Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

Sen, Anupam; Sekhar, T. Raja

doi:10.3934/cpaa.2020115

Cited by 12 publications

(9 citation statements)

References 28 publications

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“…[36,62,65]). Similar concentration effects were also observed in a system modeling the polymer flooding of a porous medium [26], in pressureles gas dynamics [67] and in thin-film equations [40,59].…”

Section: Discussionsupporting

confidence: 68%

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Kalisch

Mitrović

2022

Int. J. Appl. Comput. Math

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A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any $$n\times n$$ n × n system of conservation laws has a solution. The solution concept is an extension of the notion of singular $$\delta $$ δ -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of $$\delta $$ δ -distributions as a part of solutions to systems of conservation laws.

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Section: Discussionsupporting

confidence: 68%

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Kalisch

Mitrović

2022

Int. J. Appl. Comput. Math

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“…For one-dimensional rotated solutions, we refer reader to previous works. 23,24 By comparing with the rotated one-dimensional solution in the succeeding sections, we verify that our constructed solution is globally unique.…”

Section: Classificationmentioning

confidence: 70%

“…The above system was studied for classical wave interactions by Minhajul et al 23 and for nonclassical wave interactions by Sen and Raja Sekhar. 24 For more details on thin film flow and its modeling, we refer readers to previous works 18,[25][26][27][28][29] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

Barthwal

Sekhar

2022

Math Methods in App Sciences

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This article is concerned with formulation of three-dimensional thin film model for an antisurfactant solution and hence constructing unique global solution for a two-dimensional Riemann problem for the corresponding reduced hyperbolic form. We develop six geometrically different structures of the solution using generalized characteristic analysis method while relaxing the restriction that only one planar elementary wave is developed at the interface of each initial discontinuity. We analyze the interactions of classical and nonclassical waves in detail to construct the global solution of the corresponding 2-D Riemann problem. Further, we provide the expressions for strength, location, and propagation speed of delta shock wave at each interaction point. Moreover, we compare these solutions with the solutions of a one-dimensional rotated initial value problem and prove that our solutions are globally unique.

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“…In addition, there are still some other very effective methods [9,16,17,23,25,27] to deal with the related problems of δ-shocks. Finally, it is necessary to point out that delta shock wave has also been found in other hyperbolic models, such as in the nonlinear chromatography [20], in the shallow water dynamics [18], in the thin film [28] and also in the porous media [13].…”

Section: Introductionmentioning

confidence: 95%

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

Shan

Shen²,

Wei³

2022

Math. Model. Nat. Phenom.

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The exact Riemann solutions for the extended macroscopic production model with an adiabatic exponent are constructed in perfectly explicit forms. The asymptotic limit of Riemann solution consisting of 1-shock wave and 2-contact discontinuity tends to a delta shock solution for the pressureless gas dynamics model under the special over-compressive entropy condition as the adiabatic exponent drops to one. In contrast, the asymptotic limit of Riemann solution composed of 1-rarefaction wave and 2-contact discontinuity tends to the vacuum solution surrounded by two contact discontinuities by letting the adiabatic exponent tend to one, in which the state in the interior of the 1-rarefaction wave fan is developed into vacuum. The intrinsic phenomena of concentration and cavitation are identified and investigated carefully during this limiting process, which displays more complicated and completely different behavior compared with previous literature. In addition, some representative numerical calculations are also provided, which are in well agreement with our theoretical results.

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Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

Cited by 12 publications

References 28 publications

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

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