Stability analysis of the flow in a coflowing device

We analyze the stability of the coflow configuration. The experiments and the global stability analysis show that the emitted jet always destabilizes before the tapering conical meniscus. This implies that the parameter conditions at which polydisperse dripping arises cannot be determined from the linear stability analysis of the steady jetting mode. Transient simulations show that the linear superposition of decaying eigenmodes triggered by an initial perturbation can lead to the jet breakup. The breakup process significantly depends on the initial perturbation. These results question the validity of the linear stability analysis as applied to the coflowing and other similar configurations.

💡 Research Summary

The paper presents a comprehensive investigation of the stability of a co‑flow microfluidic device in which an inner liquid is injected coaxially within an outer liquid, forming a slender jet that may either produce a steady stream of monodisperse droplets (jetting mode) or break up close to the tip into polydisperse droplets (dripping mode). The authors combine experiments, global linear stability analysis, and transient numerical simulations to assess whether the traditional linear eigenmode approach can predict the jetting‑to‑dripping transition.

In the formulation, the inner and outer fluids are characterized by density ratio ρ = 0.962, viscosity ratio μ = 36.1, Ohnesorge number Oh_i = 0.0264, and a radius ratio R = R_o/R_i = 6.67. The control parameters are the capillary number Ca (based on the outer flow) and the flow‑rate ratio Q = Q_i/Q_o. The governing equations are the axisymmetric incompressible Navier–Stokes equations, nondimensionalized with the inner capillary radius, the visco‑capillary velocity γ/μ_i, and the capillary pressure γ/R_i. Boundary conditions include parabolic inlet profiles, no‑slip at solid walls, free‑outflow at the downstream end, and continuity of velocity and stress at the deformable interface.

For the global linear stability analysis, a perfectly steady base flow is first computed. Small perturbations of the form δU(r,z) e^{−iωt} are introduced, leading to a generalized eigenvalue problem for the complex frequency ω = ω_r + iω_i. The problem is discretized using a boundary‑fitted spectral method: Chebyshev collocation points (21 in the inner domain, 85 in the outer domain) in the radial direction and fourth‑order finite differences (80 points) in the axial direction. The MATLAB eigs routine is used to extract the leading eigenvalues.

The experimental campaign reveals a clear jetting‑to‑dripping transition at a minimum flow‑rate ratio Q_min ≈ 1.7 × 10⁻⁴ for the chosen Ca. Below this value the conical meniscus oscillates and emits a broad size distribution of droplets, whereas above Q_min a long, apparently steady jet is observed. Importantly, the meniscus remains stable even when the jet downstream breaks up.

When the same parameter set is examined with the global linear analysis, the amplitude of the critical eigenmode at the meniscus, δF(z), is essentially zero for all examined domain lengths (down to L_n = 7.5). In other words, the linear eigenvalue problem predicts a stable meniscus for every Ca and Q, contradicting the experimental observation of meniscus oscillations at Q < Q_min. This discrepancy indicates that the assumption of a perfectly steady base flow is not valid for the co‑flow configuration near the transition.

To resolve the inconsistency, the authors perform transient simulations of the linearized equations. An initial surface perturbation is imposed as a localized Gaussian‑like deformation F(z,0) − F₀(z) = β e^{−α|z − z₀|} with β = 10⁻³ and α = 10. The simulations show that, despite all eigenmodes being linearly stable (ω_i < 0), the superposition of several non‑orthogonal decaying modes can produce a short‑time amplification of the perturbation energy. This non‑modal growth leads to jet breakup well before the meniscus experiences any noticeable deformation. The breakup length and timing are highly sensitive to the location z₀ and shape of the initial perturbation, confirming that the jetting‑to‑dripping transition is governed by transient dynamics rather than by the sign of the dominant eigenvalue.

Further nonlinear simulations (full Navier–Stokes without linearization) demonstrate that once the transient amplification exceeds a finite threshold, nonlinear effects rapidly accelerate the breakup, generating the polydisperse droplets observed experimentally. The authors therefore argue that the jetting‑to‑dripping transition in co‑flow devices is a convective instability of the jet coupled with strong non‑modal transient growth, while the conical meniscus remains absolutely stable due to its axial curvature.

The paper concludes that relying solely on global linear stability analysis is insufficient for predicting the onset of dripping in co‑flowing tip‑streaming devices. Design and control strategies must incorporate transient growth analysis and consider the sensitivity to initial disturbances. This insight is likely to be relevant for a broad class of microfluidic configurations where a slender jet emerges from a confined meniscus.