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arxiv: 2604.07761 · v1 · submitted 2026-04-09 · ⚛️ physics.flu-dyn

Evidence of an inertialess Kapitza instability due to viscosity stratification

Shravya Gundavarapu , Darish Jeswin Dhas , Anubhab Roy This is my paper

Pith reviewed 2026-05-10 18:19 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Kapitza instabilityviscosity stratificationfalling filminertialess instabilityStokes flowPéclet numbersurface modelong-wave asymptotics
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0 comments X

The pith

Viscosity stratification destabilizes falling-film surfaces in the complete absence of inertia.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a continuous viscosity variation across a thin film can trigger a surface-mode instability even when the flow has zero inertia. This occurs because the viscosity field, evolving by advection and diffusion, alters the phase between perturbation vorticity and interface displacement so that the surface deformation is reinforced rather than restored. The unstable regime exists only inside a finite window of Péclet numbers; outside that window the flow remains stable. The mechanism is analogous to surfactant Marangoni instabilities but acts through bulk viscosity rather than interfacial tension.

Core claim

In the Stokes limit a linear viscosity profile across the film thickness, advected and diffused at finite Péclet number, renders the free-surface mode unstable. The growth rate increases and the band of unstable wavenumbers widens as the stratification parameter is raised, while the critical Péclet number for onset decreases. The instability is traced to a lagging vorticity perturbation that amplifies interface displacement.

What carries the argument

The phase shift of perturbation vorticity relative to interface displacement, induced by the evolving viscosity field so that the torque reinforces rather than opposes surface deformation.

If this is right

  • Stronger viscosity stratification lowers the critical Péclet number and widens the range of unstable wavenumbers.
  • Growth rates of the surface mode increase monotonically with the stratification strength inside the unstable window.
  • The same surface-mode destabilization appears in long-wave asymptotics and in full Chebyshev spectral solutions of the coupled streamfunction-viscosity eigenvalue problem.
  • The instability mechanism is structurally identical to the surfactant Marangoni case but mediated by bulk viscosity gradients instead of interfacial tension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same inertialess mechanism could operate in thermally driven coatings where temperature gradients create the viscosity profile.
  • In particle-laden films, shear-induced migration might self-sustain the linear stratification required for the instability.
  • Direct numerical simulations that allow the viscosity profile to become nonlinear would test whether the finite Péclet window survives beyond the linear assumption.

Load-bearing premise

The viscosity profile stays linear and is shaped only by advection and diffusion with no additional particle migration or nonlinear feedback.

What would settle it

A controlled experiment in a Stokes-regime falling film with an imposed linear viscosity gradient (via temperature or concentration) that shows no surface instability for any wavenumber when the Péclet number lies outside the predicted unstable window.

Figures

Figures reproduced from arXiv: 2604.07761 by Anubhab Roy, Darish Jeswin Dhas, Shravya Gundavarapu.

Figure 1
Figure 1. Figure 1: Schematic of a gravity-driven, viscosity-stratified falling film on an inclined plane with the base flow [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Variation of the Doppler-shifted wave speed [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dispersion relation showing the growth rate [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Contours of the growth rate Im(𝑐) for long-wave disturbances. (a) (𝛼, Pe) plane at 𝑘 = 0.001. (b) (𝑘, Pe) plane at 𝛼 = 0.3. 10 0 10 2 10 4 10 6 10 -3 10 -2 10 -1 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Neutral stability curves of the surface mode in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Perturbed viscosity field (color contours) overlaid with streamfunction perturbations (black contour lines) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Vorticity field (color contours) for 𝑘 = 0.001 corresponding to different combinations of stratification parameter 𝛼 and Péclet number Pe: (a) 𝛼 = 0.3, Pe = 100; (b) 𝛼 = 0.5, Pe = 100; (c) 𝛼 = 0.3, Pe = 1000; (d) 𝛼 = 0.5, Pe = 1000. 4. Mechanism of Instability We now trace the causal chain by which viscosity stratification destabilizes the surface mode, following the vorticity-interface phase framework int… view at source ↗
Figure 8
Figure 8. Figure 8: Variation of the vorticity-interface phase angle [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mechanism of the inertialess instability. (a) [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

The classical Kapitza instability of a gravity-driven falling film requires finite inertia to operate. We show that a surface-mode instability can arise in the complete absence of inertia when the film possesses a continuous viscosity stratification, a feature relevant to particle-laden films with shear-induced migration, thermally stratified coatings, and concentration-graded flows. The viscosity field, prescribed as a linear profile across the film thickness, evolves through an advection-diffusion equation characterized by a P$\'{e}$clet number. Using long-wave asymptotics and Chebyshev spectral computations, we solve the coupled eigenvalue problem for the perturbation streamfunction and viscosity fields and demonstrate that viscosity stratification destabilizes the surface mode in the zero-inertia (Stokes) limit. The instability is confined to a finite window of P$\'{e}$clet numbers. Increasing the stratification parameter lowers the critical P$\'{e}$clet number, broadens the range of unstable wavenumbers, and increases the growth rate. The instability mechanism is traced to the phase relationship between perturbation vorticity and the interface displacement: viscosity stratification shifts the vorticity to a lagging configuration, which reinforces interface deformation, following the framework of Hinch (1984). The mechanism bears a structural resemblance to the surfactant-driven Marangoni instability in creeping two-layer flows, extending this class of scalar-mediated, inertialess instabilities to bulk viscosity stratification.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that a continuous viscosity stratification in a gravity-driven falling film can destabilize the surface mode even in the complete absence of inertia (Stokes limit, Re=0). The base viscosity is taken as a linear profile across the film that evolves via an advection-diffusion equation controlled by a Péclet number Pe. Long-wave asymptotics together with Chebyshev spectral discretization are used to solve the coupled eigenvalue problem for the perturbation streamfunction and viscosity fields. The resulting instability exists only inside a finite window of Pe; increasing the stratification parameter lowers the critical Pe, widens the unstable wavenumber band, and raises the growth rate. The mechanism is traced to a lagging phase shift between perturbation vorticity and interface displacement, following the Hinch (1984) framework and bearing a structural resemblance to surfactant-driven Marangoni instabilities.

Significance. If the central result holds, the work identifies a new class of inertialess, scalar-mediated instabilities driven by bulk viscosity stratification rather than interfacial tension gradients. This is directly relevant to particle-laden films with shear-induced migration, thermally stratified coatings, and concentration-graded flows. The demonstration of a finite, tunable Pe window supplies falsifiable predictions, while the combination of long-wave asymptotics and independent spectral computations, together with an explicit link to the established Hinch vorticity-phase mechanism, strengthens the credibility of the claim.

major comments (2)
  1. [Base-state formulation (likely §2)] The linear base viscosity profile is central to the reported instability window. The manuscript should explicitly verify that this profile is a steady solution of the variable-viscosity Stokes equations plus advection-diffusion (without external forcing) and should test robustness when the base profile is allowed to become mildly nonlinear, as would occur under realistic particle migration feedback.
  2. [Linear stability formulation (likely §3)] The eigenvalue problem is solved by both long-wave asymptotics and Chebyshev spectral discretization, yet the precise boundary conditions imposed on the viscosity perturbation at the free surface and at the wall are not stated in the abstract or summary. These conditions are load-bearing for the phase-shift mechanism and must be given explicitly so that the reported finite-Pe window can be reproduced.
minor comments (1)
  1. The notation for the Péclet number (P´{e}clet) should be standardized throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and their constructive comments. We address each of the major comments below.

read point-by-point responses

  1. Referee: [Base-state formulation (likely §2)] The linear base viscosity profile is central to the reported instability window. The manuscript should explicitly verify that this profile is a steady solution of the variable-viscosity Stokes equations plus advection-diffusion (without external forcing) and should test robustness when the base profile is allowed to become mildly nonlinear, as would occur under realistic particle migration feedback.

    Authors: We agree with the referee that an explicit verification strengthens the presentation. In the base state, the velocity field is unidirectional (v_b = 0) with no streamwise dependence, causing the advection term to vanish. A linear viscosity profile has a vanishing second derivative, so the diffusion term is also zero. Therefore, the linear profile is a steady solution of the coupled system without external forcing. We will add this verification to §2 in the revised manuscript. With regard to testing robustness for mildly nonlinear base profiles, this is a valid point for applications involving particle migration. However, computing the nonlinear base state and repeating the stability analysis constitutes a significant additional effort beyond the scope of the present study. We will include a short discussion in the revised text explaining why the mechanism is expected to be robust for small deviations from linearity, while acknowledging that a comprehensive robustness study would be valuable future work. revision: partial

  2. Referee: [Linear stability formulation (likely §3)] The eigenvalue problem is solved by both long-wave asymptotics and Chebyshev spectral discretization, yet the precise boundary conditions imposed on the viscosity perturbation at the free surface and at the wall are not stated in the abstract or summary. These conditions are load-bearing for the phase-shift mechanism and must be given explicitly so that the reported finite-Pe window can be reproduced.

    Authors: We thank the referee for highlighting this. Although the boundary conditions are derived and used in the analysis, we agree they should be stated more explicitly for clarity and reproducibility. The viscosity perturbation satisfies a no-flux condition at the wall. At the free surface, the condition is obtained by linearizing the advection-diffusion equation about the interface position, resulting in a relation between the perturbation viscosity, its normal derivative, the base viscosity gradient, and the interface displacement. We will state these boundary conditions explicitly in the revised §3 and ensure they are clearly presented in the main text to allow reproduction of the finite-Pe window. We will also consider adding a brief mention in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via direct eigenvalue solution

full rationale

The central result is obtained by formulating and solving the coupled linear stability problem for the streamfunction and viscosity perturbations on a linear base viscosity profile that satisfies the steady advection-diffusion equation. Long-wave asymptotics and Chebyshev spectral discretization yield the growth rates as direct numerical outcomes of the eigenvalue problem in the Re=0 limit. The Hinch (1984) phase-shift interpretation is applied after the fact to explain the mechanism but is not used to derive or force the instability itself. No parameters are fitted to data and then relabeled as predictions, no self-citations bear the load of the existence claim, and the base state is a consistent solution of the governing equations rather than an ansatz smuggled in by citation. The reported finite Pe window therefore follows from the model equations without reduction to the inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the linear viscosity profile assumption and the validity of the long-wave Stokes equations; no new entities are postulated.

free parameters (2)
  • stratification parameter
    Strength of the linear viscosity variation across the film thickness; its value controls the instability window.
  • Péclet number
    Dimensionless ratio of advection to diffusion of the viscosity field; instability exists only inside a finite interval of this parameter.
axioms (2)
  • domain assumption Long-wave approximation is valid for the thin-film geometry
    Invoked to reduce the Navier-Stokes equations to a simpler system for asymptotics.
  • domain assumption Viscosity evolves according to a linear advection-diffusion equation with no additional source terms
    Prescribed in the abstract as the model for the stratified film.

pith-pipeline@v0.9.0 · 5550 in / 1435 out tokens · 59853 ms · 2026-05-10T18:19:55.710177+00:00 · methodology

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Reference graph

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