Article Contents
Article Contents

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

• * Corresponding author
• 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.

Mathematics Subject Classification: Primary: 35L45, 35L65; Secondary: 35Q35, 35L67.

 Citation:

• 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 $1.0$ $0.8$ $0.5$ $0.3$ $0.693$ $1.155$ $J+S$
•  [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. [2] A. Bressan, Hyperbolic Systems of Conservation Laws, Vol. 20, Oxford University Press, Oxford, 2000. [3] J. J. A. Conn, B. R. Duffy, D. Pritchard, S. K. Wilson, P. J. Halling and K. Sefiane, Fluid-dynamical model for antisurfactants, Phys. Rev. E, 93 (2016), 043121. [4] J. J. A. 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, J. Engrg. Math., 107 (2017), 167-178.  doi: 10.1007/s10665-017-9924-8. [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. [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. [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. [8] S. D. Howison, J. A. Moriarty, J. R. Ockendon, E. 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. [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. [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. [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. [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. [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. [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. [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. [16] Mi nhajul, D. 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. [17] W. Overdiep, The levelling of paints, Progress in Organic Coatings, 14 (1986), 159-175. [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. [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. [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. [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. [22] A. Sen, T. 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. [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. [24] V. D. Sharma, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves, CRC Press, 2010. doi: 10.1201/9781439836910. [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. [26] J. Smoller, Shock Waves and Reaction-diffusion Equations, Vol. 258, Springer Science & Business Media, 2012. [27] M. Sun, Interactions of elementary waves for the Aw–Rascle model, SIAM J. Appl. Math., 69 (2009), 1542-1558.  doi: 10.1137/080731402. [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. [29] S. K. Wilson, The levelling of paint films, IMA J. Appl. Math., 50 (1993), 149-166.  doi: 10.1093/imamat/50.2.149.

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