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May  2020, 19(5): 2641-2653. doi: 10.3934/cpaa.2020115

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

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India

* Corresponding author

Received  February 2019 Revised  September 2019 Published  March 2020

Fund Project: Research support from University Grant Commission, Government of India (Sr. No. 21215409 47, Ref No: 20=12=2015(ⅱ)EU-V) is gratefully acknowledged by the first author

We study the interactions between classical elementary waves and delta shock wave in quasilinear hyperbolic system of conservation laws. This governing system describes a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and P$ \acute{e} $clet numbers. This system is one of the example of non-strictly hyperbolic system whose Riemann solution consists of delta shock wave as well as classical elementary waves such as shock waves, rarefaction waves and contact discontinuities. The global structure of the perturbed Riemann solutions are constructed and analyzed case by case when delta shock wave is involved.

Citation: Anupam Sen, T. Raja Sekhar. Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2641-2653. doi: 10.3934/cpaa.2020115
References:
[1]

C. H. Chang and E. I. Franses, Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms, Colloid Surf. A - Physicochem. Eng. Asp., 100 (1995), 1-45. 

[2]

G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.

[3]

G. Q. ChenM. Slemrod and D. Wang, Vanishing viscosity method for transonic flow, Arch. Ration. Mech. Anal., 189 (2008), 159-188.  doi: 10.1007/s00205-007-0101-5.

[4]

J. J. A. Conn, Stability and Dynamics of Anti-surfactant Solutions, Ph.D Thesis, University of Strathclyde, Glasgow, 2017.

[5]

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, 11. doi: 10.1103/PhysRevE.93.043121.

[6]

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

[7]

C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Ration. Mech. Anal., 52 (1973), 1-9.  doi: 10.1007/BF00249087.

[8]

V. G. Danilov and V. M. Shelkovich, Dynamics of Propagation and interaction of $\delta$-shock waves in conservation law systems, J. Differ. Equ., 211 (2005), 333-381.  doi: 10.1016/j.jde.2004.12.011.

[9]

V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Q. Appl. Math., 63 (2005), 401-427.  doi: 10.1090/S0033-569X-05-00961-8.

[10]

B. T. Hayes and P. G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity, 9 (1996), 1547-1563.  doi: 10.1088/0951-7715/9/6/009.

[11]

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), 9. doi: 10.1063/1.4915283.

[12]

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

[13]

K. T. Joseph, A Riemann problem whose viscosity solutions contain $\delta$-measures, Asymptotic Anal., 7 (1993), 105-120. 

[14]

H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, IMA J. Appl. Math., 77 (2012), 340-350.  doi: 10.1093/imamat/hxs014.

[15]

H. Kalisch and D. Mitrovic, Singular solutions of a fully nonlinear 2$\times$ 2 system of conservation laws, Proc. Edinb. Math. Soc., 55 (2012), 711-729.  doi: 10.1017/S0013091512000065.

[16]

B. L. Keyfitz and H. C. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal., 72 (1980), 219-241.  doi: 10.1007/BF00281590.

[17]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differ. Equ., 118 (1995), 420-451.  doi: 10.1006/jdeq.1995.1080.

[18]

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.  doi: 10.1016/S0009-2509(00)00525-X.

[19]

F. A. Long and G. C. Nutting, The relative surface tension of potassium chloride solutions by a differential bubble pressure method1, J. Amer. Chem. Soc., 64 (1942), 2476-2482. 

[20]

Mi nhajulT. 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, Commun. Pure Appl. Anal., 18 (2019), 3367-3386.  doi: 10.3934/cpaa.2019152.

[21]

M. Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Meth. Appl. Sci., 27 (2004), 931-955.  doi: 10.1002/mma.480.

[22]

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.  doi: 10.3934/cpaa.2019045.

[23]

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

[24]

C. Shen, Delta shock wave solution for a symmetric Keyfitz-Kranzer system, Appl. Math. Lett., 77 (2018), 35-43.  doi: 10.1016/j.aml.2017.09.016.

[25]

W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Am. Math. Soc., 654 (1999). doi: 10.1090/memo/0654.

[26]

M. Sun, Interactions of delta shock waves for the chromatography equations, Appl. Math. Lett., 26 (2013), 631-637.  doi: 10.1016/j.aml.2013.01.002.

[27]

D. TanT. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differ. Equ., 112 (1994), 1-32.  doi: 10.1006/jdeq.1994.1093.

[28]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.  doi: 10.2307/1999646.

[29]

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

show all references

References:
[1]

C. H. Chang and E. I. Franses, Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms, Colloid Surf. A - Physicochem. Eng. Asp., 100 (1995), 1-45. 

[2]

G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.

[3]

G. Q. ChenM. Slemrod and D. Wang, Vanishing viscosity method for transonic flow, Arch. Ration. Mech. Anal., 189 (2008), 159-188.  doi: 10.1007/s00205-007-0101-5.

[4]

J. J. A. Conn, Stability and Dynamics of Anti-surfactant Solutions, Ph.D Thesis, University of Strathclyde, Glasgow, 2017.

[5]

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, 11. doi: 10.1103/PhysRevE.93.043121.

[6]

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

[7]

C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Ration. Mech. Anal., 52 (1973), 1-9.  doi: 10.1007/BF00249087.

[8]

V. G. Danilov and V. M. Shelkovich, Dynamics of Propagation and interaction of $\delta$-shock waves in conservation law systems, J. Differ. Equ., 211 (2005), 333-381.  doi: 10.1016/j.jde.2004.12.011.

[9]

V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Q. Appl. Math., 63 (2005), 401-427.  doi: 10.1090/S0033-569X-05-00961-8.

[10]

B. T. Hayes and P. G. LeFloch, Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity, 9 (1996), 1547-1563.  doi: 10.1088/0951-7715/9/6/009.

[11]

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), 9. doi: 10.1063/1.4915283.

[12]

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

[13]

K. T. Joseph, A Riemann problem whose viscosity solutions contain $\delta$-measures, Asymptotic Anal., 7 (1993), 105-120. 

[14]

H. Kalisch and D. Mitrovic, Singular solutions for the shallow-water equations, IMA J. Appl. Math., 77 (2012), 340-350.  doi: 10.1093/imamat/hxs014.

[15]

H. Kalisch and D. Mitrovic, Singular solutions of a fully nonlinear 2$\times$ 2 system of conservation laws, Proc. Edinb. Math. Soc., 55 (2012), 711-729.  doi: 10.1017/S0013091512000065.

[16]

B. L. Keyfitz and H. C. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal., 72 (1980), 219-241.  doi: 10.1007/BF00281590.

[17]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differ. Equ., 118 (1995), 420-451.  doi: 10.1006/jdeq.1995.1080.

[18]

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.  doi: 10.1016/S0009-2509(00)00525-X.

[19]

F. A. Long and G. C. Nutting, The relative surface tension of potassium chloride solutions by a differential bubble pressure method1, J. Amer. Chem. Soc., 64 (1942), 2476-2482. 

[20]

Mi nhajulT. 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, Commun. Pure Appl. Anal., 18 (2019), 3367-3386.  doi: 10.3934/cpaa.2019152.

[21]

M. Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Meth. Appl. Sci., 27 (2004), 931-955.  doi: 10.1002/mma.480.

[22]

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.  doi: 10.3934/cpaa.2019045.

[23]

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

[24]

C. Shen, Delta shock wave solution for a symmetric Keyfitz-Kranzer system, Appl. Math. Lett., 77 (2018), 35-43.  doi: 10.1016/j.aml.2017.09.016.

[25]

W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Am. Math. Soc., 654 (1999). doi: 10.1090/memo/0654.

[26]

M. Sun, Interactions of delta shock waves for the chromatography equations, Appl. Math. Lett., 26 (2013), 631-637.  doi: 10.1016/j.aml.2013.01.002.

[27]

D. TanT. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differ. Equ., 112 (1994), 1-32.  doi: 10.1006/jdeq.1994.1093.

[28]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.  doi: 10.2307/1999646.

[29]

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

Figure 1.  Wave curves in $ (h, b) $ phase plane
Figure 2.  $ J+R $ when $ 0<h_lb_l<h_rb_r $
Figure 3.  $ J+S $ when $ 0<h_rb_r<h_lb_l $
Figure 4.  $ \delta{S} $ when $ b_{l, r}>0 $, $ h_l>0 $ and $ h_r = 0 $
Figure 5.  When $ 0<h_mb_m<h_lb_l $ and $ h_r = 0 $
Figure 6.  When $ 0<h_lb_l<h_mb_m $ and $ h_r = 0 $
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