Stability equations and thickness control for n-layer coaxial microfluidic bioprinting

Coaxial microfluidic printing of multi-layer structures faces fundamental challenges in interfacial stability and layer thickness control. To address this issue, we establish a unified theoretical framework for the co-flow of power-law fluids with an arbitrary number of layers. By extending classical two-phase co-flow models, we derive analytical predictions for the steady-state interface positions and construct a generalised Rayleigh-Plateau instability criterion using linear stability analysis and front propagation theory. Our study reveals that Newtonian fluids maintain stability by absorbing perturbations through a high-viscosity outer ‘damping sheath’, whereas shear-thinning fluids rely on a synergistic mechanism combining a high-viscosity inner ‘plug’ and a low-viscosity outer ‘lubricating layer’. Validated by numerical simulations and microfluidic experiments, the interface position prediction achieves a root-mean-square error below 0.026 mm/mm. Although the stability criterion is systematically conservative due to the lubrication approximation, it can reliably eliminate all parameter combinations that inevitably lead to dripping at extremely low computational cost. By coupling the spatial amplification effect of convective perturbations with material solidification kinetics, we construct a comprehensive stability regime map. Ultimately, this study provides a robust, quantitative fluid-mechanical foundation for the rational design of multi-interface coaxial systems, with direct application to microfluidic bioprinting.