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Dynamics and Stability of Air Bubbles in a Porous Medium

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Abstract

A numerical method is developed for reliably computing the evolution of the boundary of a multiply connected water-saturated domain with air bubbles in the case when the pressure inside them depends on the bubble volume. It is assumed that the distance between the gas bubbles is comparable with their size. Gas bubbles can be near an extended phase transition boundary separating a porous medium flow and a domain saturated with a mixture of air and water vapor. The numerical method is verified by comparing the numerical solution of a test problem with its analytical solution. Caused by finite-amplitude perturbations of the phase interface, the deformation of an air bubble in an extended horizontal water-saturated porous layer with a constant pressure gradient is studied. It is shown that the instability of the bubble boundary with respect to finite perturbations leads to the splitting of the bubble. An analysis of the numerical solution shows that, although all circular bubbles move at the same velocity irrespective of their size, nevertheless, due to instability, a portion of the bubble boundary where the air displaces the fluid moves faster than an opposite portion where the fluid displaces the air. As a result, nearby bubbles are capable of merging before splitting.

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REFERENCES

  1. V. A. Shargatov, A. T. Il’Ichev, and G. G. Tsypkin, “Dynamics and stability of moving fronts of water evaporation in a porous medium,” Int. J. Heat Mass Transfer 83, 552–561 (2015).

    Article  Google Scholar 

  2. A. T. Il’ichev and G. G. Tsypkin, “Instabilities of uniform filtration flows with phase transition,” J. Exp. Theor. Phys. 107 (4), 699–711 (2008).

    Article  Google Scholar 

  3. A. T. Il’ichev and G. G. Tsypkin, “Catastrophic transition to instability of evaporation front in a porous medium,” Eur. J. Mech. 27 (6), 665–677 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. G. Tsypkin and A. T. Il’ichev, “Gravitational stability of the interface in water over steam geothermal reservoirs,” Transp. Porous Media 55 (2), 183–199 (2004).

    Article  MathSciNet  Google Scholar 

  5. A. T. Il’ichev and V. A. Shargatov, “Dynamics of water evaporation fronts,” Comput. Math. Math. Phys. 53 (9), 1350–1370 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. A. Gubin, A. V. Krivosheev, and V. A. Shargatov, “Existence of a steady-state water evaporation front in a horizontally extended low-permeability region,” Fluid Dyn. 50 (2), 240–249 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. A. Shargatov, “Instability of a liquid-vapor phase transition front in inhomogeneous wettable porous media,” Fluid Dyn. 52 (1), 146–157 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  8. Z. H. Khan and D. Pritchard, “Liquid-vapor fronts in porous media: Multiplicity and stability of front positions,” Int. J. Heat Mass Transfer 61, 1–17 (2013).

    Article  Google Scholar 

  9. C. A. J. Fletcher, Computational Techniques for Fluid Dynamics (Springer-Verlag, Berlin, 1988), Vol. 2.

    Book  MATH  Google Scholar 

  10. C. A. Brebbia, J. C. F. Telles, and W. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer-Verlag, Berlin, 1984).

    Book  MATH  Google Scholar 

  11. I. K. Lifanov, Method of Singular Integral Equations and Numerical Experiments (Yanus, Moscow, 1995) [in Russian].

    Google Scholar 

  12. N. M. Gunter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics (Gostekhteorizdat, Moscow, 1953; Ungar, New York, 1967).

  13. P. A. Krutitskii, “Method of boundary integral equations in the mixed problem for the Laplace equation with an arbitrary partition of the boundary,” Differ. Equations 37 (1), 78–89 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  14. P. A. Krutitskii, “The mixed problem for the Laplace equation in a three-dimensional multiply connected domain,” Differ. Equations 35 (9), 1193–1200 (1999).

    MathSciNet  MATH  Google Scholar 

  15. S. Li, J. S. Lowengrub, and P. H. Leo, “A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell,” J. Comput. Phys. 225 (1), 554–567 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Cristini and J. Lowengrub, “Three-dimensional crystal growth: II. Nonlinear simulation and control of the Mullins–Sekerka instability,” J. Crystal. Growth 266, 552–567 (2004).

    Article  Google Scholar 

  17. J. Caldwell, “Solutions of potential problems using the reduction to Fredholm integral equations,” J. Appl. Phys. 119, 5583–5587 (1980).

    Article  Google Scholar 

  18. C. Constanda, “On the solution of the Dirichlet problem for the two-dimensional Laplace equation,” Proc. Am. Math. Soc. 119 (3), 877–884 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  19. D. N. Nikolskii, “Evolution of a liquid-liquid interface in inhomogeneous layers,” Comput. Math. Math. Phys. 50 (7), 1205–1211 (2010).

    Article  MathSciNet  Google Scholar 

  20. D. N. Nikolskii, “Mathematical simulation of the evolution of a liquid-liquid interface in piecewise inhomogeneous layers of complex geological structure,” Comput. Math. Math. Phys. 53 (6), 858–865 (2013).

    Article  MathSciNet  Google Scholar 

  21. Yu. A. Itkulova, O. A. Abramova, N. A. Gumerov, and I. Sh. Akhatov, “Simulation of bubble dynamics in three-dimensional potential flows on heterogeneous computing systems using the fast multipole and boundary element methods,” Vychisl. Metody Program. 15 (2), 239–257 (2014).

    Google Scholar 

  22. M. C. Dallaston and S. W. McCue, “An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell,” ANZIAM J. 54, 309–326 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. C. Dallaston and S. W. McCue, “Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularization,” Nonlinearity 26, 1639–1665 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. G. L. Vasconcelos, “Multiple bubbles and fingers in a Hele-Shaw channel: Complete set of steady solutions,” J. Fluid Mech. 780, 299–326 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  25. M. M. Alimov, “Unsteady motion of a bubble in a Hele-Shaw cell,” Fluid Dyn. 51 (2), 253–265 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  26. M. M. Alimov, “Exact solution of the Muskat–Leibenzon problem for a growing elliptic bubble,” Fluid Dyn. 51 (5), 660–671 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  27. J. W. McLean and P. G. Saffman, “Stability of bubbles in a Hele-Shaw cell,” Phys. Fluids 30 (9), 2624–2635 (1987).

    Article  MATH  Google Scholar 

  28. X. Li and Y. C. Yortsos, “Bubble growth and stability in an effective porous medium,” Phys. Fluids A 6 (5), 1663–1676 (1994).

    Article  MATH  Google Scholar 

  29. K. Spayd, M. Shearer, and Z. Hu, “Stability of plane waves in two phase porous media flow,” Appl. Anal. 91 (2), 293–308 (2012).

    Article  MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

The author is grateful to G.G. Tsypkin for his interest in this work and to the reviewers for their comments.This work was supported by the Russian Science Foundation, project no. 16-11-10195.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    V. A. Shargatov

  2. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Moscow, Russia

    V. A. Shargatov

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  1. V. A. Shargatov
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Correspondence to V. A. Shargatov.

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Translated by I. Ruzanova

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Shargatov, V.A. Dynamics and Stability of Air Bubbles in a Porous Medium. Comput. Math. and Math. Phys. 58, 1172–1187 (2018). https://doi.org/10.1134/S0965542518070151

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