Hydrodynamics of high-pressure fluid flow entering and exiting a narrow pipe

In this work, we investigate the hydrodynamics of laminar flow entering and exiting an extremely narrow pipe under high-pressure conditions through numerical simulations and theoretical analysis. Two distinct flow regimes are identified: (1) a symmetric flow with mirror-like streamlines in upstream and downstream regions, and (2) an asymmetric flow characterized by vortex formation near the exiting point in the expansion plane, leading to a large recirculation zone downstream. Notably, the flow pattern is governed by the fully developed flow in the narrow pipe, which transitions between Hagen-Poiseuille flow and inviscid flow depending on hydrodynamic conditions. The transition between the two regimes is governed by two key dimensionless parameters, Re²EuR_c²/L_c² for Newtonian fluids and Re^2/nEu^2/n-1R_c^2/n/L_c^2/n for non-Newtonian fluids described by a power-law rheology, where Re is the Reynolds number, Eu is the Euler number, R_c and L_c denote the radius and length of the narrow pipe, and n denotes the rheological property of non-Newtonian fluids. These parameters quantify the competition between viscous, inertial, and pressure factors, leading to the transition from viscous-dominated to inviscid-dominated dynamics. Our theory reveals that both the scaled characteristic time for flow development and the final center velocity in the pipe follow power-law scaling relations, with a universal transition criterion Re^2/nEu^2/n-1R_c^2/n/L_c^2/n=2^1+2/n1+1/n². These findings provide new insights into flow behavior of Newtonian and non-Newtonian fluids in systems with abrupt geometric changes under high-pressure conditions, with implications for many industrial applications related to a flow through a narrow pipe.