November  2019, 18(6): 3367-3386. doi: 10.3934/cpaa.2019152

Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India

* Corresponding author

Received  October 2018 Revised  January 2019 Published  May 2019

In this article, we consider a quasilinear hyperbolic system of partial differential equations governing the dynamics of a thin film of a perfectly soluble anti-surfactant liquid. We construct elementary waves of the corresponding Riemann problem and study their interactions. Further, we provide exact solution of the Riemann problem along with numerical examples. Finally, we show that the solution of the Riemann problem is stable under small perturbation of the initial data.

Citation: Minhajul, T. Raja Sekhar, G. P. Raja Sekhar. Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3367-3386. doi: 10.3934/cpaa.2019152
References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938. doi: 10.1137/S0036139997332099. Google Scholar

[2]

A. Bressan, Hyperbolic Systems of Conservation Laws, Vol. 20, Oxford University Press, Oxford, 2000. Google Scholar

[3]

J. J. A. ConnB. R. DuffyD. PritchardS. K. WilsonP. J. Halling and K. Sefiane, Fluid-dynamical model for antisurfactants, Phys. Rev. E, 93 (2016), 043121. Google Scholar

[4]

J. J. A. ConnB. DuffyD. PritchardS. Wilson and K. Sefiane, Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution, J. Engrg. Math., 107 (2017), 167-178. doi: 10.1007/s10665-017-9924-8. Google Scholar

[5]

R. J. Duan and X. F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014. doi: 10.3934/cpaa.2013.12.985. Google Scholar

[6]

E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Vol. 118, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4612-0713-9. Google Scholar

[7]

J. F. Hernández-Sánchez, A. Eddi and J. H. Snoeijer, Marangoni spreading due to a localized alcohol supply on a thin water film, Phys. Fluids, 27 (2015), 032003.Google Scholar

[8]

S. D. HowisonJ. A. MoriartyJ. R. OckendonE. L. Terrill and S. K. Wilson, A mathematical model for drying paint layers, J. Engrg. Math., 32 (1997), 377-394. doi: 10.1023/A:1004224014291. Google Scholar

[9]

Y. Hu and W. Sheng, The Riemann problem of conservation laws in magnetogasdynamics, Commun. Pure Appl. Anal., 12 (2013), 755-769. doi: 10.3934/cpaa.2013.12.755. Google Scholar

[10]

S. Kuila and T. Raja Sekhar, Wave interactions in non-ideal isentropic magnetogasdynamics, Int. J. Appl. Comput. Math., 3 (2017), 1809-1831. doi: 10.1007/s40819-016-0195-2. Google Scholar

[11]

S. Kuila and T. Raja Sekhar, Interaction of weak shocks in drift-flux model of compressible two-phase flows, Chaos, Solitons and Fractals, 107 (2018), 222-227. doi: 10.1016/j.chaos.2017.12.030. Google Scholar

[12]

Z. Li and B. C. Y. Lu, Surface tension of aqueous electrolyte solutions at high concentrations-representation and prediction, Chem. Eng. Sci., 56 (2001), 2879-2888. Google Scholar

[13]

Y. Liu and W. Sun, Elementary wave interactions in magnetogasdynamics, Indian J. Pure Appl. Math., 47 (2016), 33-57. doi: 10.1007/s13226-016-0172-9. Google Scholar

[14]

Y. Liu and W. Sun, Wave interactions and stability of Riemann solutions of the Aw–Rascle model for generalized Chaplygin gas, Acta Appl. Math., 154 (2018), 95-109. doi: 10.1007/s10440-017-0135-0. Google Scholar

[15]

F. A. Long and G. C. Nutting, The relative surface tension of potassium chloride solutions by a differential bubble pressure method, J. Amer. Chem. Soc., 64 (1942), 2476-2482. Google Scholar

[16]

Mi nhajulD. Zeidan and T. Raja Sekhar, On the wave interactions in the drift-flux equations of two-phase flows, Appl. Math. Comput., 327 (2018), 117-131. doi: 10.1016/j.amc.2018.01.021. Google Scholar

[17]

W. Overdiep, The levelling of paints, Progress in Organic Coatings, 14 (1986), 159-175. Google Scholar

[18]

T. Raja Sekhar and Minhajul, Elementary wave interactions in blood flow through artery, J. Math. Phys., 58 (2017), 101502. doi: 10.1063/1.5004666. Google Scholar

[19]

T. Raja Sekhar and V. D. Sharma, Interaction of shallow water waves, Stud. Appl. Math., 121 (2008), 1-25. doi: 10.1111/j.1467-9590.2008.00402.x. Google Scholar

[20]

T. Raja Sekhar and V. D. Sharma, Riemann problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl., 11 (2010), 619-636. doi: 10.1016/j.nonrwa.2008.10.036. Google Scholar

[21]

A. Sen and T. Raja Sekhar, Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation, Commun. Pure Appl. Anal., 18 (2019), 931-942. Google Scholar

[22]

A. SenT. Raja Sekhar and V. D. Sharma, Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws, Quart. Appl. Math., 75 (2017), 539-554. doi: 10.1090/qam/1466. Google Scholar

[23]

V. Sharanya and G. P. Raja Sekhar, Thermocapillary migration of a spherical drop in an arbitrary transient Stokes flow, Phys. Fluids, 27 (2015), 063104.Google Scholar

[24]

V. D. Sharma, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves, CRC Press, 2010. doi: 10.1201/9781439836910. Google Scholar

[25]

C. Shen, Wave interactions and stability of the Riemann solutions for the chromatography equations, J. Math. Anal. Appl., 365 (2010), 609-618. doi: 10.1016/j.jmaa.2009.11.037. Google Scholar

[26]

J. Smoller, Shock Waves and Reaction-diffusion Equations, Vol. 258, Springer Science & Business Media, 2012. Google Scholar

[27]

M. Sun, Interactions of elementary waves for the Aw–Rascle model, SIAM J. Appl. Math., 69 (2009), 1542-1558. doi: 10.1137/080731402. Google Scholar

[28]

M. Sun, A note on the interactions of elementary waves for the AR traffic flow model without vacuum, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1503-1512. doi: 10.1016/S0252-9602(11)60336-6. Google Scholar

[29]

S. K. Wilson, The levelling of paint films, IMA J. Appl. Math., 50 (1993), 149-166. doi: 10.1093/imamat/50.2.149. Google Scholar

show all references

References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938. doi: 10.1137/S0036139997332099. Google Scholar

[2]

A. Bressan, Hyperbolic Systems of Conservation Laws, Vol. 20, Oxford University Press, Oxford, 2000. Google Scholar

[3]

J. J. A. ConnB. R. DuffyD. PritchardS. K. WilsonP. J. Halling and K. Sefiane, Fluid-dynamical model for antisurfactants, Phys. Rev. E, 93 (2016), 043121. Google Scholar

[4]

J. J. A. ConnB. DuffyD. PritchardS. Wilson and K. Sefiane, Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution, J. Engrg. Math., 107 (2017), 167-178. doi: 10.1007/s10665-017-9924-8. Google Scholar

[5]

R. J. Duan and X. F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014. doi: 10.3934/cpaa.2013.12.985. Google Scholar

[6]

E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Vol. 118, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4612-0713-9. Google Scholar

[7]

J. F. Hernández-Sánchez, A. Eddi and J. H. Snoeijer, Marangoni spreading due to a localized alcohol supply on a thin water film, Phys. Fluids, 27 (2015), 032003.Google Scholar

[8]

S. D. HowisonJ. A. MoriartyJ. R. OckendonE. L. Terrill and S. K. Wilson, A mathematical model for drying paint layers, J. Engrg. Math., 32 (1997), 377-394. doi: 10.1023/A:1004224014291. Google Scholar

[9]

Y. Hu and W. Sheng, The Riemann problem of conservation laws in magnetogasdynamics, Commun. Pure Appl. Anal., 12 (2013), 755-769. doi: 10.3934/cpaa.2013.12.755. Google Scholar

[10]

S. Kuila and T. Raja Sekhar, Wave interactions in non-ideal isentropic magnetogasdynamics, Int. J. Appl. Comput. Math., 3 (2017), 1809-1831. doi: 10.1007/s40819-016-0195-2. Google Scholar

[11]

S. Kuila and T. Raja Sekhar, Interaction of weak shocks in drift-flux model of compressible two-phase flows, Chaos, Solitons and Fractals, 107 (2018), 222-227. doi: 10.1016/j.chaos.2017.12.030. Google Scholar

[12]

Z. Li and B. C. Y. Lu, Surface tension of aqueous electrolyte solutions at high concentrations-representation and prediction, Chem. Eng. Sci., 56 (2001), 2879-2888. Google Scholar

[13]

Y. Liu and W. Sun, Elementary wave interactions in magnetogasdynamics, Indian J. Pure Appl. Math., 47 (2016), 33-57. doi: 10.1007/s13226-016-0172-9. Google Scholar

[14]

Y. Liu and W. Sun, Wave interactions and stability of Riemann solutions of the Aw–Rascle model for generalized Chaplygin gas, Acta Appl. Math., 154 (2018), 95-109. doi: 10.1007/s10440-017-0135-0. Google Scholar

[15]

F. A. Long and G. C. Nutting, The relative surface tension of potassium chloride solutions by a differential bubble pressure method, J. Amer. Chem. Soc., 64 (1942), 2476-2482. Google Scholar

[16]

Mi nhajulD. Zeidan and T. Raja Sekhar, On the wave interactions in the drift-flux equations of two-phase flows, Appl. Math. Comput., 327 (2018), 117-131. doi: 10.1016/j.amc.2018.01.021. Google Scholar

[17]

W. Overdiep, The levelling of paints, Progress in Organic Coatings, 14 (1986), 159-175. Google Scholar

[18]

T. Raja Sekhar and Minhajul, Elementary wave interactions in blood flow through artery, J. Math. Phys., 58 (2017), 101502. doi: 10.1063/1.5004666. Google Scholar

[19]

T. Raja Sekhar and V. D. Sharma, Interaction of shallow water waves, Stud. Appl. Math., 121 (2008), 1-25. doi: 10.1111/j.1467-9590.2008.00402.x. Google Scholar

[20]

T. Raja Sekhar and V. D. Sharma, Riemann problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl., 11 (2010), 619-636. doi: 10.1016/j.nonrwa.2008.10.036. Google Scholar

[21]

A. Sen and T. Raja Sekhar, Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation, Commun. Pure Appl. Anal., 18 (2019), 931-942. Google Scholar

[22]

A. SenT. Raja Sekhar and V. D. Sharma, Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws, Quart. Appl. Math., 75 (2017), 539-554. doi: 10.1090/qam/1466. Google Scholar

[23]

V. Sharanya and G. P. Raja Sekhar, Thermocapillary migration of a spherical drop in an arbitrary transient Stokes flow, Phys. Fluids, 27 (2015), 063104.Google Scholar

[24]

V. D. Sharma, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves, CRC Press, 2010. doi: 10.1201/9781439836910. Google Scholar

[25]

C. Shen, Wave interactions and stability of the Riemann solutions for the chromatography equations, J. Math. Anal. Appl., 365 (2010), 609-618. doi: 10.1016/j.jmaa.2009.11.037. Google Scholar

[26]

J. Smoller, Shock Waves and Reaction-diffusion Equations, Vol. 258, Springer Science & Business Media, 2012. Google Scholar

[27]

M. Sun, Interactions of elementary waves for the Aw–Rascle model, SIAM J. Appl. Math., 69 (2009), 1542-1558. doi: 10.1137/080731402. Google Scholar

[28]

M. Sun, A note on the interactions of elementary waves for the AR traffic flow model without vacuum, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1503-1512. doi: 10.1016/S0252-9602(11)60336-6. Google Scholar

[29]

S. K. Wilson, The levelling of paint films, IMA J. Appl. Math., 50 (1993), 149-166. doi: 10.1093/imamat/50.2.149. Google Scholar

Figure 1.  Elementary wave curves passing through a fixed state $ (b_l, h_l) $ in the $ (b, h) $-plane. Three elementary wave curves are identified, namely a shock wave curve (labelled as $ S $), a rarefaction wave curve (labelled as $ R $) and a contact discontinuity curve (labelled as $ J $)
Figure 2.  Solution structure of Riemann problem in the $ (x, t) $-plane. Three constant states, namely $ (h_l, b_l) $, $ (h_{\ast}, b_{\ast}) $ and $ (h_r, b_r) $ are separated by the elementary waves
Figure 3.  Exact solution of thickness parameter $ h $ and concentration gradient $ b $ at $ t = 0.95 $ with $ b_l = 0.8 $, $ h_l = 1.0 $, $ b_r = 1.8 $ and $ h_r = 1.0 $
Figure 4.  Exact solution of thickness parameter $ h $ and concentration gradient $ b $ at $ t = 0.95 $ with $ b_l = 0.8 $, $ h_l = 1.0 $, $ b_r = 0.3 $ and $ h_r = 0.5 $
Figure 5.  Wave interactions when $ b_lh_l>b_mh_m>b_rh_r $
Figure 6.  Wave interactions when $ b_lh_l\leq b_mh_m\leq b_rh_r $
Figure 7.  Wave interactions when $ b_lh_l\leq b_mh_m $ and $ b_rh_r<b_mh_m $
Figure 8.  Wave interactions when $ b_lh_l\leq b_mh_m $ and $ b_rh_r<b_mh_m $
Figure 10.  Wave interactions when $ b_lh_l>b_mh_m $ and $ b_mh_m\leq b_rh_r $
Figure 9.  Wave interactions when $ b_lh_l>b_mh_m $ and $ b_mh_m\leq b_rh_r $
Table 1.  Initial data and solution for the Riemann problem
Test $ h_l $ $ b_l $ $ h_r $ $ b_r $ $ b_{\ast} $ $ h_{\ast} $ Result
1 $ 1.0 $ $ 0.8 $ $ 1.0 $ $ 1.8 $ $ 1.20 $ $ 0.667 $ $ J+R $
2 $ 1.0 $ $ 0.8 $ $ 0.5 $ $ 0.3 $ $ 0.693 $ $ 1.155 $ $ J+S $
Test $ h_l $ $ b_l $ $ h_r $ $ b_r $ $ b_{\ast} $ $ h_{\ast} $ Result
1 $ 1.0 $ $ 0.8 $ $ 1.0 $ $ 1.8 $ $ 1.20 $ $ 0.667 $ $ J+R $
2 $ 1.0 $ $ 0.8 $ $ 0.5 $ $ 0.3 $ $ 0.693 $ $ 1.155 $ $ J+S $
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