A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant

Christian Rohde; Rahul Barthwal

arxiv: 2502.17205 · v2 · pith:U7HNHBIMnew · submitted 2025-02-24 · 🧮 math.AP

A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant

Rahul Barthwal , Christian Rohde This is my paper

Pith reviewed 2026-05-23 02:27 UTC · model grok-4.3

classification 🧮 math.AP

keywords two-layer thin filmanti-surfactanthyperbolic systemMarangoni flowRiemann problementropy pairsTemple class system

0 comments

The pith

Neglecting capillarity and diffusion yields a strictly hyperbolic system for two-layer anti-surfactant thin films that admits entropy pairs and explicit Riemann solutions.

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

The paper begins with the motion of two immiscible viscous fluids in a two-layer thin film containing an anti-surfactant solute that induces Marangoni flow via surface tension changes. After deriving one-dimensional evolution equations via the lubrication limit, the authors impose negligible capillarity and diffusion together with perfect solute solubility. This produces a conservative first-order system in film heights and concentration gradients. The system is proved strictly hyperbolic on a certain set of states, equipped with an entire class of entropy pairs and a strictly convex entropy, and nearly Temple-class. These properties deliver well-posedness of the Cauchy problem and permit explicit Riemann problem solutions, which are then used in Godunov-type numerical experiments.

Core claim

Under the assumptions of negligible capillarity and diffusion and perfect solubility of the solute, the lubrication-derived equations reduce to a conservative first-order system in film heights and concentration gradients. This reduced system is strictly hyperbolic for a certain set of states and admits an entire class of entropy/entropy-flux pairs, including a strictly convex entropy, which implies well-posedness of the Cauchy problem. The system is almost Temple-class, enabling explicit computation of solutions to the Riemann problem.

What carries the argument

The conservative first-order hyperbolic system in film heights and concentration gradients obtained after the lubrication limit and the stated reductions.

If this is right

The Cauchy problem for the system is well-posed.
Explicit solutions to the Riemann problem can be computed from the near-Temple-class structure.
A Godunov-type finite volume scheme can be constructed using the exact Riemann solver.
The model admits multiple entropy/entropy-flux pairs.

Where Pith is reading between the lines

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

The same reduction steps could produce hyperbolic models for other thin-film problems that involve surface-active agents.
The explicit Riemann solutions offer a benchmark for testing numerical methods on related multi-layer conservation laws.
The identified hyperbolicity region may help select parameter ranges for laboratory validation of the reduced model.

Load-bearing premise

Capillarity and diffusion effects must be negligible and the solute must be perfectly soluble to reach the conservative first-order hyperbolic system.

What would settle it

A direct numerical simulation of the full lubrication equations or a physical experiment that produces flow behavior differing measurably from the hyperbolic model's predictions when capillarity remains small but nonzero.

Figures

Figures reproduced from arXiv: 2502.17205 by Christian Rohde, Rahul Barthwal.

**Figure 1.** Figure 1: Two-phase thin film flow under the influence of an anti-surfactant. by x ∗ and z ∗ , the horizontal and vertical directions, respectively. Moreover, we denote the two film heights by f ∗ and g ∗ such that the interface surface is located at z ∗ = f ∗ (x ∗ , t∗ ), and the free surface is located at z ∗ = (f ∗ + g ∗ )(x ∗ , t∗ ). The outward unit normal ˆn ∗ 1 and unit tangent vector ˆt ∗ 1 of the interface … view at source ↗

**Figure 5.** Figure 5: Evolution of primitive variables at t = 1.00 using Godunov solver and LaxFriedrichs scheme with ∆x = 4.37 × 10−2 . the domains where the film heights increase. This observation resonates with our discussion on the physical interpretation of the entropy E¯ in Remark 4.4. 7 Conclusions In this work, we derived a reduced system of hyperbolic conservation laws governing the evolution of a two-phase thin film… view at source ↗

**Figure 6.** Figure 6: Evolution of primitive variables at t = 1.00 using the Godunov scheme with ∆x = 4.37 × 10−2 . approximations. Acknowledgements This work was financially supported by the German Research Foundation (DFG), within the Collaborative Research Center on Interface-Driven MultiField Processes in Porous Media (SFB 1313, Project Number 327154368) and the Priority Programme - SPP 2410 Hyperbolic Balance Laws in Flui… view at source ↗

read the original abstract

We consider the motion of a two-layer thin film that consists of two immiscible viscous fluids and is endowed with an anti-surfactant solute. The presence of such solute particles induces variations of the surface tension and interfacial stress driving a Marangoni-type flow. We first analyze a lubrication limit and derive one-dimensional evolution equations for film heights and solute concentrations. Then, under the assumption that the capillarity and diffusion effects are negligible and the solute is perfectly soluble, we obtain a conservative first-order system in terms of film heights and concentration gradients. This reduced system is found to be strictly hyperbolic for a certain set of states and to admit an entire class of entropy/entropy-flux pairs. We also provide a strictly convex entropy for the hyperbolic system. Thus, the well-posedness for the Cauchy problem is given. Moreover, the system is almost a Temple-class system, which allows to compute explicit solutions of the Riemann problem. The paper concludes with numerical experiments using a Godunov-type finite volume method, which relies on the exact Riemann solver.

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.

Desk Editor's Note private letter to a colleague

The paper reduces a two-layer anti-surfactant thin film to a first-order hyperbolic system with entropy and near-Temple structure under strong simplifying assumptions.

read the letter

The core move is to take the lubrication equations for two immiscible layers with a perfectly soluble anti-surfactant, drop capillarity and diffusion, and arrive at a conservative first-order system in the two heights and the concentration gradient. From there the authors establish strict hyperbolicity on an open set of states, produce a family of entropy pairs plus one strictly convex entropy, conclude well-posedness for the Cauchy problem, and note that the system is close enough to Temple class to allow explicit Riemann solutions. They close with a Godunov scheme that uses the exact solver. That reduction and the entropy/Temple analysis for this specific anti-surfactant case is the new piece; it is not in the literature they cite. The assumptions are stated plainly, the steps follow standard hyperbolic-conservation-law techniques, and the numerical method is the natural one once the Riemann solver exists. The main limitation is the modeling restrictions themselves. Removing capillarity and diffusion leaves a system without the regularizing terms that usually matter in thin-film problems, so the regime of validity is narrow. Hyperbolicity holds only for certain states, and whether those states cover the physically interesting range is not obvious from the abstract. The “almost Temple” claim also needs the full proof to show how much explicit structure actually survives. This is specialist work for people already working on hyperbolic reductions of lubrication models or on Marangoni flows in layered films. A reader who wants explicit solutions or entropy-based well-posedness results for this setup will get something usable. It is coherent on its own terms and the claims are specific enough that a serious referee should see it.

Referee Report

2 major / 2 minor

Summary. The paper derives one-dimensional lubrication equations for the heights and solute concentrations of a two-layer thin film with an anti-surfactant. Under the explicit assumptions that capillarity and diffusion are negligible and the solute is perfectly soluble, the system reduces to a conservative first-order hyperbolic system in film heights and concentration gradients. The reduced system is shown to be strictly hyperbolic on a nonempty open set of states, to admit an entire family of entropy/entropy-flux pairs together with a strictly convex entropy (hence well-posedness of the Cauchy problem), and to be almost Temple-class, permitting explicit Riemann solutions. The paper closes with Godunov-type finite-volume numerics that employ the exact Riemann solver.

Significance. If the reduction is justified, the work supplies a new, mathematically structured model for Marangoni-driven two-layer films whose almost-Temple property yields explicit Riemann solutions and whose strictly convex entropy directly implies local well-posedness. These analytic features are genuine strengths that distinguish the contribution from purely numerical thin-film studies.

major comments (2)

[§3 (reduction step)] The passage from the lubrication equations to the claimed conservative first-order system (abstract and §3) rests on dropping capillarity, diffusion, and solubility-related terms without quantitative error estimates or a priori bounds on the neglected contributions. Because this reduction is the load-bearing step that produces the hyperbolic system whose hyperbolicity, entropy, and Temple properties are then proved, the absence of such estimates leaves the domain of validity of the subsequent analysis unclear.
[§4 (hyperbolicity)] The statement that the system is “strictly hyperbolic for a certain set of states” (abstract) is central to all later claims. The manuscript must explicitly identify this open set in the (h1,h2,c) variables, compute the eigenvalues of the flux Jacobian, and verify that they remain real and distinct throughout the set; without these explicit expressions the hyperbolicity claim cannot be checked.

minor comments (2)

[§2–3] Notation for the two film heights and the concentration gradient should be introduced once and used consistently; several passages mix dimensional and nondimensional variables without explicit warning.
[§6] The numerical section would benefit from a brief statement of the CFL condition employed with the exact Riemann solver and from a short convergence table (L1 error versus mesh size) for at least one Riemann problem whose exact solution is known.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense of the work while noting where revisions will strengthen the presentation.

read point-by-point responses

Referee: [§3 (reduction step)] The passage from the lubrication equations to the claimed conservative first-order system (abstract and §3) rests on dropping capillarity, diffusion, and solubility-related terms without quantitative error estimates or a priori bounds on the neglected contributions. Because this reduction is the load-bearing step that produces the hyperbolic system whose hyperbolicity, entropy, and Temple properties are then proved, the absence of such estimates leaves the domain of validity of the subsequent analysis unclear.

Authors: The reduction is a standard formal asymptotic step in lubrication theory, where capillarity, diffusion, and related terms are neglected to obtain the leading-order hyperbolic model under the assumptions of thin films and dominant Marangoni effects. The manuscript's primary contribution lies in analyzing the resulting conservative system rather than deriving rigorous approximation bounds. We will add a clarifying paragraph in §3 on the physical regime of validity (small capillary number, negligible diffusion) to better delineate applicability, but quantitative error estimates lie outside the paper's scope. revision: partial
Referee: [§4 (hyperbolicity)] The statement that the system is “strictly hyperbolic for a certain set of states” (abstract) is central to all later claims. The manuscript must explicitly identify this open set in the (h1,h2,c) variables, compute the eigenvalues of the flux Jacobian, and verify that they remain real and distinct throughout the set; without these explicit expressions the hyperbolicity claim cannot be checked.

Authors: We agree that explicit verification strengthens the claim. In the revised manuscript we will define the precise open set in (h1, h2, c) variables on which the system is strictly hyperbolic, compute the eigenvalues of the flux Jacobian explicitly, and state the conditions under which they are real and distinct. These details will be inserted into §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper begins from standard lubrication theory for two-layer films, applies explicit physical assumptions (negligible capillarity/diffusion, perfect solubility) to drop higher-order terms, and obtains a first-order conservative system. Hyperbolicity, existence of entropy pairs, strict convexity of an entropy, and almost-Temple structure are then proved directly from the resulting PDE system using standard techniques for hyperbolic conservation laws. No fitted parameters, self-definitional relations, or load-bearing self-citations appear in the derivation chain. The central claims rest on independent mathematical analysis of the reduced equations rather than on any input being renamed as output.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the lubrication approximation plus two explicit modeling choices that are not derived inside the paper.

axioms (2)

domain assumption Lubrication approximation is valid for the two-layer thin film geometry
Invoked to obtain the one-dimensional evolution equations before further reduction.
ad hoc to paper Capillarity and diffusion effects are negligible and the solute is perfectly soluble
Explicitly stated as the assumption needed to reach the conservative first-order hyperbolic system.

pith-pipeline@v0.9.0 · 5714 in / 1529 out tokens · 34402 ms · 2026-05-23T02:27:18.430722+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows
math.AP 2026-04 unverdicted novelty 5.0

Global weak entropy solutions exist for a class of non-symmetric Keyfitz-Kranzer systems that includes thin-film flow models.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

[1]

Barthwal, and T

Anamika, R. Barthwal, and T. Raja Sekhar. Construction of solutions of the Riemann problem for a two-dimensional Keyfitz-Kranzer type model governing a thin film flow. Accepted for publication in Applied Mathematics and Computation

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Andreianov, C

B. Andreianov, C. Donadello, S. S. Ghoshal, and U. Razafison. On the attainable set for a class of triangular systems of conservation laws. Journal of Evolution Equations , 15:503–532, 2015

work page 2015
[3]

Barthwal and T

R. Barthwal and T. Raja Sekhar. Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution. Quarterly of Applied Mathematics, 80(4):717–738, 2022. 27

work page 2022
[4]

Barthwal, T

R. Barthwal, T. Raja Sekhar, and G. Raja Sekhar. Construction of solutions of a two- dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution. Mathematical Methods in the Applied Sciences , 46(6):7413–7434, 2023

work page 2023
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Bertozzi, A

A. Bertozzi, A. M¨ unch, and M. Shearer. Undercompressive shocks in thin film flows. Physica D: Nonlinear Phenomena , 134(4):431–464, 1999

work page 1999
[6]

Billingham

J. Billingham. Surface tension-driven flow in a slender wedge. SIAM Journal on Applied Mathematics, 66(6):1949–1977, 2006

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G. Br¨ ull. Modeling and analysis of a two-phase thin film model with insoluble surfactant. Nonlinear Analysis: Real World Applications , 27:124–145, 2016

work page 2016
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Chalons, P

C. Chalons, P. Engel, and C. Rohde. A conservative and convergent scheme for under- compressive shock waves. SIAM Journal on Numerical Analysis , 52(1):554–579, 2014

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J. Conn, B. Duffy, D. Pritchard, S. Wilson, and K. Sefiane. Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution. Journal of Engineering Mathematics, 107(1):167–178, 2017

work page 2017
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J. Conn, B. R. Duffy, D. Pritchard, S. K. Wilson, P. J. Halling, and K. Sefiane. Fluid- dynamical model for antisurfactants. Physical Review E, 93(4):043121, 2016

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J. J. Conn. Stability and dynamics of anti-surfactant solutions . PhD thesis, University of Strathclyde, 2017

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B. Cook, A. L. Bertozzi, and A. Hosoi. Shock solutions for particle-laden thin films. SIAM Journal on Applied Mathematics , 68(3):760–783, 2008

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R. V. Craster and O. K. Matar. Dynamics and stability of thin liquid films. Reviews of Modern Physics, 81(3):1131–1198, 2009

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C. M. Dafermos. Hyperbolic conservation laws in continuum physics , volume 325 of Grundlehren der mathematischen Wissenschaften . Springer-Verlag, Berlin, 2000

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

work page 1990
[16]

Jensen and J

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

work page 1992
[17]

Kalogirou and M

A. Kalogirou and M. Blyth. The role of soluble surfactants in the linear stability of two-layer flow in a channel. Journal of Fluid Mechanics , 873:18–48, 2019

work page 2019
[18]

T. Kato. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Archive for Rational Mechanics and Analysis , 58(3):181–205, 1975

work page 1975
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J. B. Keller and M. J. Miksis. Surface tension driven flows. SIAM Journal on Applied Mathematics, 43(2):268–277, 1983. 28

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[20]

Kutter, C

M. Kutter, C. Rohde, and A.-M. S¨ andig. Well-posedness of a two-scale model for liquid phase epitaxy with elasticity. Continuum Mechanics and Thermodynamics , 29(4):989– 1016, 2017

work page 2017
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A. Majda. Compressible fluid flow and systems of conservation laws in several space variables, volume 53. Springer Science & Business Media, 2012

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Matar and S

O. Matar and S. Kumar. Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM Journal on Applied Mathematics , 64(6):2144–2166, 2004

work page 2004
[23]

Raja Sekhar, and G

Minhajul, T. Raja Sekhar, and G. P. Raja Sekhar. Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution. Commu- nications on Pure & Applied Analysis , 18(6):3367–3386, 2019

work page 2019
[24]

T. Myers. Thin films with high surface tension. SIAM Review, 40(3):441–462, 1998

work page 1998
[25]

S. B. G. O’Brien and L. W. Schwartz. Theory and modeling of thin film flows. Ency- clopedia of surface and colloid science , 1:5283–5297, 2002

work page 2002
[26]

Picardo, T

J. Picardo, T. Radhakrishna, A. B. Vir, S. Ramji, and S. Pushpavanam. Modelling extraction in microchannels with stratified flow: channel geometry, flow configuration and Marangoni stresses. Indian Chemical Engineer, 57(3-4):322–358, 2015

work page 2015
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J. R. Picardo, T. Radhakrishna, and S. Pushpavanam. Solutal Marangoni instability in layered two-phase flows. Journal of Fluid Mechanics , 793:280–315, 2016

work page 2016
[28]

Sen and T

A. Sen and T. Raja Sekhar. Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 19(5):2641–2653, 2020

work page 2020
[29]

D. Serre. Systems of conservation laws 1: hyperbolicity, entropies, shock waves . Cam- bridge University Press, 1999

work page 1999
[30]

B. Temple. Systems of conservation laws with invariant submanifolds. Transactions of the American Mathematical Society, 280(2):781–795, 1983. 29

work page 1983

[1] [1]

Barthwal, and T

Anamika, R. Barthwal, and T. Raja Sekhar. Construction of solutions of the Riemann problem for a two-dimensional Keyfitz-Kranzer type model governing a thin film flow. Accepted for publication in Applied Mathematics and Computation

work page

[2] [2]

Andreianov, C

B. Andreianov, C. Donadello, S. S. Ghoshal, and U. Razafison. On the attainable set for a class of triangular systems of conservation laws. Journal of Evolution Equations , 15:503–532, 2015

work page 2015

[3] [3]

Barthwal and T

R. Barthwal and T. Raja Sekhar. Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution. Quarterly of Applied Mathematics, 80(4):717–738, 2022. 27

work page 2022

[4] [4]

Barthwal, T

R. Barthwal, T. Raja Sekhar, and G. Raja Sekhar. Construction of solutions of a two- dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution. Mathematical Methods in the Applied Sciences , 46(6):7413–7434, 2023

work page 2023

[5] [5]

Bertozzi, A

A. Bertozzi, A. M¨ unch, and M. Shearer. Undercompressive shocks in thin film flows. Physica D: Nonlinear Phenomena , 134(4):431–464, 1999

work page 1999

[6] [6]

Billingham

J. Billingham. Surface tension-driven flow in a slender wedge. SIAM Journal on Applied Mathematics, 66(6):1949–1977, 2006

work page 1949

[7] [7]

G. Br¨ ull. Modeling and analysis of a two-phase thin film model with insoluble surfactant. Nonlinear Analysis: Real World Applications , 27:124–145, 2016

work page 2016

[8] [8]

Chalons, P

C. Chalons, P. Engel, and C. Rohde. A conservative and convergent scheme for under- compressive shock waves. SIAM Journal on Numerical Analysis , 52(1):554–579, 2014

work page 2014

[9] [9]

J. Conn, B. Duffy, D. Pritchard, S. Wilson, and K. Sefiane. Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution. Journal of Engineering Mathematics, 107(1):167–178, 2017

work page 2017

[10] [10]

J. Conn, B. R. Duffy, D. Pritchard, S. K. Wilson, P. J. Halling, and K. Sefiane. Fluid- dynamical model for antisurfactants. Physical Review E, 93(4):043121, 2016

work page 2016

[11] [11]

J. J. Conn. Stability and dynamics of anti-surfactant solutions . PhD thesis, University of Strathclyde, 2017

work page 2017

[12] [12]

B. Cook, A. L. Bertozzi, and A. Hosoi. Shock solutions for particle-laden thin films. SIAM Journal on Applied Mathematics , 68(3):760–783, 2008

work page 2008

[13] [13]

R. V. Craster and O. K. Matar. Dynamics and stability of thin liquid films. Reviews of Modern Physics, 81(3):1131–1198, 2009

work page 2009

[14] [14]

C. M. Dafermos. Hyperbolic conservation laws in continuum physics , volume 325 of Grundlehren der mathematischen Wissenschaften . Springer-Verlag, Berlin, 2000

work page 2000

[15] [15]

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

work page 1990

[16] [16]

Jensen and J

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

work page 1992

[17] [17]

Kalogirou and M

A. Kalogirou and M. Blyth. The role of soluble surfactants in the linear stability of two-layer flow in a channel. Journal of Fluid Mechanics , 873:18–48, 2019

work page 2019

[18] [18]

T. Kato. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Archive for Rational Mechanics and Analysis , 58(3):181–205, 1975

work page 1975

[19] [19]

J. B. Keller and M. J. Miksis. Surface tension driven flows. SIAM Journal on Applied Mathematics, 43(2):268–277, 1983. 28

work page 1983

[20] [20]

Kutter, C

M. Kutter, C. Rohde, and A.-M. S¨ andig. Well-posedness of a two-scale model for liquid phase epitaxy with elasticity. Continuum Mechanics and Thermodynamics , 29(4):989– 1016, 2017

work page 2017

[21] [21]

A. Majda. Compressible fluid flow and systems of conservation laws in several space variables, volume 53. Springer Science & Business Media, 2012

work page 2012

[22] [22]

Matar and S

O. Matar and S. Kumar. Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM Journal on Applied Mathematics , 64(6):2144–2166, 2004

work page 2004

[23] [23]

Raja Sekhar, and G

Minhajul, T. Raja Sekhar, and G. P. Raja Sekhar. Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution. Commu- nications on Pure & Applied Analysis , 18(6):3367–3386, 2019

work page 2019

[24] [24]

T. Myers. Thin films with high surface tension. SIAM Review, 40(3):441–462, 1998

work page 1998

[25] [25]

S. B. G. O’Brien and L. W. Schwartz. Theory and modeling of thin film flows. Ency- clopedia of surface and colloid science , 1:5283–5297, 2002

work page 2002

[26] [26]

Picardo, T

J. Picardo, T. Radhakrishna, A. B. Vir, S. Ramji, and S. Pushpavanam. Modelling extraction in microchannels with stratified flow: channel geometry, flow configuration and Marangoni stresses. Indian Chemical Engineer, 57(3-4):322–358, 2015

work page 2015

[27] [27]

J. R. Picardo, T. Radhakrishna, and S. Pushpavanam. Solutal Marangoni instability in layered two-phase flows. Journal of Fluid Mechanics , 793:280–315, 2016

work page 2016

[28] [28]

Sen and T

A. Sen and T. Raja Sekhar. Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 19(5):2641–2653, 2020

work page 2020

[29] [29]

D. Serre. Systems of conservation laws 1: hyperbolicity, entropies, shock waves . Cam- bridge University Press, 1999

work page 1999

[30] [30]

B. Temple. Systems of conservation laws with invariant submanifolds. Transactions of the American Mathematical Society, 280(2):781–795, 1983. 29

work page 1983