Newtonian die-swell phenomenon revisited

Abstract

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.