matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/2411-9326-2024-4-82-105matmess-139Research ArticleМАТЕМАТИКАTwo-phase radial viscous fingering problem in a Hele-Shaw cell with surface tension. I. Classical solvabilityTwo-phase radial viscous fingering problem in a Hele-Shaw cell with surface tension. I. Classical solvabilityTaniA.TaniA.

Atusi Tani 

Yokohama 223-8522 

Atusi Tani 

Yokohama 223-8522 

tani@math.keio.ac.jpTaniH.TaniH.

Hisasi Tani 

Yokohama, 220-6001 

Hisasi Tani 

Yokohama, 220-6001 

hisasitani@gmail.comDepartment of Mathematics, Keio UniversityЯпонияDepartment of Mathematics, Keio UniversityJapanJANUSЯпонияJANUSJapan20243012202431482105Copyright © Tani A., Tani H., 20242024Tani A., Tani H.Tani A., Tani H.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/139

We study the problem of radial fingering in immiscible liquid/liquid flow in a Hele-Shaw cell under injection or suction of liquid of less viscosity. The addition of new effects of viscosity and surface tension at the liquid/liquid interface brings our theory into a much better agreement with experiments than other theories. In this paper the classical solvability of the original Hele-Shaw problem (a nonlinear problem with a free boundary for elliptic equations) is established by means of its parabolic regularization with a small parameter ε (> 0) in the time-derivative term and the nonhomogeneous term (the parabolic regularized Hele-Shaw problem) and by vanishing along some subsequence of {ε > 0}. The similar result for a one-phase problem has been already studied in “H. Tani, Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, Sib. J. Pure Appl. Math., 16, 79–92 (2016);” “A. Tani and H. Tani, On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, J. Appl. Mech. Tech. Phys., 65, No. 5, 178–191 (2024).”

We study the problem of radial fingering in immiscible liquid/liquid flow in a Hele-Shaw cell under injection or suction of liquid of less viscosity. The addition of new effects of viscosity and surface tension at the liquid/liquid interface brings our theory into a much better agreement with experiments than other theories. In this paper the classical solvability of the original Hele-Shaw problem (a nonlinear problem with a free boundary for elliptic equations) is established by means of its parabolic regularization with a small parameter ε (> 0) in the time-derivative term and the nonhomogeneous term (the parabolic regularized Hele-Shaw problem) and by vanishing along some subsequence of {ε > 0}. The similar result for a one-phase problem has been already studied in “H. Tani, Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, Sib. J. Pure Appl. Math., 16, 79–92 (2016);” “A. Tani and H. Tani, On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension, J. Appl. Mech. Tech. Phys., 65, No. 5, 178–191 (2024).”

radial viscous fingeringtwo-phase Hele-Shaw problemsurface tensionclassical solutionradial viscous fingeringtwo-phase Hele-Shaw problemsurface tensionclassical solutionReferencesHele-Shaw H. S., “The flow of water,” Nature, 58, 33–36 (1898).Hele-Shaw H. S., “The flow of water,” Nature, 58, 33–36 (1898).Hill S., “Channeling in packed columns,” Chem. Eng. Sci., 1, 247–253 (1952).Hill S., “Channeling in packed columns,” Chem. Eng. Sci., 1, 247–253 (1952).Saffman P. G. and Taylor G. I., “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid,” Proc. R. Soc. Lond., Ser. A, 245, 312–329 (1958).Saffman P. G. and Taylor G. I., “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid,” Proc. R. Soc. Lond., Ser. A, 245, 312–329 (1958).Bensimon D., Kadanoff L. P., Liang S., Shraiman B. I., and Tang C., “Viscous flow in two dimensions,” Rev. Mod. Phys., 58, 977–999 (1986).Bensimon D., Kadanoff L. P., Liang S., Shraiman B. I., and Tang C., “Viscous flow in two dimensions,” Rev. Mod. Phys., 58, 977–999 (1986).Kessler D. A., Koplik J., and Levine H., “Pattern selection in fingering growth phenomena,” Adv. Phys., 37, 255–339 (1988).Kessler D. A., Koplik J., and Levine H., “Pattern selection in fingering growth phenomena,” Adv. Phys., 37, 255–339 (1988).McCloud K. V. and Mahrer J. V., “Experimental perturbations to Saffman–Taylor flow,” Phys. Rep., 260, 139–185 (1995).McCloud K. V. and Mahrer J. V., “Experimental perturbations to Saffman–Taylor flow,” Phys. Rep., 260, 139–185 (1995).Paterson L., “Radial fingering in a Hele-Shaw cell,” J. Fluid Mech., 113, 513–529 (1981).Paterson L., “Radial fingering in a Hele-Shaw cell,” J. Fluid Mech., 113, 513–529 (1981).Park C.-W. and Homsy G., “Two-phase displacement in Hele-Shaw cell: theory, J. Fluid Mech., 139, 291–308 (1984).Park C.-W. and Homsy G., “Two-phase displacement in Hele-Shaw cell: theory, J. Fluid Mech., 139, 291–308 (1984).Homsy G., “Viscous fingering in porous media,” Annu. Rev. Fluid Mech., 19, 271–311 (1987).Homsy G., “Viscous fingering in porous media,” Annu. Rev. Fluid Mech., 19, 271–311 (1987).Martyushev L. M. and Birzina A. I., “Specific feature of the loss of stability during radial displacement of fluid in the Hele-Shaw cell,” J. Phys., Condens. Matter, 20, 045201–08 (2008).Martyushev L. M. and Birzina A. I., “Specific feature of the loss of stability during radial displacement of fluid in the Hele-Shaw cell,” J. Phys., Condens. Matter, 20, 045201–08 (2008).Tani H. and Tani A., “Effect of the wetting layer on the fingering pattern in a Hele-Shaw cell,” J. Phys. Soc. Japan, 83, 034401 (2014).Tani H. and Tani A., “Effect of the wetting layer on the fingering pattern in a Hele-Shaw cell,” J. Phys. Soc. Japan, 83, 034401 (2014).Kim H., Funada T., Joseph D. D., and Homsy G., “Viscous potential flow analysis of radial fingering in a Hele-Shaw cell,” Phys. Fluids, 21, 074106-1–9 (2009).Kim H., Funada T., Joseph D. D., and Homsy G., “Viscous potential flow analysis of radial fingering in a Hele-Shaw cell,” Phys. Fluids, 21, 074106-1–9 (2009).Tani H., “Weakly nonlinear analysis on radial growing interface with the effect of viscous normal stress,” J. Mol. Liquids, 200, 38–41 (2014).Tani H., “Weakly nonlinear analysis on radial growing interface with the effect of viscous normal stress,” J. Mol. Liquids, 200, 38–41 (2014).Miranda J. A. and Widom M., “Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis,” Phys., D120, 315–328 (1998).Miranda J. A. and Widom M., “Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis,” Phys., D120, 315–328 (1998).Gustafsson B. and Vasil’ev A., Conformal and Potential Analysis in Hele-Shaw Cells, Birkha¨user (2000).Gustafsson B. and Vasil’ev A., Conformal and Potential Analysis in Hele-Shaw Cells, Birkha¨user (2000).Bazali˘ı B. V., “On a proof of the classical solvability of the Hele-Shaw problem with a free surface [in Russian],” Ukr. Mat. Zh., 50, 1452–1462 (1998).Bazali˘ı B. V., “On a proof of the classical solvability of the Hele-Shaw problem with a free surface [in Russian],” Ukr. Mat. Zh., 50, 1452–1462 (1998).Antontsev S. N., Gon˛calves C. R., and Meirmanov A. M., “Exact estimates for the classical solutions to the free boundary problem in the Hele-Shaw cell,” Adv. Differ. Equ., 8, No. 10, 1259–1280 (2003).Antontsev S. N., Gon˛calves C. R., and Meirmanov A. M., “Exact estimates for the classical solutions to the free boundary problem in the Hele-Shaw cell,” Adv. Differ. Equ., 8, No. 10, 1259–1280 (2003).Tani A. and Tani H., “Classical solvability of the radial viscous fingering problem in a HeleShaw cell,” Mat. Zametki SVFU, 25, No. 3, 92–114 (2018).Tani A. and Tani H., “Classical solvability of the radial viscous fingering problem in a HeleShaw cell,” Mat. Zametki SVFU, 25, No. 3, 92–114 (2018).Tani A. and Tani H., “On the uniqueness of the classical solutions of the radial viscous fingering problems in a Hele-Shaw cell,” J. Sib. Fed. Univ., Math. Phys., 14, No. 4, 475–482 (2021).Tani A. and Tani H., “On the uniqueness of the classical solutions of the radial viscous fingering problems in a Hele-Shaw cell,” J. Sib. Fed. Univ., Math. Phys., 14, No. 4, 475–482 (2021).Tani H., “Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension,” Sib. J. Pure Appl. Math., 16, 79–92 (2016).Tani H., “Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension,” Sib. J. Pure Appl. Math., 16, 79–92 (2016).Tani A. and Tani H., “On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension,” J. Appl. Mech. Tech. Phys., 65, No. 5, 178–191 (2024).Tani A. and Tani H., “On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension,” J. Appl. Mech. Tech. Phys., 65, No. 5, 178–191 (2024).Tani A. and Tani H., “Classical solvability of the two-phase radial viscous fingering problem in a Hele-Shaw cell,” in: Mathematical Fluid Dynamics, Present and Future (Y. Shibata and Y. Suzuki, eds.), Springer Proc. Math. Stat., pp. 317–348, 183 (2016).Tani A. and Tani H., “Classical solvability of the two-phase radial viscous fingering problem in a Hele-Shaw cell,” in: Mathematical Fluid Dynamics, Present and Future (Y. Shibata and Y. Suzuki, eds.), Springer Proc. Math. Stat., pp. 317–348, 183 (2016).Escher J. and Simonett G., “Classical solutions of multidimensional Hele-Shaw model,” SIAM J. Math. Anal., 28, 1028–1047 (1997).Escher J. and Simonett G., “Classical solutions of multidimensional Hele-Shaw model,” SIAM J. Math. Anal., 28, 1028–1047 (1997).Escher J. and Simonett G., “Classical solutions for Hele-Shaw models with surface tension,” Adv. Differ. Equ., 2, 619–642 (1997).Escher J. and Simonett G., “Classical solutions for Hele-Shaw models with surface tension,” Adv. Differ. Equ., 2, 619–642 (1997).Antontsev S. N., Gon˛calves C. R., and Meirmanov A. M., “Local existence of classical solutions to the well-posed Hele-Shaw problem,” Port. Math., 59, 435–452 (2002).Antontsev S. N., Gon˛calves C. R., and Meirmanov A. M., “Local existence of classical solutions to the well-posed Hele-Shaw problem,” Port. Math., 59, 435–452 (2002).Howison S. D., “A Note on the two-phase Hele-Shaw problem,” J. Fluid Mech., 409, 243–249 (2000).Howison S. D., “A Note on the two-phase Hele-Shaw problem,” J. Fluid Mech., 409, 243–249 (2000).Bazali˘ı B. V., “On estimates for the solution of a model conjugation problem in the theory of problems with a free boundary [in Russian],” Differ. Uravn., 33, 1374–1381 (1997).Bazali˘ı B. V., “On estimates for the solution of a model conjugation problem in the theory of problems with a free boundary [in Russian],” Differ. Uravn., 33, 1374–1381 (1997).Bazali˘ı B. V., “Stefan problem for the Laplace equation with regard for the curvature of the free boundary [in Russian],” Ukr. Mat. Zh., 49, 1299–1315 (1997).Bazali˘ı B. V., “Stefan problem for the Laplace equation with regard for the curvature of the free boundary [in Russian],” Ukr. Mat. Zh., 49, 1299–1315 (1997).Ladyzhenskaya O. A., Solonnikov V. A., and Ural’tseva N. N., Linear and Quasi-Linear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).Ladyzhenskaya O. A., Solonnikov V. A., and Ural’tseva N. N., Linear and Quasi-Linear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).Solonnikov V. A., “On boundary value problems for linear parabolic systems of differential equations of general form [in Russian],” Tr. Mat. Inst. Steklov, 83, 3–163 (1965).Solonnikov V. A., “On boundary value problems for linear parabolic systems of differential equations of general form [in Russian],” Tr. Mat. Inst. Steklov, 83, 3–163 (1965).Tani A., “Two-phase free boundary problem for compressible viscous fluid motion,” J. Math. Kyoto Univ., 24, 243–267 (1984).Tani A., “Two-phase free boundary problem for compressible viscous fluid motion,” J. Math. Kyoto Univ., 24, 243–267 (1984).

The authors declare that there are no conflicts of interest present.