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arxiv: 2509.05383 · v2 · pith:AMAWKH4Bnew · submitted 2025-09-05 · ❄️ cond-mat.soft · physics.flu-dyn

Self-similar rupture of thin films of power-law fluid

Michael C Dallaston , Steven A Kedda , Scott W McCue This is my paper

Pith reviewed 2026-05-21 23:23 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords thin film rupturepower-law fluidsself-similar solutionsbifurcation diagramsnaking bifurcationshear-thinning limitnon-Newtonian fluids
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The pith

Self-similar rupture solutions for power-law fluid films form a bifurcation diagram with snaking branches around n=1 and infinitely many solutions as n approaches zero.

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

Models of Newtonian thin liquid films are known to rupture in finite time in a self-similar way due to surface tension and van der Waals forces. This paper extends those similarity solutions to films of power-law fluids, where viscosity depends on shear rate according to an exponent n. The resulting family of solutions, organized in a bifurcation diagram by n, shows branches that merge through a snaking bifurcation near n=1. In the limit of strong shear-thinning as n goes to zero, there appear countably infinite solutions, each with an exponentially small inner region. Time-dependent simulations of the governing equations are drawn to the primary branch of these solutions.

Core claim

The authors compute similarity solutions for the rupture of thin films of power-law fluids and find that the bifurcation diagram indexed by the power-law exponent n has a highly nontrivial structure, with branches merging via a snaking bifurcation around n=1. They also find a countably infinite number of solutions in the extreme shear-thinning limit n to 0, where similarity solutions possess an exponentially small inner region. Numerical simulations show attraction to the single primary branch, and the asymptotic structure for small n is determined.

What carries the argument

The self-similar ansatz applied to the thin film evolution equations for power-law fluids, which generates a bifurcation diagram in the exponent n that exhibits snaking and infinite branches.

If this is right

  • The snaking bifurcation implies that for certain ranges of n near 1, multiple distinct self-similar rupture profiles exist.
  • As n approaches 0, the number of possible self-similar solutions becomes infinite, each distinguished by an exponentially small inner structure.
  • The time-dependent dynamics select the primary similarity solution over the others.
  • The small-n asymptotic analysis provides an explicit description of the inner region for shear-thinning fluids.

Where Pith is reading between the lines

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

  • This bifurcation structure could mean that slight variations in fluid properties near Newtonian behavior lead to qualitatively different rupture outcomes.
  • Similar snaking might appear in other non-Newtonian thin film models, such as those with yield stress.
  • The infinite solutions in the n to 0 limit suggest a possible accumulation of modes that could influence the approach to rupture in highly shear-thinning materials.

Load-bearing premise

The governing model equations remain valid all the way to the rupture singularity and the self-similar form accurately describes the late-time behavior without needing extra physical effects to regularize the singularity.

What would settle it

A direct comparison of the predicted similarity profiles against high-resolution numerical solutions of the time-dependent thin film equation for a power-law fluid at a value of n near 1, checking if secondary branches are ever selected.

Figures

Figures reproduced from arXiv: 2509.05383 by Michael C Dallaston, Scott W McCue, Steven A Kedda.

Figure 1
Figure 1. Figure 1: A film of power-law fluid of thickness h(x, t) wets a solid surface and flows due to surface tension and disjoining pressure applied to the fluid-air interface, while shear stress τxy vanishes at the interface. In the next section we summarise the formulation of the power-law thin-film equation with disjoining pressure and surface tension. In section 3 we describe our numerical method for solving this equa… view at source ↗
Figure 2
Figure 2. Figure 2: Results of simulations of the PDE (4) for (a, c, e) the shear-thinning case n = 0.5, and (b, d, f) the shear-thickening case. In (a,b), the profiles h(x, t) are plotted for times approaching rupture (h → 0 at a point). In (c,d), the profiles are scaled on to the similarity variables η, H from (7), and shown to asymptote to the numerically computed similarity solution (9) (dotted line) described in Section … view at source ↗
Figure 3
Figure 3. Figure 3: Bifurcation diagram of similarity solutions (solutions of (9)) for rupture of a power-law fluid depending on the power-law exponent n. Solid lines are those computed using the continuation procedure described in section 4.2, while the dashed lines are found using a rescaled method that converges for small n described in section 4.3. The dots indicate n → 0 asymptotic solutions found in section 5. Stars ind… view at source ↗
Figure 4
Figure 4. Figure 4: The (a) similarity profiles H and (b) fluxes Q of the four similarity solutions that exist at n = 0.5. The primary branch is denoted as solid, while the branches (in order of decreasing H(0) are depicted as dashed, dash-dotted, and dotted lines, respectively. Oscillations in the similarity profiles may be observed for this value of n. higher branches are seen to be oscillatory in η. While oscillations are … view at source ↗
Figure 5
Figure 5. Figure 5: The (a) similarity profiles H and (b) flux Q on the primary branch (H(0) largest when n = 1) for (i) n = 0.5, (ii) n = 0.1, (iii) n = 0.01. The dashed line indicates the numerically computed leading-order asymptotic solution for n → 0 found in section 5. plot solutions on the primary branch for decreasing n ∈ {0.5, 0.1, 0.01}, demonstrating the convergence of the solution to an asymptotic limit as n → 0. A… view at source ↗
Figure 6
Figure 6. Figure 6: Numerically calculated leading-order similarity profiles in the small-n limit. (a) the selection curve (H′ (0) as dependent on touchdown location η ∗ ), that demonstrates a countably infinite number of solutions to (16) (corresponding to points where H′ (0) = 0). (b) the first four similarity profiles H0 (in order of increasing η ∗ ), depicted as solid, dashed, dash-dotted, and dotted lines, respectively. … view at source ↗
read the original abstract

Models that describe Newtonian liquid films evolving due to the competing effects of surface tension and attractive intermolecular or van der Waals forces are known to rupture in finite time in a self-similar manner. We extend the computation of similarity solutions to non-Newtonian power-law liquid films. The resulting bifurcation diagram, indexed by power law exponent $n$, has a highly nontrivial structure with branches merging via a snaking bifurcation around $n=1$. A countably infinite number of solutions are also found in the extreme shear-thinning ($n\to 0$) limit, in which similarity solutions possess an exponentially small inner region. Numerical simulations of the time-dependent model are shown to be attracted to the single primary branch of similarity solutions. The asymptotic structure of solutions in the small $n$ limit is also determined.

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 / 2 minor

Summary. The paper extends similarity solutions for finite-time rupture of thin films driven by van der Waals forces from the Newtonian case to power-law fluids with exponent n. It presents a bifurcation diagram in n showing snaking bifurcations near n=1, a countably infinite family of solutions as n→0 with an exponentially small inner region, asymptotic analysis of that limit, and time-dependent simulations that are attracted to the primary branch.

Significance. If the numerical evidence and model assumptions hold, the work provides a nontrivial extension of self-similar rupture theory to non-Newtonian fluids, revealing complex bifurcation structure and infinite branches that could inform models of coating flows and shear-thinning materials. The combination of numerical bifurcation tracking, time-dependent verification, and small-n asymptotics is a clear strength.

major comments (2)
  1. The central claims rest on the power-law lubrication equations remaining uniformly valid down to h=0. For n<1 the effective viscosity vanishes at high shear rates, which risks violating the slow-variation assumption of the thin-film reduction before the self-similar regime is reached; the opposite occurs for n>1. This assumption is load-bearing for both the existence of the reported branches and the attraction of time-dependent solutions, yet the manuscript does not provide a quantitative check (e.g., via local Reynolds number or aspect-ratio estimates near the singularity).
  2. Numerical evidence for the snaking diagram and the countably infinite branches as n→0 is presented without reported convergence studies, mesh-refinement tests, or error bars on the bifurcation curves. Given the exponentially small inner region for small n and the sensitivity near the snaking point, these checks are necessary to confirm that the branch merging and multiplicity are not numerical artifacts.
minor comments (2)
  1. The abstract and introduction should explicitly state the range of n examined and the precise form of the power-law constitutive relation used in the lubrication equations.
  2. Figure captions for the bifurcation diagram should indicate the numerical method (e.g., continuation package or discretization scheme) and any continuation parameter used to trace the branches.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their positive assessment of the work and for highlighting important points regarding the validity of the model and the numerical evidence. We address each major comment below and plan to incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: The central claims rest on the power-law lubrication equations remaining uniformly valid down to h=0. For n<1 the effective viscosity vanishes at high shear rates, which risks violating the slow-variation assumption of the thin-film reduction before the self-similar regime is reached; the opposite occurs for n>1. This assumption is load-bearing for both the existence of the reported branches and the attraction of time-dependent solutions, yet the manuscript does not provide a quantitative check (e.g., via local Reynolds number or aspect-ratio estimates near the singularity).

    Authors: We agree that a quantitative validation of the lubrication approximation near the singularity would be valuable. In the revised manuscript, we will include order-of-magnitude estimates of the local Reynolds number and film aspect ratio based on the similarity solutions for representative values of n. This will help delineate the parameter regime where the model remains valid. We note that similar assumptions are made in the Newtonian case (n=1), which has been extensively studied, but we will make this explicit for the power-law extension. revision: yes

  2. Referee: Numerical evidence for the snaking diagram and the countably infinite branches as n→0 is presented without reported convergence studies, mesh-refinement tests, or error bars on the bifurcation curves. Given the exponentially small inner region for small n and the sensitivity near the snaking point, these checks are necessary to confirm that the branch merging and multiplicity are not numerical artifacts.

    Authors: We acknowledge the importance of demonstrating numerical convergence. The bifurcation computations were performed using a continuation method with adaptive spatial discretization, but we did not report detailed mesh refinement studies in the original submission. In the revision, we will add a section or appendix detailing convergence tests, including results from successively refined meshes and estimated errors on the bifurcation points, particularly for the snaking behavior near n=1 and the small-n limit. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical continuation of self-similar ODEs from stated PDEs

full rationale

The paper begins from the standard power-law lubrication equations, introduces the self-similar ansatz in the usual way to obtain a boundary-value ODE problem, and computes the bifurcation diagram in n by numerical continuation. Time-dependent PDE simulations are run separately to check attraction to the primary branch. No fitted parameters are relabeled as predictions, no load-bearing step reduces to a self-citation whose content is itself unverified, and the central claims (snaking structure, infinite branches as n→0) are outputs of solving the stated equations rather than inputs by construction. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claims rest on the standard thin-film lubrication approximation and the power-law constitutive relation.

axioms (1)
  • domain assumption The thin-film lubrication equations remain valid for power-law fluids near rupture.
    Invoked implicitly when the similarity solutions are computed from the model.

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Barenblatt.Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics

    G.I. Barenblatt.Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Cambridge University Press, 1996

    work page 1996

  2. [2]

    Becker, G

    J. Becker, G. Gr¨ un, R. Seemann, H. Mantz, K. Jacobs, K.R. Mecke, and R. Blossey. Complex dewetting scenarios captured by thin-film models.Nature materials, 2:59–63, 2003

    work page 2003

  3. [3]

    D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley. Wetting and spreading.Rev. Modern Phys., 81:739–805, 2009. Self-similar rupture of thin films of power-law fluid20

    work page 2009

  4. [4]

    Brewster, S.J

    M.E. Brewster, S.J. Chapman, A.D. Fitt, and C.P. Please. Asymptotics of slow flow of very small exponent power-law shear-thinning fluids in a wedge.Eur. J. Appl. Math., 6:559–571, 1995

    work page 1995

  5. [5]

    Chapman, M.C

    S.J. Chapman, M.C. Dallaston, S. Kalliadasis, P.H. Trinh, and T.P. Witelski. The role of exponential asymptotics and complext singularities in self-similarity, transitions, and branch merging of nonlinear dynamics.Physica D, 453:133802, 2023

    work page 2023

  6. [6]

    Chapman, A.D

    S.J. Chapman, A.D. Fitt, and C.P. Please. Extrusion of power-law shear-thinning fluids with small exponent.Int. J. Nonlin. Mech., 32:187–199, 1997

    work page 1997

  7. [7]

    Chapman, P.H

    S.J. Chapman, P.H. Trinh, and T.P. Witelski. Exponential asymptotics for thin film rupture. SIAM J. Appl. Math., 73:232–253, 2013

    work page 2013

  8. [8]

    Craster and O.K

    R.V. Craster and O.K. Matar. Dynamics and stability of thin liquid films.Rev. Modern Phys., 81:1131, 2009

    work page 2009

  9. [9]

    Dallaston

    M.C. Dallaston. Capillary levelling of thin liquid films of power-law rheology.ANZIAM J., 66:62–76, 2024

    work page 2024

  10. [10]

    Dallaston, M.A

    M.C. Dallaston, M.A. Fontelos, M.A. Herrada, J.M. Lopez-Herrera, and J. Eggers. Regular and complex singularities of the generalized thin film equation in two dimensions.J. Fluid Mech., 917:A20, 2021

    work page 2021

  11. [11]

    Dallaston, M.A

    M.C. Dallaston, M.A. Fontelos, D. Tseluiko, and S. Kalliadasis. Discrete self-similarity in interfacial hydrodynamics and the formation of iterated structures.Phys. Rev. Lett., 120:034505, 2018

    work page 2018

  12. [12]

    Dallaston, D

    M.C. Dallaston, D. Tseluiko, Z. Zheng, M.A. Fontelos, and S. Kalliadasis. Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows.Nonlinearity, 30:2647, 2017

    work page 2017

  13. [13]

    Doedel, A.R

    E.J. Doedel, A.R. Champneys, F. Dercole, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Oldeman, R.C. Paffenroth, B Sandstede, XJ Wang, and CH Zhang. Auto-07p.Montreal Concordia University,

    work page

  14. [14]

    http://indy.cs.concordia.ca/auto/

    work page

  15. [15]

    Eggers and M.A

    J. Eggers and M.A. Fontelos.Singularities: formation, structure, and propagation, volume 53. Cambridge University Press, 2015

    work page 2015

  16. [16]

    R. R. Eley. Applied rheology in the protective and decorative coatings industry.Rheology Reviews, 2005:173–240, 2005

    work page 2005

  17. [17]

    Erneux and S.H

    T. Erneux and S.H. Davis. Nonlinear rupture of free films.Phys Fluids A, 5:1117–1122, 1993

    work page 1993

  18. [18]

    Flitton and J.R

    J.C. Flitton and J.R. King. Surface-tension-driven dewetting of Newtonian and power-law fluids. J. Eng Math., 50:241–266, 2004

    work page 2004

  19. [19]

    Fowler and C

    A.C. Fowler and C. Johnson. Ice-sheet surging and ice-stream formation.Ann. Glaciol., 23, 1996

    work page 1996

  20. [20]

    Fowler, G

    A.C. Fowler, G. Kember, and S.G.B. O’Brien. Small exponent asymptotics.IMA J. Appl. Math., 64:23–38, 2000

    work page 2000

  21. [21]

    Garg, P.M

    V. Garg, P.M. Kamat, C.R. Anthony, S.S. Thete, and O.A. Basaran. Self-similar rupture of thin films of power-law fluids on a substrate.J. Fluid Mech., 826:455–483, 2017

    work page 2017

  22. [22]

    Garg, S.S

    V. Garg, S.S. Thete, C.R. Anthony, and O.A. Basaran. Local dynamics during thinning and rupture of liquid sheets of power-law fluids.J. Fluid Mech., 942:A15, 2022

    work page 2022

  23. [23]

    Gaver and J.B

    D.P. Gaver and J.B. Grotberg. The dynamics of a localized surfactant on a thin film.J. Fluid Mech., 213:127–148, 1990

    work page 1990

  24. [24]

    Ghatak, R

    A. Ghatak, R. Khanna, and A. Sharma. Dynamics and morphology of holes in dewetting of thin films.J. Colloid Interface Sci., 212:483–494, 1999

    work page 1999

  25. [25]

    Glasner and T.P

    K.B. Glasner and T.P. Witelski. Coarsening dynamics of dewetting films.Phys. Rev. E, 67:016302, 2003

    work page 2003

  26. [26]

    R.S.R. Gorla. Rupture of thin power-law liquid film on a cylinder.J. Appl. Mech., 68(2):294–297, 2001

    work page 2001

  27. [27]

    H.E. Huppert. The propagation of two-dimensional and axisymmetric viscous gravity currents Self-similar rupture of thin films of power-law fluid21 over a rigid horizontal surface.J. Fluid Mech., 121:43–58, 1982

    work page 1982

  28. [28]

    H.E. Huppert. Gravity currents: a personal perspective.J. Fluid Mech., 554:299–322, 2006

    work page 2006

  29. [29]

    Hwang and S-H

    C-C. Hwang and S-H. Chang. Rupture theory of thin power-law liquid film.J. Appl. Phys., 74:2965–2967, 1993

    work page 1993

  30. [30]

    Ida and M.J

    M.P. Ida and M.J. Miksis. Thin film rupture.Appl. Math. Lett., 9:35–40, 1996

    work page 1996

  31. [31]

    Iyer and D.W

    R.R. Iyer and D.W. Bousfield. The leveling of coating defects with shear thinning rheology. Chem. Eng. Sci., 51:4611–4617, 1996

    work page 1996

  32. [32]

    Jensen and J.B

    O.E. Jensen and J.B. Grotberg. Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture.J. Fluid Mech., 240:259–288, 1992

    work page 1992

  33. [33]

    Kedda.Mathematical Models of Self-similarity in Thin Film Evolution

    S.A. Kedda.Mathematical Models of Self-similarity in Thin Film Evolution. PhD thesis, Queensland University of Technology, 2025

    work page 2025

  34. [34]

    Kedda, M.C

    S.A. Kedda, M.C. Dallaston, and S.W. McCue. Long-time emergent dynamics of liquid films undergoing thermocapillary instability.Phys. Rev. E, 110:035104, 2024

    work page 2024

  35. [35]

    Kember, A.C

    G.C. Kember, A.C. Fowler, J.D. Evans, and S.B.G. O’Brien. Exponential asymptotics with a small exponent.Quart. Appl. Mech., 58:561–576, 2000

    work page 2000

  36. [36]

    J.R. King. Two generalisations of the thin film equation.Math. Comput. Model., 34:737–756, 2001

    work page 2001

  37. [37]

    Thin-film flow of a power-law liquid falling down an inclined plate.J

    S Miladinova, G Lebon, and E Toshev. Thin-film flow of a power-law liquid falling down an inclined plate.J. Non-Newton. Fluid, 122:69–78, 2004

    work page 2004

  38. [38]

    Moreno-Boza, A

    D. Moreno-Boza, A. Mart´ ınez-Calvo, and A. Sevilla. Stokes theory of thin-film rupture.Phys. Rev. Fluids, 5:014002, 2020

    work page 2020

  39. [39]

    T. G. Myers. Thin films with high surface tension.Siam Rev., 40:441–462, 1998

    work page 1998

  40. [40]

    T. G. Myers. Application of non-Newtonian models to thin film flow.Physical Review E, 72:066302, 2005

    work page 2005

  41. [41]

    Noble and J-P

    P. Noble and J-P. Vila. Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations.J. Fluid Mech., 735:29–60, 2013

    work page 2013

  42. [42]

    O’Brien, E.G

    S.B.G. O’Brien, E.G. Gath, A.C. Fowler, and G. Kember. Asymptotics with small exponent in a model for ice-sheet surging. InMath. Proc. R. Irish Acad., pages 67–80, 1998

    work page 1998

  43. [43]

    Oron, S.H

    A. Oron, S.H. Davis, and S.G. Bankoff. Long-scale evolution of thin liquid films.Rev. Mod. Phys., 69:931–980, 1997

    work page 1997

  44. [44]

    Richardson and J.R

    G. Richardson and J.R. King. The Saffman–Taylor problem for an extremely shear-thinning fluid. Quart. J. Mech. Appl. Math., 60:161–200, 2007

    work page 2007

  45. [45]

    Richardson and JR King

    G. Richardson and JR King. The Hele–Shaw injection problem for an extremely shear-thinning fluid.Eur. J. Appl. Math., 26:563–594, 2015

    work page 2015

  46. [46]

    Ruckenstein and R.K

    E. Ruckenstein and R.K. Jain. Spontaneous rupture of thin liquid films. InWetting Theory, pages 588–603. CRC press, 2018

    work page 2018

  47. [47]

    Shklyaev, A.A

    S. Shklyaev, A.A. Alabuzhev, and M. Khenner. Long-wave Marangoni convection in a thin film heated from below.Phys. Rev. E, 85:016328, 2012

    work page 2012

  48. [48]

    Shklyaev, A.V

    S. Shklyaev, A.V. Straube, and A. Pikovsky. Superexponential droplet fractalization as a hierarchical formation of dissipative compactons.Phys. Rev. E., 82:020601, 2010

    work page 2010

  49. [49]

    Teletzke, D.H

    G.F. Teletzke, D.H. Ted, and L.E. Scriven. How liquids spread on solids.Chem. Eng. Comm., 55:41–82, 1987

    work page 1987

  50. [50]

    Thete, C

    S.S. Thete, C. Anthony, O.A. Basaran, and P. Doshi. Self-similar rupture of thin free films of power-law fluids.Phys. Rev. E, 92:023014, 2015

    work page 2015

  51. [51]

    Tseluiko, J

    D. Tseluiko, J. Baxter, and U. Thiele. A homotopy continuation approach for analysing finite-time singularities in thin liquid films.IMA J. Appl. Math., 78:762–776, 2013

    work page 2013

  52. [52]

    Witelski and A.J

    T.P. Witelski and A.J. Bernoff. Stability of self-similar solutions for van der Waals driven thin film rupture.Phys. Fluids, 11:2443–2445, 1999. Self-similar rupture of thin films of power-law fluid22

    work page 1999

  53. [53]

    Witelski and A.J

    T.P. Witelski and A.J. Bernoff. Dynamics of three-dimensional thin film rupture.Physica D, 147:155–176, 2000

    work page 2000

  54. [54]

    Zhang and J.R

    W.W. Zhang and J.R. Lister. Similarity solutions for van der Waals rupture of a thin film on a solid substrate.Phys. Fluids, 11:2454–2462, 1999

    work page 1999

  55. [55]

    Zhang, O.K

    Y.L. Zhang, O.K. Matar, and R.V. Craster. Analysis of tear film rupture: effect of non-Newtonian rheology.J. Colloid Interface Sci., 262:130–148, 2003

    work page 2003