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Annular liquid jets: exact constraint absorption and finite-time loss of transversality

Francisco R. Villatoro

arXiv:2609.05457v1cs.CEmath.NA

TL;DR

The paper addresses the failure of differentiated free-end constraints in one-dimensional annular-jet models. It absorbs the algebraic constraint into the dependent variables, verifies the resulting Chebyshev solutions, and finds thickness-dependent convergence, pressure-only resonance, and finite-time loss of transversality.

  • Problem

    Existing formulations differentiate the algebraic convergence constraint, leaving it unenforced and making the steady length equation degenerate.

  • Method

    The paper absorbs the constraint identically, discretises the reduced system by Chebyshev collocation, and verifies unsteady solutions with off-grid residuals and an independent solver.

  • Results

    Finite-thickness convergence length decreases strictly with nozzle thickness ratio when the free end is transversal, forced response is resonant in pressure coefficient but not convergence length, and transversality fails in finite time.

  • Takeaways & Limitations

    The free boundary can be reduced to a single scalar only while its zero remains transversal; constraint absorption provides a geometrically convergent, a posteriori-verifiable formulation before that failure.

Abstract

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An annular liquid jet encloses a volume of gas and, when surface tension dominates inertia, closes on the symmetry axis at a finite distance. In the one-dimensional model of this configuration that distance is not a boundary condition but an algebraic constraint on the state at the free end, and the established schemes differentiate it in time and march the resulting equation for the domain length. We show that this index reduction leaves the constraint unenforced and, in the steady limit, degenerate, and we avoid it by absorbing the constraint identically in the choice of dependent variable. The reduced system carries no constraint and is discretised by Chebyshev collocation. Convergence is geometric, published steady data are recovered, and a residual evaluated off the collocation nodes verifies the unsteady solution a posteriori. Three results follow. For jets of finite thickness the convergence length decreases strictly with the thickness-to-radius ratio at the nozzle whenever the free end is transversal, which contradicts a published table and localises the discrepancy in the treatment of the free end. The forced response is resonant in the pressure coefficient but not in the convergence length. And under forcing the free end ceases to be transversal in finite time, so that the reduction of the free boundary to a single scalar fails, at parameter values published as periodic responses. In one of them the computed failure time coincides with a near-vertical segment of the published record, and coarse discretisations do not detect the failure at all.

I. INTRODUCTION

Annular liquid jets can close around an enclosed gas volume when surface tension dominates inertia, but their free-end convergence length is an algebraic constraint that existing formulations mishandle. This paper absorbs that constraint into the variables, verifies the resulting spectral solutions, and establishes consequences for thickness, forcing, and free-end transversality.

  • Motivation: Surface-tension-dominated annular sheets close on the symmetry axis at a finite convergence length, whereas inertia-dominated sheets remain open.The closed configuration is relevant to fusion-wall protection, chemical reactors, and coaxial swirl injectors.
  • Problem formulation: The convergence point is an algebraic free-end constraint, not a prescribed boundary condition, distinguishing annular jets from liquid curtains with imposed length.Existing treatments differentiate this relation and march the resulting equation for the domain length.
  • Approach: The paper absorbs the constraint by writing the mean radius as the distance to the free end times a new unknown, producing an unconstrained reduced system.The method removes the removable tip singularity analytically and uses Chebyshev collocation without projection, stabilisation, or extrapolated tip values.
  • Verification: Off-grid residuals and decay of spectral coefficients provide a posteriori verification beyond the equations enforced at collocation nodes.An independently written upwind finite-difference solver supplies an additional check.
  • Results: The paper establishes strict thickness monotonicity of convergence length, pressure-only resonance under forcing, and finite-time loss of free-end transversality.These results concern the model and the validity of reducing the free boundary to a single scalar.

B. Criticality of the steady system

The steady annular-jet system becomes singular when the liquid reaches the propagation speed of sinuous capillary waves. A compatibility condition is required for the singularity to be removable, linking annular criticality to transcritical behavior in planar sheets.

  • Criticality: The steady coefficient matrix has determinant We u − J, which vanishes at the nozzle when We = cos θ0.This is the annular counterpart of the transcritical condition in planar sheets.
  • Compatibility: At the critical surface, the singularity is removable only when the right-hand side lies in the coefficient matrix’s range.The resulting compatibility condition is the annular analogue of a regularity condition imposed at a critical station.
  • Relation to prior formulations: The apparent literature difference between cos θ0 and cos^2 θ0 arises from velocity conventions and a documented factor in the streamline formulation.Both forms describe the same critical condition after translating variables.

C. The tip relation as an algebraic constraint

After mapping the moving domain to a fixed interval, closure becomes an algebraic relation between the tip state and convergence length. Differentiating that relation produces a degenerate, drift-prone formulation, whereas absorbing it into the dependent variable enforces closure identically.

  • Tip constraint: Mapping η = z/L(t) places the convergence point at η = 1, where the tip relation imposes g(m(τ,1), R(τ,1)) ≡ 2R^2 − βm = 0.For a membrane, this reduces to R(τ,1) = 0.
  • Differentiated formulation: The differentiated tip relation forms an index-two differential-algebraic system with L as its single algebraic variable.The differentiated equation contains a denominator based on the spatial derivative of the constraint.
  • Failure mode: Marching the differentiated equation allows local consistency error to drift the state off the constraint manifold.In the steady limit it reduces to dL/dt = 0 for every L, so the discarded algebraic relation alone would determine the convergence length.
  • Constraint absorption: Absorbing the constraint makes R(τ,1) = 0 hold identically and removes the constraint from the reduced variables.The scalar free-boundary reduction remains valid only while the zero is transversal.
  • Reduced length equation: The resulting length equation is algebraically equivalent to the differentiated equation for β = 0 but avoids indeterminate ratios and extrapolated tip values.Its finite-time loss is tied to the later loss of transversality.

A. Reduced system and the removable singularity

The reduced equations are evaluated on Chebyshev nodes with the tip singularity removed analytically, while enclosed-volume integration is checked for aliasing. These choices yield a regular discretisation whose quadrature error is negligible relative to interpolation error.

  • Reduced system: The mapped transport velocity is a = (u − η dL/dt)/L, and the reduced equations use it after substituting the absorbed constraint.This expresses the dynamics on the fixed interval [0,1].
  • Removable singularity: The source singularity at η = 1 is removable because its numerator vanishes there, so the tip value is evaluated using an analytic l’Hôpital limit.A regular quotient is used instead of directly forming Jη/Rη where both terms may vanish.
  • Collocation: Chebyshev–Gauss–Lobatto values are differentiated with the associated matrix, while nozzle data are imposed algebraically and equations are collocated at the remaining N nodes.The discretisation yields 4N + 1 ordinary differential equations for the nodal values and L.
  • Enclosed volume: 2.1 × 10^-8, 2.4 × 10^-13, and round-off are the relative differences between Clenshaw–Curtis and Gauss–Legendre volume quadrature at N = 8, 16, and 24.The comparison is for the steady state at Fr = 10, We = 50, Cpn = 0.5.
  • Quadrature check: Four to six orders of magnitude separate aliasing error from approximation error at every tested resolution.Changing quadrature alters the degeneracy time by only 2 × 10^-9, so aliasing is not the limiting factor.

C. Steady solver

The steady solver combines continuation with Chebyshev collocation and validates solutions using off-grid residuals and an independent finite-difference method. The resulting scheme converges geometrically, reproduces published steady data, and avoids the instability caused by enforcing the constraint inconsistently.

  • Steady solver: Continuation must adapt resolution near criticality because an unresolved nozzle scale stalls the parameter ladder.The coarse-grid continuation fails below a critical margin of about 2×10^-2; adapting resolution extends continuation to a margin of 10^-2.
  • A posteriori verification: Off-grid residuals assess the interpolating solution because collocation enforces the equations identically at the nodes.The residual is evaluated at 2001 interior points, excluding the nozzle boundary artefact.
  • Independent scheme: The independent finite-difference formulation becomes unstable when inconsistent tip stencils let R(τ, 1) drift.The tip slope moved from −1.70 to +2.4 before failure; imposing the constraint at the last node removes the drift.
  • Spectral convergence: The spectral steady solution reaches 6 × 10^-13 error at N = 24, after which Newton tolerance sets the error floor.The error falls by about twelvefold per two added modes before saturation.
  • Published steady data: Published convergence lengths agree within 1.2×10^-4 to 2.8×10^-3 in relative terms, consistent with the reference scheme's first-order discretisation.Three cases differing only in Cp max have the same convergence length, confirming Cpn as the steady pressure parameter.

B. A case with no steady solution

At a published parameter set, the model has no steady solution because nozzle compatibility fails on the critical surface. Approaching that surface creates a collapsing nozzle layer that coarse discretisations cannot resolve and can replace with a plausible but spurious finite convergence length.

  • Critical compatibility: The published case with We = 1 has no steady solution because compatibility requires Cpn = 1, not the specified Cpn = 0.At the nozzle, the coefficient matrix is singular and its right-hand side lies outside its range.
  • Critical limit: As the critical margin δ → 0+, the interface slope diverges and a nozzle layer of width O(δ) develops.The convergence length approaches 0.099752 even though the nozzle layer collapses.
  • Discretisation failure: A first-order uniform-grid scheme returns the finite value 0.3262 for We = 1.0650 instead of detecting the absent steady solution at We = 1.The unresolved layer produces a superficially plausible result, while the model has no steady solution at the published parameter set.
  • Critical-surface approach: The critical-surface approach is quantified by the margin We − cos θ0, and the collocation solver fails to converge at the smallest tested margin.Table III compares collocation at N = 64 with a Dormand–Prince march.

C. Unsteady verification

The finite-thickness analysis shows that convergence length decreases with nozzle thickness under transversality, contrary to published tabulations, while identifying the free-end treatment as the discrepancy’s source.

  • Monotonicity: The convergence length is strictly decreasing with β whenever the free-end zero is transversal.This held throughout the sampled parameter set; the largest sampled h′(L) was −2.6 × 10−3.
  • Small-thickness limit: L(0) − L(β) scales as c β1/2 near the membrane limit, so finite-thickness jets do not approach the membrane regularly.Measured exponents were 0.5034 and 0.5029 in the two parameter families.
  • Limitations: The monotonicity is a property of the posed model, although the thin-sheet approximation fails near closure for every β > 0.The result also assumes transversality, which can fail under forcing.
  • Published-data discrepancy: The published table predicts increasing convergence length with β, whereas the model predicts decreasing length and values about six per cent below the membrane value.The discrepancy cannot be a property of the equations and is localised in the free-end treatment.
  • Published-data discrepancy: The marched formulation differentiates the tip constraint but does not re-impose it, and its steady equation is satisfied by any L.At β = 0, extrapolated and exact tip values coincide to leading order, explaining agreement with membrane data.

VI. FORCED RESPONSE

The forced-response analysis normalises dynamics using a closed-loop quasi-static gain and validates the unsteady solver against independent steady calculations across multiple parameter sets.

  • A. Quasi-static gains: The relevant response is the closed-loop gain because changing L changes enclosed volume and therefore Cpn.At Fr = 10, We = 50, θ0 = 0 and Cp max = 1, the closed-loop dL/du0 is 7.035783 and the normalised gain is 0.560359.
  • A. Quasi-static gains: The pressure coefficient varies antiphase with nozzle velocity because increased velocity lengthens the jet, increases enclosed volume and lowers Cpn.The reported pressure derivative is dCpn/du0 = −0.582084.
  • A. Quasi-static gains: Low-frequency dynamic gains recover independently computed static gains within 0.72% for convergence length and 0.42% for pressure coefficient across six parameter sets.The two calculations share no code path, providing cross-validation of the unsteady machinery.

B. Frequency response

The forced response is resonant in the pressure coefficient but shows no maximum in convergence length over the accessible frequency range. Under forcing, the free end can lose transversality in finite time, invalidating the scalar free-boundary reduction.

  • Frequency response: 4.6-fold increase: the convergence-length gain rises monotonically between St = 0.005 and St = 0.405, with no maximum detected.The frequency range is bounded because integration fails at higher Strouhal number.
  • Frequency response: The pressure response peaks near the weakly damped mode’s amplitude-peak frequency, while its magnitude exceeds a single-mode estimate because of remaining spectral contributions and non-normality.The relevant eigenvalue is −0.31038 + 0.63253 i, with natural frequency 0.7046 and damping ratio ζ = 0.4405.
  • Frequency response: Phase lags increase with Strouhal number for both convergence length and pressure coefficient, after correcting the pressure phase for its quasi-static antiphase.The convergence-length lag rises from 6.11 to 7.37, while the pressure-coefficient lag rises from 3.38 to 4.68 over the reported range.
  • Finite-time loss of transversality: At A = 0.5 and Stg = 0.5, transversality fails at t∗ ≃ 10.7668, despite the case being published as a periodic response.The event time converges with resolution and the corresponding convergence length is L(t∗) = 12.9704.
  • Finite-time loss of transversality: At the event, the interface approaches the axis tangentially with R ∼ (L − z)^2, so the scalar L(t) parametrisation fails without implying a singularity of the underlying free-surface problem.The one-dimensional model cannot determine whether the near-cylindrical continuation pinches off or reopens.
  • Finite-time loss of transversality: N = 16 misses the event and returns a periodic solution, whereas finer resolutions collapse onto the same event time; an independent upwind scheme cross-checks the result.The coarse solution reaches t = 50 with minimum S(τ, 1) = 0.447.

B. Nozzle forcing and a published signature

Nozzle-forcing transients depend strongly on the start-up protocol, while finite-time loss of transversality also occurs at smaller forcing amplitudes. A published near-vertical convergence-length segment coincides with the computed degeneracy, whereas high-frequency mass collapse is identified as a discretisation failure rather than a second model degeneracy.

  • Nozzle forcing and a published signature: The same nozzle-forcing parameters produce t∗ ≈ 11.23 with abrupt switch-on and t∗ ≈ 31.81 with a two-cycle ramp.Thus the transient time depends strongly on how the solution is started.
  • Nozzle forcing and a published signature: At smaller forcing, transversality loss converges to t∗ = 22.88 ± 0.02 for abrupt switch-on and t∗ = 32.80728 with a two-cycle ramp.The protocols differ by 9.92, or 43% of the earlier event time.
  • Nozzle forcing and a published signature: At Stg = 0.5, the critical forcing amplitude lies in 0.45 < Acrit ≤ 0.50: A = 0.45 recovers, whereas A = 0.50 and A = 0.55 produce converged events.Coarse resolutions can falsely report an event near this threshold.
  • Nozzle forcing and a published signature: The published record’s near-vertical segment at t ≈ 22.5 coincides with the computed transversality loss and may be the degeneracy’s signature.The first-order scheme appears to resolve the event as a steep finite descent rather than a formulation failure.
  • Nozzle forcing and a published signature: At a = 0.02 and St = 0.50, mass remains bounded below by 0.42 while transversality reaches zero at t = 11.309.This identifies loss of transversality, rather than mass collapse, as the mechanism in that case.
  • Nozzle forcing and a published signature: At high Strouhal number, the apparent mass-collapse failure is reported as an underresolved oscillatory discretisation failure, not as a second model degeneracy.The authors do not identify which instability causes the numerical failure.

VIII. CONCLUSIONS

The paper replaces differentiated free-end constraints with an exact constraint-absorbing formulation and uses it to establish thickness monotonicity, selective resonance, and finite-time loss of transversality. The conclusions also identify broader applicability and open directions for pressure–volume stability.

  • VIII. CONCLUSIONS: Absorbing the algebraic convergence constraint removes constraint drift, steady-limit degeneracy, and extrapolated tip values simultaneously.The same formulation was independently reproduced in a finite-difference solver, supporting a formulation-level rather than scheme-specific diagnosis.
  • VIII. CONCLUSIONS: For transversal free ends, convergence length decreases strictly with nozzle thickness-to-radius ratio, contradicting a published table and localising the discrepancy at the free end.The membrane limit is described as a square-root singular limit rather than a regular one.
  • VIII. CONCLUSIONS: The forced response is resonant in pressure coefficient but not in convergence length over the accessible frequency range.The pressure gain peaks at the weakly damped semidiscrete mode’s frequency.
  • VIII. CONCLUSIONS: Finite-time loss of transversality occurs at parameters previously reported as periodic responses, while coarse discretisations can remove the event entirely.Event times are stable under threshold, tolerance, integrator, resolution, and an unrelated numerical scheme; one published record contains a matching near-vertical segment.
  • VIII. CONCLUSIONS: The same index-two constraint and two failure modes may arise in other slender free-surface flows ending at moving closure points.Examples named by the paper include water bells, converging swirling sheets, and coating curtains collected at finite stations.
  • VIII. CONCLUSIONS: A systematic stability study of enclosed-gas pressure–volume feedback, using Cp max as a bifurcation parameter, is identified as a needed next step.The proposed study builds on the formulation and verification procedure developed here.
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