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doi: 10.3934/dcdsb.2020307

Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell

Xuming Xie ,

Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

* Corresponding author: Xuming Xie

Received  May 2020 Revised  August 2020 Published  October 2020

Hele-Shaw cells where the top plate is lifted uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on an interfacial flow with kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap. We obtain the local existence of analytic solution of the moving boundary problem when the initial data is analytic. The methodology is to use complex analysis and reduce the free boundary problem to a Riemann-Hilbert problem and an abstract Cauchy-Kovalevskaya evolution problem.

Keywords: Interfacial flow, kinetic undercooling regulation, Riemann-Hilbert problem, abstract Cauchy-Kovalevskaya problem.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76S05.
Citation: Xuming Xie. Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020307
References:
[1]

P. H. A. Anjos, E. O. Dias and J. A. Miranda, Kinetic undercooling in Hele-Shaw flows, Phys. Rev. E, 92 (2015), 043019. doi: 10.1103/PhysRevE.92.043019.  Google Scholar

[2]

J. M. Back, S. W. McCue, M. H.-N. Hsieh and T. J. Moroney, The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem, Appl. Math. Comp., 229 (2014), 41-52.  doi: 10.1016/j.amc.2013.12.003.  Google Scholar

[3]

G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, McGraw-Hill, 1966.  Google Scholar

[4]

G. Carvalho, H. Gadêlha and J. Miranda, Elastic fingering in rotating Hele–Shaw flows, Phys. Rev. E, 89 (2014), 053019. doi: 10.1103/PhysRevE.89.053019.  Google Scholar

[5]

S. J. Chapman, On the role of Stokes lines in the selection of Saffman-Taylor fingers with small surface tension, Eur. J. Appl Math., 10 (1999), 513-534.  doi: 10.1017/S0956792599003848.  Google Scholar

[6]

S. J. Chapman and J. R. King, The selection of Saffman-Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46 (2003), 1-32.  doi: 10.1023/A:1022860705459.  Google Scholar

[7]

C. Y. Chen, C. H. Chen and J. Miranda, Numerical study of miscible fingering in a timedependent gap Hele-Shaw cell, Phys. Rev. E, 71 (2005), 056304. Google Scholar

[8]

R. Combescot, T. Dombre, V. Hakim, Y. Pomeau and A. Pumir, Shape selection for Saffman-Taylor fingers, Physical Review Letter, 56 (1986), 2036-2039.  doi: 10.1103/PhysRevLett.56.2036.  Google Scholar

[9]

R. Combescot, V. Hakim, T. Dombre, Y. Pomeau and A. Pumir, Analytic theory of Saffman-Taylor fingers, Physical Review A, 37 (1988), 1270-1283.  doi: 10.1103/PhysRevA.37.1270.  Google Scholar

[10]

R. Combescot and T. Dombre, Selection in the Saffman-Taylor bubble and asymmetrical finger problem, Phys. Rev. A, 38 (1988), 2573-2581.  doi: 10.1103/PhysRevA.38.2573.  Google Scholar

[11]

M. C. Dallaston and S. W. McCue, Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularization, European Journal of Applied Mathematics, 25 (2014), 707-727.  doi: 10.1017/S0956792514000230.  Google Scholar

[12]

E. O. Dias and J. A. Miranda, Control of radial fingering patterns: A weakly nonlinear approach, Phys. Rev. E, 81 (2010), 016312. doi: 10.1103/PhysRevE.81.016312.  Google Scholar

[13]

E. O. Dias and J. A. Miranda, Determining the number of fingers in the lifting Hele-Shaw problem, Phys. Rev. E, 88 (2013), 043002. doi: 10.1103/PhysRevE.88.043002.  Google Scholar

[14]

E. O. Dias and J. A. Miranda, Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows, Phys. Rev. E, 87 (2013), 053015. doi: 10.1103/PhysRevE.87.053015.  Google Scholar

[15]

J. D. Evans and J. R. King, Asymptotic results for the Stefan problem with kinetic undercooling, Q. J. Mech. Appl. Math., 53 (2000), 449-473.  doi: 10.1093/qjmam/53.3.449.  Google Scholar

[16]

B. P. J. Gardiner, S. W. McCue, M. C. Dallaston and T. J. Moroney, Saffman-Taylor fingers with kinetic undercooling, Physical Review E, 91 (2015), 023016. doi: 10.1103/PhysRevE.91.023016.  Google Scholar

[17]

A. He, J. Lowengrub and A. Belmonte, Modeling an elastic fingering instability in a reactive Hele-Shaw flow, SIAM J. Appl. Math., 72 (2012), 842-856.  doi: 10.1137/110844313.  Google Scholar

[18]

D. C. Hong and J. S. Langer, Analytic theory for the selection of Saffman-Taylor fingers, Phys. Rev. Lett., 56 (1986), 2032-2035.   Google Scholar

[19]

D. Kessler and H. Levine, The theory of Saffman-Taylor finger, Phys. Rev. A, 33 (1986), 2634-2639.   Google Scholar

[20]

J. R. King and J. D. Evans, Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem, SIAM J. Appl. Math., 65 (2005), 1677-1707.  doi: 10.1137/04060528X.  Google Scholar

[21] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar
[22]

J. W. Mclean and P. G. Saffman, The effect of surface tension on the shape of fingers in a Hele Shaw cell, J. Fluid Mech., 102 (1981), 455-469.  doi: 10.1017/S0022112081002735.  Google Scholar

[23]

J. Nase, D. Derks and A. Lindner, Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell, Phys. Fluids, 23 (2011), 123101. doi: 10.1063/1.3659140.  Google Scholar

[24]

L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differential Geometry, 6 (1972), 561-576.  doi: 10.4310/jdg/1214430643.  Google Scholar

[25]

T. Nishida, A note on a theorem of Nirenberg, J. Differential Geometry, 12 (1977), 629-633.  doi: 10.4310/jdg/1214434231.  Google Scholar

[26]

N. B. Pleshchinskii and M. Reissig, Hele-Shaw flows with nonlinear kinetic undercooling regularization, Nonlinear Anal., 50 (2002), 191-203.  doi: 10.1016/S0362-546X(01)00745-3.  Google Scholar

[27]

L. Reis and J. Miranda, Controlling fingering instabilities in nonflat Hele-Shaw geometries, Phys. Rev. E, 84 (2011), 066313. Google Scholar

[28]

M. Reissig, S. V. Rogosin and F. Hubner, Analytical and numerical treatment of a complex model for Hele-Shaw moving boundary value problems with kinetic undercooling regularization, Eur. J. Appl. Math., 10 (1999), 561-579.  doi: 10.1017/S0956792599003939.  Google Scholar

[29]

L. A. Romero, The Fingering Problem in a Hele-Shaw Cell, Ph.D thesis, California Institute of Technology, 1981. Google Scholar

[30]

P. G. Saffman, Viscous fingering in Hele-shaw cells, J. Fluid Mech., 173 (1986), 73-94.  doi: 10.1017/S0022112086001088.  Google Scholar

[31]

P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous liquid, Proc. R. Soc. London Ser. A, 245 (1958), 312-329.  doi: 10.1098/rspa.1958.0085.  Google Scholar

[32]

M. J. Shelley, F.-R. Tian and K. Wlodarski, Hele-Shaw flow and pattern formation in a time-dependent gap, Nonlinearity, 10 (1997), 1471-1495.  doi: 10.1088/0951-7715/10/6/005.  Google Scholar

[33]

S. Sinha, T. Dutta and S. Tarafdar, Adhesion and fingering in the lifting Hele–Shaw cell: Role of the substrate, Eur. Phys. J. E, 25 (2008), 267-275.  doi: 10.1140/epje/i2007-10289-9.  Google Scholar

[34]

B. I. Shraiman, Velocity selection and the Saffman-Taylor problem, Phys. Rev. Lett., 56 (1986), 2028-2031.  doi: 10.1103/PhysRevLett.56.2028.  Google Scholar

[35]

S. Tanveer, The effect of surface tension on the shape of a Hele-Shaw cell bubble, Physics of Fluids, 29 (1986), 3537-3548.  doi: 10.1063/1.865831.  Google Scholar

[36]

S. Tanveer, Analytic theory for the selection of a symmetric Saffman-Taylor finger in a Hele–Shaw cell, Phys. Fluids, 30 (1987), 1589-1605.  doi: 10.1063/1.866225.  Google Scholar

[37]

S. Tanveer and X. Xie, Analyticity and nonexistence of classical steady Hele-Shaw fingers, Communications on Pure and Applied Mathematics, 56 (2003), 353-402.  doi: 10.1002/cpa.3030.  Google Scholar

[38]

G. Taylor and P. G. Saffman, A note on the motion of bubbles in a Hele-Shaw cell and porous medium, Q. J. Mech. Appl. Math., 12 (1959), 265-279.  doi: 10.1093/qjmam/12.3.265.  Google Scholar

[39]

F. R. Tian, On the breakdown of Hele-Shaw solutions with non-zero surface tension, Nonlinear Sci., 5 (1995), 479-494.  doi: 10.1007/BF01209023.  Google Scholar

[40]

F. R. Tian, A Cauchy integral approach to Hele-Shaw Problems with a free boundary: The zero surface tension case, Arch. Rat. Mech. Anal., 135 (1996), 175-196.  doi: 10.1007/BF02198454.  Google Scholar

[41]

F.-R. Tian, Hele-Shaw problems in multidimensional spaces, Nonlinear Sci., 10 (2000), 275-290.  doi: 10.1007/s003329910011.  Google Scholar

[42]

J.-M. Vanden-Broeck, Fingers in a Hele-Shaw cell with surface tension, Phys. Fluids, 26 (1983), 2033-2034.  doi: 10.1063/1.864406.  Google Scholar

[43]

X. Xie and S. Tanveer, Rigorous results in steady finger selection in viscous fingering, Arch. Rational Mech. Anal., 166 (2003), 219-286.  doi: 10.1007/s00205-002-0235-4.  Google Scholar

[44]

X. Xie, Rigorous results in existence and selection of Saffman-Taylor fingers by kinetic undercooling, European Journal of Applied Mathematics, 30 (2019), 63-116.  doi: 10.1017/S0956792517000390.  Google Scholar

[45]

M. Zhao, X. Li, W. Ying and A.Belmonte, J. Lowengrub and S. Li, Computation of a shrinking interface in a Hele-Shaw cell, SIAM J. Sci. Comput., 40 (2018), B1206–B1228. doi: 10.1137/18M1172533.  Google Scholar

show all references

References:
[1]

P. H. A. Anjos, E. O. Dias and J. A. Miranda, Kinetic undercooling in Hele-Shaw flows, Phys. Rev. E, 92 (2015), 043019. doi: 10.1103/PhysRevE.92.043019.  Google Scholar

[2]

J. M. Back, S. W. McCue, M. H.-N. Hsieh and T. J. Moroney, The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem, Appl. Math. Comp., 229 (2014), 41-52.  doi: 10.1016/j.amc.2013.12.003.  Google Scholar

[3]

G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, McGraw-Hill, 1966.  Google Scholar

[4]

G. Carvalho, H. Gadêlha and J. Miranda, Elastic fingering in rotating Hele–Shaw flows, Phys. Rev. E, 89 (2014), 053019. doi: 10.1103/PhysRevE.89.053019.  Google Scholar

[5]

S. J. Chapman, On the role of Stokes lines in the selection of Saffman-Taylor fingers with small surface tension, Eur. J. Appl Math., 10 (1999), 513-534.  doi: 10.1017/S0956792599003848.  Google Scholar

[6]

S. J. Chapman and J. R. King, The selection of Saffman-Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46 (2003), 1-32.  doi: 10.1023/A:1022860705459.  Google Scholar

[7]

C. Y. Chen, C. H. Chen and J. Miranda, Numerical study of miscible fingering in a timedependent gap Hele-Shaw cell, Phys. Rev. E, 71 (2005), 056304. Google Scholar

[8]

R. Combescot, T. Dombre, V. Hakim, Y. Pomeau and A. Pumir, Shape selection for Saffman-Taylor fingers, Physical Review Letter, 56 (1986), 2036-2039.  doi: 10.1103/PhysRevLett.56.2036.  Google Scholar

[9]

R. Combescot, V. Hakim, T. Dombre, Y. Pomeau and A. Pumir, Analytic theory of Saffman-Taylor fingers, Physical Review A, 37 (1988), 1270-1283.  doi: 10.1103/PhysRevA.37.1270.  Google Scholar

[10]

R. Combescot and T. Dombre, Selection in the Saffman-Taylor bubble and asymmetrical finger problem, Phys. Rev. A, 38 (1988), 2573-2581.  doi: 10.1103/PhysRevA.38.2573.  Google Scholar

[11]

M. C. Dallaston and S. W. McCue, Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularization, European Journal of Applied Mathematics, 25 (2014), 707-727.  doi: 10.1017/S0956792514000230.  Google Scholar

[12]

E. O. Dias and J. A. Miranda, Control of radial fingering patterns: A weakly nonlinear approach, Phys. Rev. E, 81 (2010), 016312. doi: 10.1103/PhysRevE.81.016312.  Google Scholar

[13]

E. O. Dias and J. A. Miranda, Determining the number of fingers in the lifting Hele-Shaw problem, Phys. Rev. E, 88 (2013), 043002. doi: 10.1103/PhysRevE.88.043002.  Google Scholar

[14]

E. O. Dias and J. A. Miranda, Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows, Phys. Rev. E, 87 (2013), 053015. doi: 10.1103/PhysRevE.87.053015.  Google Scholar

[15]

J. D. Evans and J. R. King, Asymptotic results for the Stefan problem with kinetic undercooling, Q. J. Mech. Appl. Math., 53 (2000), 449-473.  doi: 10.1093/qjmam/53.3.449.  Google Scholar

[16]

B. P. J. Gardiner, S. W. McCue, M. C. Dallaston and T. J. Moroney, Saffman-Taylor fingers with kinetic undercooling, Physical Review E, 91 (2015), 023016. doi: 10.1103/PhysRevE.91.023016.  Google Scholar

[17]

A. He, J. Lowengrub and A. Belmonte, Modeling an elastic fingering instability in a reactive Hele-Shaw flow, SIAM J. Appl. Math., 72 (2012), 842-856.  doi: 10.1137/110844313.  Google Scholar

[18]

D. C. Hong and J. S. Langer, Analytic theory for the selection of Saffman-Taylor fingers, Phys. Rev. Lett., 56 (1986), 2032-2035.   Google Scholar

[19]

D. Kessler and H. Levine, The theory of Saffman-Taylor finger, Phys. Rev. A, 33 (1986), 2634-2639.   Google Scholar

[20]

J. R. King and J. D. Evans, Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem, SIAM J. Appl. Math., 65 (2005), 1677-1707.  doi: 10.1137/04060528X.  Google Scholar

[21] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar
[22]

J. W. Mclean and P. G. Saffman, The effect of surface tension on the shape of fingers in a Hele Shaw cell, J. Fluid Mech., 102 (1981), 455-469.  doi: 10.1017/S0022112081002735.  Google Scholar

[23]

J. Nase, D. Derks and A. Lindner, Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell, Phys. Fluids, 23 (2011), 123101. doi: 10.1063/1.3659140.  Google Scholar

[24]

L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differential Geometry, 6 (1972), 561-576.  doi: 10.4310/jdg/1214430643.  Google Scholar

[25]

T. Nishida, A note on a theorem of Nirenberg, J. Differential Geometry, 12 (1977), 629-633.  doi: 10.4310/jdg/1214434231.  Google Scholar

[26]

N. B. Pleshchinskii and M. Reissig, Hele-Shaw flows with nonlinear kinetic undercooling regularization, Nonlinear Anal., 50 (2002), 191-203.  doi: 10.1016/S0362-546X(01)00745-3.  Google Scholar

[27]

L. Reis and J. Miranda, Controlling fingering instabilities in nonflat Hele-Shaw geometries, Phys. Rev. E, 84 (2011), 066313. Google Scholar

[28]

M. Reissig, S. V. Rogosin and F. Hubner, Analytical and numerical treatment of a complex model for Hele-Shaw moving boundary value problems with kinetic undercooling regularization, Eur. J. Appl. Math., 10 (1999), 561-579.  doi: 10.1017/S0956792599003939.  Google Scholar

[29]

L. A. Romero, The Fingering Problem in a Hele-Shaw Cell, Ph.D thesis, California Institute of Technology, 1981. Google Scholar

[30]

P. G. Saffman, Viscous fingering in Hele-shaw cells, J. Fluid Mech., 173 (1986), 73-94.  doi: 10.1017/S0022112086001088.  Google Scholar

[31]

P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous liquid, Proc. R. Soc. London Ser. A, 245 (1958), 312-329.  doi: 10.1098/rspa.1958.0085.  Google Scholar

[32]

M. J. Shelley, F.-R. Tian and K. Wlodarski, Hele-Shaw flow and pattern formation in a time-dependent gap, Nonlinearity, 10 (1997), 1471-1495.  doi: 10.1088/0951-7715/10/6/005.  Google Scholar

[33]

S. Sinha, T. Dutta and S. Tarafdar, Adhesion and fingering in the lifting Hele–Shaw cell: Role of the substrate, Eur. Phys. J. E, 25 (2008), 267-275.  doi: 10.1140/epje/i2007-10289-9.  Google Scholar

[34]

B. I. Shraiman, Velocity selection and the Saffman-Taylor problem, Phys. Rev. Lett., 56 (1986), 2028-2031.  doi: 10.1103/PhysRevLett.56.2028.  Google Scholar

[35]

S. Tanveer, The effect of surface tension on the shape of a Hele-Shaw cell bubble, Physics of Fluids, 29 (1986), 3537-3548.  doi: 10.1063/1.865831.  Google Scholar

[36]

S. Tanveer, Analytic theory for the selection of a symmetric Saffman-Taylor finger in a Hele–Shaw cell, Phys. Fluids, 30 (1987), 1589-1605.  doi: 10.1063/1.866225.  Google Scholar

[37]

S. Tanveer and X. Xie, Analyticity and nonexistence of classical steady Hele-Shaw fingers, Communications on Pure and Applied Mathematics, 56 (2003), 353-402.  doi: 10.1002/cpa.3030.  Google Scholar

[38]

G. Taylor and P. G. Saffman, A note on the motion of bubbles in a Hele-Shaw cell and porous medium, Q. J. Mech. Appl. Math., 12 (1959), 265-279.  doi: 10.1093/qjmam/12.3.265.  Google Scholar

[39]

F. R. Tian, On the breakdown of Hele-Shaw solutions with non-zero surface tension, Nonlinear Sci., 5 (1995), 479-494.  doi: 10.1007/BF01209023.  Google Scholar

[40]

F. R. Tian, A Cauchy integral approach to Hele-Shaw Problems with a free boundary: The zero surface tension case, Arch. Rat. Mech. Anal., 135 (1996), 175-196.  doi: 10.1007/BF02198454.  Google Scholar

[41]

F.-R. Tian, Hele-Shaw problems in multidimensional spaces, Nonlinear Sci., 10 (2000), 275-290.  doi: 10.1007/s003329910011.  Google Scholar

[42]

J.-M. Vanden-Broeck, Fingers in a Hele-Shaw cell with surface tension, Phys. Fluids, 26 (1983), 2033-2034.  doi: 10.1063/1.864406.  Google Scholar

[43]

X. Xie and S. Tanveer, Rigorous results in steady finger selection in viscous fingering, Arch. Rational Mech. Anal., 166 (2003), 219-286.  doi: 10.1007/s00205-002-0235-4.  Google Scholar

[44]

X. Xie, Rigorous results in existence and selection of Saffman-Taylor fingers by kinetic undercooling, European Journal of Applied Mathematics, 30 (2019), 63-116.  doi: 10.1017/S0956792517000390.  Google Scholar

[45]

M. Zhao, X. Li, W. Ying and A.Belmonte, J. Lowengrub and S. Li, Computation of a shrinking interface in a Hele-Shaw cell, SIAM J. Sci. Comput., 40 (2018), B1206–B1228. doi: 10.1137/18M1172533.  Google Scholar

Figure 1.  Lifting plate Hele-Shaw flow
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