Wavemaker and endogeneity of gravitationally stretched weakly viscoelastic jets

TL;DR

The paper develops a unified one-dimensional, full-curvature slender-jet model for gravitationally stretched viscoelastic jets and analyzes global stability via a linearized eigenvalue framework on spatially developing base states. Using direct–adjoint diagnostics, it demonstrates that Newtonian jets exhibit a localized inlet-driven wavemaker, while moderate viscoelasticity broadens the receptive region downstream and introduces a polymeric endogeneity via elastic tension, yielding a coupled capillary–elastic mechanism that governs the jetting–dripping transition. The authors validate the Newtonian baseline against established results and show that increasing Deborah number $De$ generally reduces the critical Weber number $We_c$ and the onset frequency $\omega_i$, with the magnitude of this shift depending on $Bo$ and $\Gamma$. This work provides mechanistic insight into receptivity and sensitivity in stretched viscoelastic jets and offers a framework for future extensions to nozzle stresses, resolvent analyses, and additional physics such as surfactants or confinement.

Abstract

Highly stretched capillary jets produced by gravity are central to drop generation, micro-thread formation, and extensional-rheometry concepts. For Newtonian fluids, the transition from steady jetting to self-excited oscillations in a gravitationally stretched jet is predicted accurately by one-dimensional slender-jet equations that retain the exact interfacial curvature and admit a global eigenvalue analysis Rubio-Rubio et al. 2013. Separately, weakly viscoelastic jets governed by Oldroyd--B/Giesekus constitutive laws exhibit elastocapillary regimes and beads-on-a-string dynamics that are well captured by one-dimensional free-surface models Ardekani et al. 2010. Here we formulate a unified one-dimensional model for gravitationally stretched viscoelastic jets, combining full-curvature capillarity with a Giesekus stress closure, and we analyse its global linear stability on spatially developing base states. We first benchmark the Newtonian limit, reproducing marginal spectra and base-flow profiles, and then quantify how elasticity shifts the critical jetting--dripping boundary by tracking the leading global Hopf eigenpair across the rheological parametric space. For experimentally relevant moderate elasticity, characterised by order-unity Deborah numbers, polymeric tension modifies both the critical Weber number and the selected oscillation frequency, and endogeneity decompositions reveal that marginality results from a balance between capillary/kinematic contributions and an additional elastic-stress feedback pathway. To interpret and predict the onset mechanism, we compute wavemakers and receptivity/structural-sensitivity fields from direct--adjoint eigenfunctions, showing that viscoelasticity broadens the sensitivity region downstream while the adjoint remains strongly localized near the inlet, thereby identifying the near-nozzle region as the dominant receptive location.

Figures (4)

• Figure 1: a) Direct spectrum, where hollowed out squares mark the leading eigenvalues, b) spatial evolution of the base polymeric stresses, and c) base jet radius, for $\mathrm{We} = 0.003$, $\mathrm{Bo} = 1.81$, $\Gamma = 5.83$ and selected rheology parameters (see legend). Dark blue crosses in a) and c) are Newtonian results from RubioRubio2013, included for comparison.
• Figure 2: Marginally stable jet for $\mathrm{We} = 7\times 10^{-4}$, $\mathrm{Bo} = 1.8$, $\Gamma = 5.8$, $\beta = 0.5$, $\mathrm{De} = 5$. a) Direct spectrum, for which the leading mode (marked with a square) is $\omega \simeq -1.211\times 10^{-6} \pm 0.08984\mathrm{i}$. b) Comparison between time marching and the leading linear eigenmode dynamics. c) and d) Direct and adjoint normalised eigenfunctions.
• Figure 3: Wavemaker and endogeneity spatial distributions for three separate cases: $(\mathrm{We}, \mathrm{Bo}, \Gamma) = (3\times 10^{-3},1.81, 5.83)$ (Newtonian), $(\mathrm{We}, \mathrm{Bo}, \Gamma, \beta, \mathrm{De}) = (1.85 \times 10^{-3}, 1.8, 5.8, 0.6, 3)$, and $(\mathrm{We}, \mathrm{Bo}, \Gamma, \beta, \mathrm{De}) = (7 \times 10^{-4}, 1.8, 5.8, 0.5, 5)$. For endogeneity computations, solid lines indicate real part, and dashed lines indicate imaginary part.
• Figure 4: Marginal curves $\mathrm{We}_c(\mathrm{Bo})$ and corresponding frequencies $\omega_{i,c}(\mathrm{Bo})$ for $\Gamma = 0.84, 5.83$, and $18.5$.