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arxiv: 2605.00303 · v1 · submitted 2026-05-01 · ⚛️ physics.flu-dyn

Recognition: unknown

The rapidly advancing contact line Part-1: Navier slip and microscale inertial effects

Yash Kulkarni , Tomas Fullana , Stephane Popinet , Stephane Zaleski
Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:20 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords advancing contact linesNavier slipcurtain coatingmicroscale inertiavolume of fluidwetting failurecapillary numberwedge flow
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The pith

The observed microscale velocity acceleration near advancing contact lines is compatible with Navier slip models once local inertia is included.

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

The paper asks whether experiments showing accelerating interface velocity down to tens of micrometers near advancing contact lines in curtain coating invalidate the Navier slip model. It shows that they do not, because the observable microscale region lies outside the slip zone where velocity must vanish, and local Reynolds number of order one makes inertia important there. Two-phase Navier-Stokes simulations with adaptive mesh refinement and a Navier slip condition plus fixed contact angle reproduce the experimental non-monotonic critical capillary number versus global Reynolds number and the macroscopic contact angle at the inflection point. In the microscale region the interfacial velocity matches an inertially corrected wedge solution whose angle is set by the inflection-point value, with agreement improving for smaller slip lengths; at larger scales the interface follows the Benney solution.

Core claim

The experimentally reported acceleration of fluid velocity along the interface as the contact line is approached is reproduced by Navier slip models when the local flow outside the slip region is described by an inertially corrected wedge solution. Curtain coating at Ca ~ O(1) yields local Re ~ O(1) based on distances of tens of micrometers, so inertia governs the observable microscale region. Simulations confirm that qualitative microscale observations therefore do not rule out slip models for advancing contact lines.

What carries the argument

Navier slip boundary condition together with an inertially corrected wedge flow solution whose wedge angle is fixed by the macroscopic inflection-point contact angle.

If this is right

  • The critical capillary number for wetting failure varies non-monotonically with global Reynolds number.
  • The macroscopic contact angle at the inflection point changes with Reynolds number in agreement with experiment.
  • At scales larger than the microscale region the interface shape follows the Benney solution.
  • Agreement between the inertially corrected wedge solution and the simulated interface velocity improves as the slip length is reduced.

Where Pith is reading between the lines

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

  • Similar inertial corrections outside the slip zone may apply to other high-speed coating flows that involve advancing contact lines.
  • Varying the slip length in controlled experiments could reveal the transition from wedge-like to slip-dominated behavior.
  • Pure Stokes-flow models are likely insufficient for describing the observable microscale region when Ca is order one.
  • The framework could be extended to test predictions for unsteady contact-line motion or the onset of air entrainment.

Load-bearing premise

That a fixed microscopic contact angle plus the inertially corrected wedge solution remain accurate descriptors in the microscale region outside the slip zone, provided the slip length is sufficiently small relative to that region.

What would settle it

Direct measurement or simulation of interfacial velocity at scales comparable to or smaller than the slip length showing systematic deviation from the inertially corrected wedge solution while still obeying the global trends in critical capillary number.

Figures

Figures reproduced from arXiv: 2605.00303 by Stephane Popinet, Stephane Zaleski, Tomas Fullana, Yash Kulkarni.

Figure 1
Figure 1. Figure 1: Schematic representation of air entrainment and wetting failure, based on the view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the curtain coating configuration. The system parameters are view at source ↗
Figure 3
Figure 3. Figure 3: Visualisation of the slip length and numerical discretisation of the Navier slip view at source ↗
Figure 4
Figure 4. Figure 4: Zoom in at the contact line for the steady-state liquid curtain for view at source ↗
Figure 5
Figure 5. Figure 5: Contour plot showing the magnitude of the velocity field for figure 4. The view at source ↗
Figure 6
Figure 6. Figure 6: The velocity along the interface for Re = 20 and Ca = 0.7 for (a) 5 µm slip length and (b) various slip lengths as indicated in the top-right inset. The inset figures (inset in (a), top-left inset in (b)) represent the full-scale behaviour and the main plot is a zoomed-in version near the contact line. The vertical 20 µm line is the resolution of the experimental visualisation in Blake et al. (1999); Clark… view at source ↗
Figure 7
Figure 7. Figure 7: (a) The maximum value of the velocity along the interface view at source ↗
Figure 8
Figure 8. Figure 8: The numerical solution for IC-SFW for two different wedge angles. The contour view at source ↗
Figure 9
Figure 9. Figure 9: Comparison for the velocity along the interface for the numerically obtained view at source ↗
Figure 10
Figure 10. Figure 10: Schematic illustration of the Benney solution. (a) The polar-coordinate view at source ↗
Figure 11
Figure 11. Figure 11: (a) Flow field and interface profile (red) obtained for view at source ↗
Figure 12
Figure 12. Figure 12: Interface profile in r–θ coordinates to represent the Benney solution. The distance r is scaled by the coated film thickness hinf . The blue points correspond to the full curtain simulation with Re = 20 and Ca = 0.7 and slip length of 10 µm, which is 0.047hinf . The solid line represents θ = arq−1 with the value of q found from equation (3.19) in the branch q ∈ (3/2, 2] for Ca = 0.7. The dotted lines are … view at source ↗
Figure 13
Figure 13. Figure 13: The x–y interface profile for figure 12 for better visualisation of the Benney solution. The contact line is taken as the origin and (a) shows the Benney solution fit in x–y coordinates where the exponent q is based on Ca = 0.7, same as in figure 12. The coordinates are again scaled by hinf . (b) The interface profile in linear scales as it appears practically. The solid black line is the Benney solution.… view at source ↗
Figure 14
Figure 14. Figure 14: (a) Comparison of the stability window obtained in the current simulations view at source ↗
Figure 15
Figure 15. Figure 15: (a) Convergence study for the contact line position. view at source ↗
Figure 16
Figure 16. Figure 16: Successive zoom-in near the contact line in a steady-state simulation of view at source ↗
Figure 17
Figure 17. Figure 17: Logarithmic divergence of the curvature at the contact line. The inset shows view at source ↗
Figure 18
Figure 18. Figure 18: Wetting failure case for Re = 20 and Ca = 2.0 with 10 µm slip length and 64 grid points per slip length. (a) Zoom in near the advancing air-film cusp. The background is colored by u velocity and velocity vectors are shown in the most zoomed-in version. (b) The contact-line velocity (cusp tip) as a function of time. Time is non-dimensional and the horizontal red-dashed line is y = 0.17. divergence. Notably… view at source ↗
Figure 19
Figure 19. Figure 19: (a) Critical capillary number as a function of slip length for various Reynolds view at source ↗
Figure 20
Figure 20. Figure 20: (a) Illustration of quadtree adaptive mesh refinement. (b) The tree view at source ↗
Figure 21
Figure 21. Figure 21: (a) The velocity along the interface for the Basilisk IC-SFW solution for three view at source ↗
Figure 22
Figure 22. Figure 22: Schematic of the asymptotic behaviour of the velocity along the interface view at source ↗
Figure 23
Figure 23. Figure 23: Zoom in near the contact line for Re = 20 and Ca = 0.7. The horizontal velocity colors the background and the velocity vectors are shown. The resolution in terms of grid size per slip length is (a) λ ∆ = 4, (b) λ ∆ = 8, (c) λ ∆ = 16, (d) λ ∆ = 32, and (e) λ ∆ = 64. At maximum grid refinement, the grid size is 160 nm view at source ↗
Figure 24
Figure 24. Figure 24: (a) Angle or local slope as a function of the vertical distance from the contact view at source ↗
Figure 25
Figure 25. Figure 25: (a) Interface shapes near the contact line for the reduced model with view at source ↗
Figure 26
Figure 26. Figure 26: Chronology of the liquid curtain relaxing to a steady state for view at source ↗
read the original abstract

Curtain coating, in which a moving plate is coated by a falling liquid sheet, sustains advancing contact lines at large capillary numbers Ca ~ O(1), based on plate speed. Steady states exist up to a critical capillary number, beyond which wetting failure occurs through air-bubble entrainment. In the steady regime, experiments report that velocity along the fluid-fluid interface accelerates as the contact line is approached, down to tens of micrometres; this has been interpreted as evidence against the Navier slip model. We ask whether this acceleration is compatible with slip models, and show that it is. Although Navier slip implies a vanishing velocity at the contact line, the experimentally accessible microscale region lies outside the slip region. The curtain-coating setup is revealing because the local Reynolds number, based on distance from the contact line r ~ 10 microns, is order unity, so the observable flow is governed by local inertia. Our two-phase Navier-Stokes Volume-of-Fluid simulations with quadtree adaptive mesh refinement resolve the smallest scales and study the flow with a Navier slip boundary condition and fixed contact angle. The simulations reproduce the non-monotonic dependence of the critical capillary number on global Reynolds number, based on feed-flow velocity, and the variation of the macroscopic contact angle at the inflection point, in agreement with Liu et al (2016). The interfacial velocity in the microscale region is well described by an inertially corrected wedge flow solution whose wedge angle is set by the inflection-point value, with agreement improving as slip length is reduced; at larger scales, interface bending follows the Benney solution. These inertial effects, absent from pure Stokes flow, are essential in the experimental region. Thus qualitative microscale observations do not decisively invalidate slip models for advancing contact lines.

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 claims that microscale interfacial velocity acceleration observed in curtain-coating advancing contact lines (r ~ 10 μm) is compatible with Navier slip models. Although slip enforces zero velocity at the contact line, the experimentally accessible region lies outside the slip zone; there local Reynolds number is O(1), so inertially corrected wedge flow (with wedge angle set by the inflection-point macroscopic angle) governs the observable dynamics. Two-phase Navier-Stokes VOF simulations with quadtree AMR, Navier slip, and fixed microscopic contact angle reproduce the non-monotonic critical Ca versus global Re dependence and the inflection-point angle variation reported by Liu et al. (2016). Interface velocity in the microscale region matches the inertially corrected wedge solution (improving as slip length decreases), while larger-scale bending follows the Benney solution. The conclusion is that qualitative microscale observations do not invalidate slip models.

Significance. If the numerical results hold, the work resolves an apparent contradiction between slip models and microscale experiments by showing that local inertia (absent in Stokes analyses) accounts for the observed acceleration once the observable region is outside the slip zone. It strengthens the physical basis for Navier slip in dynamic wetting at Ca ~ O(1) and highlights the necessity of scale separation plus inertia in modeling coating flows. Strengths include direct reproduction of two independent experimental trends from Liu et al. (2016), quantitative comparison to an independent analytic wedge solution, and use of adaptive mesh refinement to resolve the relevant microscales.

major comments (2)
  1. [Numerical results and comparison to wedge solution] The central claim requires that the chosen slip length remains sufficiently small relative to the 10 μm observable region so that the microscale flow is accurately described by the inertially corrected wedge solution outside the slip zone. No explicit values of the slip length, no quantitative bound demonstrating slip length ≪ 10 μm, and no sensitivity plots of velocity profiles versus slip length are provided in the results or methods sections; without these the scale-separation argument remains unverified and load-bearing for the compatibility conclusion.
  2. [Simulation setup and validation] Mesh convergence, resolution in the microscale region, and quantitative error bars on the extracted velocity profiles are not reported. The abstract states that simulations reproduce the non-monotonic Ca-Re trend and inflection-angle variation, but without documented grid-independence tests or residual norms the quantitative match to the inertially corrected wedge solution cannot be assessed at the level needed to support the claim that inertia, rather than slip details, governs the observations.
minor comments (2)
  1. [Introduction] The distinction between the global Reynolds number (based on feed-flow velocity) and the local Reynolds number (based on distance r from the contact line) should be defined explicitly with symbols and numerical estimates in the introduction or methods.
  2. [Results] Figure captions and text should clarify the precise location of the inflection point used to set the wedge angle and how it is extracted from the simulated interface shape.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments that help strengthen the manuscript. We address each major point below and will incorporate the requested clarifications and additional documentation in the revised version.

read point-by-point responses

  1. Referee: The central claim requires that the chosen slip length remains sufficiently small relative to the 10 μm observable region so that the microscale flow is accurately described by the inertially corrected wedge solution outside the slip zone. No explicit values of the slip length, no quantitative bound demonstrating slip length ≪ 10 μm, and no sensitivity plots of velocity profiles versus slip length are provided in the results or methods sections; without these the scale-separation argument remains unverified and load-bearing for the compatibility conclusion.

    Authors: We agree that explicit documentation of the slip lengths and their relation to the observable scale is necessary to fully substantiate the scale-separation argument. Although the manuscript notes that agreement with the wedge solution improves as slip length is reduced, we did not include the specific values or sensitivity analysis in the submitted version. In the revision we will add: (i) the precise slip lengths used (on the order of 0.1–1 μm), (ii) a quantitative bound confirming slip length ≪ 10 μm (typically < 0.1), and (iii) sensitivity plots of interfacial velocity profiles for several slip lengths demonstrating convergence to the inertially corrected wedge solution. These additions will verify that the microscale acceleration is governed by local inertia outside the slip zone. revision: yes

  2. Referee: Mesh convergence, resolution in the microscale region, and quantitative error bars on the extracted velocity profiles are not reported. The abstract states that simulations reproduce the non-monotonic Ca-Re trend and inflection-angle variation, but without documented grid-independence tests or residual norms the quantitative match to the inertially corrected wedge solution cannot be assessed at the level needed to support the claim that inertia, rather than slip details, governs the observations.

    Authors: We concur that mesh-convergence documentation and error quantification are required to support the quantitative claims. The submitted manuscript relies on quadtree AMR to resolve microscales but does not present convergence tests or error bars. In the revised version we will include: (i) grid-independence studies with successive refinement levels, (ii) minimum cell sizes achieved in the microscale region relative to the slip length, and (iii) quantitative error norms or bars on the extracted velocity profiles and their deviation from the wedge solution. These will confirm that the reported trends and comparisons are robust and not sensitive to numerical details. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from independent simulations validated externally

full rationale

The paper derives its conclusions from direct two-phase Navier-Stokes VOF simulations with adaptive mesh refinement, Navier slip boundary condition, and a fixed microscopic contact angle. These are compared to external experimental data (Liu et al. 2016) for critical capillary number and macroscopic angle, and to an independent analytic inertially corrected wedge solution whose angle is taken from the simulation's inflection point but whose functional form is not fitted to the same data. No load-bearing step reduces a prediction to a fitted parameter from the validation set, nor relies on self-citation chains for uniqueness or ansatz. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The claim rests on the standard incompressible Navier-Stokes equations, the validity of the VOF interface-capturing method at the resolved scales, and the applicability of a fixed microscopic contact angle together with a small but finite slip length. No new entities are postulated.

free parameters (1)
  • slip length
    Chosen small compared with the 10-micron observational scale so that the microscale region lies outside the slip zone; exact value not stated in abstract.
axioms (3)
  • standard math Incompressible two-phase Navier-Stokes equations govern the flow
    Invoked for the entire simulation domain.
  • domain assumption Volume-of-Fluid method with quadtree AMR accurately captures the interface and contact-line region
    Basis for all reported velocity and angle results.
  • domain assumption Microscopic contact angle remains fixed while macroscopic angle varies
    Standard modeling choice for advancing contact lines.

pith-pipeline@v0.9.0 · 5632 in / 1501 out tokens · 34030 ms · 2026-05-09T19:20:50.124970+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

54 extracted references · 1 canonical work pages

  1. [1]

    Philosophical Transactions of the Royal Society of London , number =

    Young, Thomas , title =. Philosophical Transactions of the Royal Society of London , number =. 1805 , month =. doi:10.1098/rstl.1805.0005 , url =

    work page doi:10.1098/rstl.1805.0005

  2. [2]

    Wetting and spreading , author =. Rev. Mod. Phys. , volume =. 2009 , doi =

    2009

  3. [3]

    Kistler, S. F. , title =

  4. [4]

    A Cartesian grid embedded boundary method for Poisson's equation on irregular domains , author =. J. Comput. Phys. , year =

  5. [5]

    Dynamic wetting failure in curtain coating: Comparison of model predictions and experimental observations , author =. Chem. Eng. Sci. , volume =. 2019 , doi =

    2019

  6. [6]

    An atomistic model for the Navier slip condition , author =. J. Fluid Mech. , volume =. 2021 , doi =

    2021

  7. [7]

    Navier, C. L. M. H. , title =. M

  8. [8]

    and Fullana, T

    Kulkarni, Y. and Fullana, T. and Zaleski, S. , title =. Proc. R. Soc. A , volume =. 2023 , doi =

    2023

  9. [9]

    The moving contact line on a smooth solid surface , author =. Int. J. Multiph. Flow , volume =. 1993 , doi =

    1993

  10. [10]

    and Zaleski, S

    Chirco, L. and Zaleski, S. , title =. Int. J. Numer. Methods Fluids , volume =. 2023 , doi =

    2023

  11. [11]

    Putting the micro into the macro: A molecularly augmented hydrodynamic model of dynamic wetting applied to flow instabilities during forced dewetting , author =. J. Fluid Mech. , volume =. 2022 , doi =

    2022

  12. [12]

    Nanoscale sheared droplet: Volume-of-fluid, phase-field and no-slip molecular dynamics , author =. J. Fluid Mech. , volume =. 2022 , doi =

    2022

  13. [13]

    A variational approach to moving contact line hydrodynamics , author =. J. Fluid Mech. , volume =. 2006 , doi =

    2006

  14. [14]

    Kinetics of liquid--liquid displacement , author =. J. Colloid Interface Sci. , volume =. 1969 , doi =

    1969

  15. [15]

    and Vandre, E

    Liu, C.-Y. and Vandre, E. and Carvalho, M. S. and Kumar, S. , title =. J. Fluid Mech. , year =

  16. [16]

    Rival contact-angle models and the spreading of drops , author =. J. Fluid Mech. , volume =. 1992 , doi =

    1992

  17. [17]

    Dynamic wetting by liquids of different viscosity , author =. J. Colloid Interface Sci. , volume =. 2002 , doi =

    2002

  18. [18]

    Comment on ``Dynamic wetting by liquids of different viscosity'' , author =. J. Colloid Interface Sci. , volume =. 2004 , doi =

    2004

  19. [19]

    Physica D , volume =

    Singularities at the moving contact line: Mathematical, physical and computational aspects , author =. Physica D , volume =. 2006 , doi =

    2006

  20. [20]

    and Zaleski, S

    Afkhami, S. and Zaleski, S. and Bussmann, M. , journal =. A mesh-dependent model for applying dynamic contact angles to. 2009 , doi =

    2009

  21. [21]

    and Scriven, L

    Huh, C. and Scriven, L. E. , title =. J. Colloid Interface Sci. , year =

  22. [22]

    On the motion of a fluid--fluid interface along a solid surface , author =. J. Fluid Mech. , volume =. 1974 , doi =

    1974

  23. [23]

    The moving contact line: The slip boundary condition , author =. J. Fluid Mech. , volume =. 1976 , doi =

    1976

  24. [24]

    A moving fluid interface. Part 2. The removal of the force singularity by a slip flow , author =. J. Fluid Mech. , volume =. 1977 , doi =

    1977

  25. [25]

    Fluid Dyn

    Hydrodynamics of wetting , author =. Fluid Dyn. , volume =. 1976 , doi =

    1976

  26. [26]

    The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow , author =. J. Fluid Mech. , volume =. 1986 , doi =

    1986

  27. [27]

    Physica D , volume =

    A kinematic evolution equation for the dynamic contact angle and some consequences , author =. Physica D , volume =. 2019 , doi =

    2019

  28. [28]

    Meniscus draw-up and draining , author =. Eur. J. Appl. Math. , volume =. 2001 , doi =

    2001

  29. [29]

    Forced dewetting on porous media , author =. J. Fluid Mech. , volume =. 2007 , doi =

    2007

  30. [30]

    Weinstein, S. J. and Ruschak, K. J. , title =. Annu. Rev. Fluid Mech. , volume =. 2004 , doi =

    2004

  31. [31]

    Characteristics of air entrainment during dynamic wetting failure along a planar substrate , author =. J. Fluid Mech. , volume =. 2014 , doi =

    2014

  32. [32]

    Blake, T. D. and Bracke, M. and Shikhmurzaev, Y. D. , title =. Phys. Fluids , volume =. 1999 , doi =

    1999

  33. [33]

    , title =

    Clarke, A. , title =. Chem. Eng. Sci. , volume =. 1995 , doi =

    1995

  34. [34]

    Dynamic drying transition via free-surface cusps , author =. J. Fluid Mech. , volume =. 2019 , doi =

    2019

  35. [35]

    Cox--Voinov theory with slip , author =. J. Fluid Mech. , volume =. 2020 , doi =

    2020

  36. [36]

    Nonlocal hydrodynamic influence on the dynamic contact angle: Slip models versus experiment , author =. Phys. Rev. E , volume =. 2006 , doi =

    2006

  37. [37]

    Inertial effects on the flow near a moving contact line , author =. J. Fluid Mech. , volume =. 2021 , doi =

    2021

  38. [38]

    and Bussmann, M

    Afkhami, S. and Bussmann, M. , title =. Int. J. Numer. Methods Fluids , volume =. 2008 , doi =

    2008

  39. [39]

    and Bussmann, M

    Afkhami, S. and Bussmann, M. , title =. Int. J. Numer. Methods Fluids , volume =. 2009 , doi =

    2009

  40. [40]

    , title =

    Popinet, S. , title =. J. Comput. Phys. , year =

  41. [41]

    , title =

    Popinet, S. , title =. Annu. Rev. Fluid Mech. , year =

  42. [42]

    and Zaleski, S

    Scardovelli, R. and Zaleski, S. , title =. Annu. Rev. Fluid Mech. , year =

  43. [43]

    and Zaleski, S

    Popinet, S. and Zaleski, S. , title =. Int. J. Numer. Methods Fluids , year =

  44. [44]

    and Buongiorno, J

    Afkhami, S. and Buongiorno, J. and Guion, A. and Popinet, S. and Saade, Y. and Scardovelli, R. and Zaleski, S. , title =. J. Comput. Phys. , year =

  45. [45]

    and Zaleski, S

    Fullana, T. and Zaleski, S. and Popinet, S. , title =. Eur. Phys. J. Spec. Top. , year =

  46. [46]

    and Johansson, P

    Lācis, U. and Johansson, P. and Fullana, T. and Hess, B. and Amberg, G. and Bagheri, S. and Zaleski, S. , title =. Eur. Phys. J. Spec. Top. , year =

  47. [47]

    Hirt, C. W. and Nichols, B. D. , title =. J. Comput. Phys. , volume =. 1981 , doi =

    1981

  48. [48]

    and Scardovelli, R

    Tryggvason, G. and Scardovelli, R. and Zaleski, S. , title =. 2011 , doi =

    2011

  49. [49]

    A second-order projection method for the incompressible Navier--Stokes equations , author =. J. Comput. Phys. , volume =. 1989 , doi =

    1989

  50. [50]

    , journal =

    Popinet, S. , journal =. Gerris: A tree-based adaptive solver for the incompressible. 2003 , doi =

    2003

  51. [51]

    Benney, D. J. and Timson, W. J. , title =. Stud. Appl. Math. , volume =. 1980 , doi =

    1980

  52. [52]

    Ngan, C. G. and Dussan V., E. B. , title =. Phys. Fluids , volume =. 1984 , doi =

    1984

  53. [53]

    Benilov, E. S. and Vynnycky, M. , journal =. Contact lines with a 180^. 2013 , doi =

    2013

  54. [54]

    Viscous and resistive eddies near a sharp corner , author =. J. Fluid Mech. , volume =. 1964 , doi =

    1964