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 ($\text{Re}$) and the Capillary number ($\text{Ca}$) based on the nozzle bulk velocity ($U_b$). At high $\text{Re}$, the jet contracts downstream of the nozzle, while at low $\text{Re}$, the jet expands, which is known as the Newtonian die-swell phenomenon. To study the influence of $\text{Re}$ on 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 ($P_{\text{exc}}$) and a contribution from the excess integral wall shear stress ($T_{\text{exc}}$) relative to, respectively, the pressure drop and integral wall shear stress for Poiseuille flow. The results of the numerical simulations for $\text{Re}\in [4.2:47.4]$ show that $P_{\text{exc}}$ is larger than zero for $\text{Re}\lesssim 6$, while negative for $\text{Re}\gtrsim 6$. Furthermore, $T_{\text{exc}}$ is always negative and is the dominant contribution to the overall viscous force for $\text{Re}\lesssim 12$. Normalized by the integral wall shear stress for Poiseuille flow, $T_{\text{exc}}$ appears to scale with ${\text{Re}}^{-1/3}$, implying that axial velocity perturbations near the nozzle wall scale with ${\text{Re}}^{-1/3}$ too when normalized by $U_b$. Self-similar behavior is also found for the axial velocity perturbation at the nozzle centerline, which scales with ${\text{Re}}^{-2/3}$ when normalized by $U_b$. The observed power-law scalings in $\text{Re}$ are 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 $\text{Re}$ is 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 $\text{Re}$, 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.

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• Interfacial flows
• Laminar flows