Exact solutions to interfacial flows with kinetic undercooling in a  Hele-Shaw cell of time-dependent gap

Xuming Xie; Sahar Almashaan

doi:10.26637/mjm11S/002

Abstract

Hele-Shaw cells where the top plate is moving uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on interfacial flows with linear and nonlinear kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap. We obtain some exact solutions of the moving boundary problems when the initial shape is a circle, an ellipse or an annular domain. For the nonlinear case, a linear stability analysis is also presented for the circular solutions. The methodology is to use complex analysis and PDE theory.

Keywords:

Hele-Shaw flow, nonlinear kinetic undercooling, exact solution, Schwarz function, Laplace equation

Mathematics Subject Classification:

35Q35, 76S05

Authors Imprint References Funding Related Articles Downloads

Xuming Xie Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA. https://orcid.org/0000-0002-1160-7612
Sahar Almashaan Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA. https://orcid.org/0000-0002-8964-8000

Pages: 27-42
Date Published: 01-10-2023
Vol. 11 No. S (2023): Malaya Journal of Matematik (MJM): Special Issue Dedicated to Professor Gaston M. N'Guérékata’s 70th Birthday

P. H. A. A NJOS , E. O. D IAS AND J. A. M IRANDA , Kinetic undercooling in Hele-Shaw flows, Phys. Rev. E, 92 DOI: https://doi.org/10.1103/PhysRevE.92.043019

(2015), 043019.

J. M. B ACK , S. W. M C C UE , M. N. H SIEH , AND T. J. M ORONEY , The effect of surface tension and kinetic

undercooling on a radially-symmetric melting problem, Appl. Math. Comp., 229 (2014), 41-52. DOI: https://doi.org/10.1016/j.amc.2013.12.003

G. C ARVALHO , H. G AD ˆ ELHA , AND J. M IRANDA , Elastic fingering in rotating Hele–Shaw flows, Phys. Rev. E, 89 (2014), 053019. DOI: https://doi.org/10.1103/PhysRevE.89.053019

S. J. C HAPMAN , 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: https://doi.org/10.1017/S0956792599003848

S. J. C HAPMAN AND J. R. K ING , The selection of Saffman-Taylor fingers by kinetic undercooling, Journal of

Engineering Mathematics, 46 (2003), 1-32.

C.Y. C HEN , C.H. C HEN , AND J. M IRANDA , Numerical study of miscible fingering in a time dependent gap Hele-Shaw cell, Phys. Rev. E, 71 (2005), 056304. DOI: https://doi.org/10.1103/PhysRevE.71.056304

R. C OMBESCOT , V. H AKIM , T. D OMBRE , Y. P OMEAU AND A. P UMIR , Shape selection for Saffman-Taylor fingers, Physical Review Letter, 56 (1986), 2036-2039. DOI: https://doi.org/10.1103/PhysRevLett.56.2036

R. C OMBESCOT , V. H AKIM , T. D OMBRE ; Y. P OMEAU AND A. P UMIR , Analytic theory of Saffman-Taylor

fingers, Physical Review A, 37(1988), 1270-1283. DOI: https://doi.org/10.1103/PhysRevA.37.1270

M. C. D ALLASTON AND S. W. M C C UE , Corner and finger formation in Hele-Shaw flow with kinetic

undercooling regularization, European Journal of Applied Mathematics, 25 (2014), 707-727. DOI: https://doi.org/10.1017/S0956792514000230

P.J. D AVIS , The Schwarz function and its applications, The Mathematical Association of America, 1974.

E. D IAS AND J. M IRANDA , Control of radial fingering patterns: A weakly nonlinear approach, Phys. Rev. E, 81 (2010), 016312. DOI: https://doi.org/10.1103/PhysRevE.81.016312

E. D IAS AND J. M IRANDA , Determining the number of fingers in the lifting Hele-Shaw problem, Phys. Rev. E, 88 (2013), 043002. DOI: https://doi.org/10.1103/PhysRevE.88.043002

E. D IAS AND J. M IRANDA , Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows,

Phys. Rev. E, 87 (2013), 053015.

J.D. E VANS AND J.R. K ING , Asymptotic results for the Stefan problem with kinetic undercooling, Q.J. Mech. Appl. Math., 53 (2000), 449-473. DOI: https://doi.org/10.1093/qjmam/53.3.449

B.P.J G ARDINER , S.W M C C UE , M.C D ALLASTON AND T.J M ORONEY , Saffman-Taylor fingers with kinetic undercooling, Physical Review E, 91 (2015), 023016. DOI: https://doi.org/10.1103/PhysRevE.91.023016

A. H E , J. L OWENGRUB , AND A. B ELMONTE , Modeling an elastic fingering instability in a reactive Hele-Shaw flow, SIAM J. Appl. Math., 72 (2012), pp. 842–856. DOI: https://doi.org/10.1137/110844313

D.C. H ONG AND J.S. L ANGER , Analytic theory for the selection of Saffman-Taylor fingers, Phys. Rev. Lett., 56 (1986), 2032-2035. DOI: https://doi.org/10.1103/PhysRevLett.56.2032

S.D. H OWISON , Complex variable methods in a Hele-Shaw moving boundary problems, Euro. J. Appl. Math. 3 (1992), 209-224. DOI: https://doi.org/10.1017/S0956792500000802

D. K ESSLER AND H. L EVINE , The theory of Saffman-Taylor finger, Phys. Rev. A, 33 (1986), 2634-2639. DOI: https://doi.org/10.1103/PhysRevA.33.2634

J.R. K ING AND J.D. E VANS , Regularization by kinetic undercooling of blow-up in the ill-posed Stefan

problem, SIAM J. Appl. Math., 65 (2005), 1677-1707. DOI: https://doi.org/10.1137/04060528X

K. M ALAIKAH , T. V. S AVINA , AND A. A. N EPOMNYASHCHY , Hele-Shaw flow with a time-dependent gap: The Schwarz function approach to the interior problem. Contemporary Mathematics (667) 2016, p199-210. DOI: https://doi.org/10.1090/conm/667/13540

J.W. M CLEAN AND P.G. S AFFMAN , The effect of surface tension on the shape of fingers in a Hele Shaw cell, J. Fluid Mech., 102 (1981), 455-469. DOI: https://doi.org/10.1017/S0022112081002735

J. N ASE , D. D ERKS , AND A. L INDNER , Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell, Phys. Fluids, 23 (2011), 123101. DOI: https://doi.org/10.1063/1.3659140

N. B. P LESHCHINSKII AND M. R EISSIG , Hele-Shaw flows with nonlinear kinetic undercooling regularization, Nonlinear Anal., 50 (2002), 191-203. DOI: https://doi.org/10.1016/S0362-546X(01)00745-3

M. R EISSIG , S. V. R OGOSIN , AND F. H UBNER , Analytical and numerical treatment of a complex model for Hele-Shaw moving boundary value problems with kinetic undercooling regularization, Eur. J. Appl. Math.,

(1999), 561-579.

L. A. R OMERO , The fingering problem in a Hele-Shaw cell, Ph.D thesis, California Institute of Technology,

P.G. S AFFMAN , Viscous Fingering in Hele-shaw cells, J. Fluid Mech., 173 (1986), 73-94. DOI: https://doi.org/10.1017/S0022112086001088

P.G. S AFFMAN AND G.I.T AYLOR , The penetration of a fluid into a porous medium of Hele-Shaw cell

containing a more viscous fluid, Proc. R. Soc. London Ser. A, 245 (1958), 312-329. DOI: https://doi.org/10.1098/rspa.1958.0085

M. S HELLEY , F. T IAN , AND K. W LODARSKI , Hele-Shaw flow and pattern formation in a time-dependent gap, Nonlinearity, 10 (1997), 1471–1495. DOI: https://doi.org/10.1088/0951-7715/10/6/005

S. S INHA , T. D UTTA , AND S. T ARAFDAR , Adhesion and fingering in the lifting Hele–Shaw cell: Role of the substrate, Eur. Phys. J. E, 25 (2008), 267–275. DOI: https://doi.org/10.1140/epje/i2007-10289-9

B.I. S HRAIMAN , On velocity selection and the Saffman-Taylor problem, Phys. Rev. Lett., 56 (1986), 2028- DOI: https://doi.org/10.1103/PhysRevLett.56.2028

S. T ANVEER , The effect of surface tension on the shape of a Hele-Shaw cell bubble, Physics of Fluids, 29 DOI: https://doi.org/10.1063/1.865831

(1986), 3537- 3548.

S. T ANVEER , Analytic theory for the selection of symmetric Saffman-Taylor finger. Phys. Fluids, 30 (1987), DOI: https://doi.org/10.1063/1.866225

-1605.

S.T ANVEER AND X. X IE , Analyticity and Nonexistence of Classical Steady Hele-Shaw Fingers,

Communications on pure and applied mathematics, 56 (2003), 353-402. DOI: https://doi.org/10.1002/cpa.3030

G.I. T AYLOR AND P.G. S AFFMAN , A note on the motion of bubbles in a Hele-Shaw cell and Porous medium, Q. J. Mech. Appl. Math., 17 (1959), 265-279. DOI: https://doi.org/10.1093/qjmam/12.3.265

T IAN , F. R. , On the breakdown of Hele-Shaw solutions with non-zero surface tension I, Nonlinear Sci., 5 DOI: https://doi.org/10.1007/BF01209023

(1995), 479-494.

T IAN , F. R. , 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: https://doi.org/10.1007/BF02198454

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

J. M. V ANDEN -B ROECK , Fingers in a Hele-Shaw cell with surface tension, Phys. Fluids, 26 (1983), 2033- DOI: https://doi.org/10.1063/1.864406

X. X IE AND S. T ANVEER , Rigorous results in steady finger selection in viscous fingering, Arch. rational

mech. anal., 166 (2003), 219-286. DOI: https://doi.org/10.1007/s00205-002-0235-4

X. X IE , Rigorous results in existence and selection of Saffman-Taylor fingers by kinetic undercooling,

European Journal of Applied Mathematics, 30 (2019), 63-116. DOI: https://doi.org/10.1017/S0956792517000390

X. X IE , Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-

Shaw cell. Discrete & Continuous Dynamical Systems - B,26 (2021), 4663-4680. DOI: https://doi.org/10.3934/dcdsb.2020307

X. X IE , Classical solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-

Shaw cell, submitted, 2022.

M. Z HAO , X. L I , W. Y ING , A.B ELMONTE , J. L OWENGRUB , AND S. L I , Computation of a shrinking interface in a Hele-Shaw cell, SIAM J. Sci. Comput., 40 (2018), B1206–B1228. DOI: https://doi.org/10.1137/18M1172533