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September  2021, 26(9): 4663-4680. doi: 10.3934/dcdsb.2020307

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

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

* Corresponding author: Xuming Xie

Received  May 2020 Revised  August 2020 Published  September 2021 Early access  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.

Citation: Xuming Xie. Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4663-4680. 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.

[2]

J. M. BackS. W. McCueM. 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.

[3]

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

[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.

[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.

[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.

[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.

[8]

R. CombescotT. DombreV. HakimY. Pomeau and A. Pumir, Shape selection for Saffman-Taylor fingers, Physical Review Letter, 56 (1986), 2036-2039.  doi: 10.1103/PhysRevLett.56.2036.

[9]

R. CombescotV. HakimT. DombreY. Pomeau and A. Pumir, Analytic theory of Saffman-Taylor fingers, Physical Review A, 37 (1988), 1270-1283.  doi: 10.1103/PhysRevA.37.1270.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

A. HeJ. 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.

[18]

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

[19]

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

[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.

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

[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.

[24]

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

[25]

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

[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.

[27]

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

[28]

M. ReissigS. 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.

[29]

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

[30]

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

[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.

[32]

M. J. ShelleyF.-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.

[33]

S. SinhaT. 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.

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

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.

[2]

J. M. BackS. W. McCueM. 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.

[3]

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

[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.

[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.

[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.

[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.

[8]

R. CombescotT. DombreV. HakimY. Pomeau and A. Pumir, Shape selection for Saffman-Taylor fingers, Physical Review Letter, 56 (1986), 2036-2039.  doi: 10.1103/PhysRevLett.56.2036.

[9]

R. CombescotV. HakimT. DombreY. Pomeau and A. Pumir, Analytic theory of Saffman-Taylor fingers, Physical Review A, 37 (1988), 1270-1283.  doi: 10.1103/PhysRevA.37.1270.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

A. HeJ. 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.

[18]

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

[19]

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

[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.

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

[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.

[24]

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

[25]

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

[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.

[27]

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

[28]

M. ReissigS. 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.

[29]

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

[30]

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

[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.

[32]

M. J. ShelleyF.-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.

[33]

S. SinhaT. 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.

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

Figure 1.  Lifting plate Hele-Shaw flow
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