible alternative to often-used mechanical excitation mechanisms to control jet breakup. ... Read more