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

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  • 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:

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  • 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 $
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