A gradient stable scheme for a phase field model for the moving contact line problem

In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1,2,4]. The nonlinear version of the scheme is sem...

Full description

Saved in:

Bibliographic Details
Published in:	Journal of computational physics Vol. 231; no. 4; pp. 1372 - 1386
Main Authors:	Gao, Min, Wang, Xiao-Ping
Format:	Journal Article
Language:	English
Published:	Kidlington Elsevier Inc 20.02.2012 Elsevier
Subjects:	Accuracy Boundaries Cahn–Hilliard equation Computational techniques Contact Convex splitting Exact sciences and technology Free energy Mathematical analysis Mathematical methods in physics Mathematical models Moving contact line Navier-Stokes equations Phase field Physics Splitting Navier–Stokes equations Moving contact line Convex splitting Cahn–Hilliard equation Phase field Cahn-Hilliard equation Total energy Cahn Hilliard equation Boundary conditions Models Calculation Free energy Navier-Stokes equations Calculation methods
ISSN:	0021-9991, 1090-2716
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1,2,4]. The nonlinear version of the scheme is semi-implicit in time and is based on a convex splitting of the Cahn–Hilliard free energy (including the boundary energy) together with a projection method for the Navier–Stokes equations. We show, under certain conditions, the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Numerical tests are carried out to verify the accuracy and stability of the scheme. The behavior of the solution near the contact line is examined. It is verified that, when the interface intersects with the boundary, the consistent splitting scheme [21,22] for the Navier Stokes equations has the better accuracy for pressure.
AbstractList	In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. The nonlinear version of the scheme is semi-implicit in time and is based on a convex splitting of the Cahn-Hilliard free energy (including the boundary energy) together with a projection method for the Navier-Stokes equations. We show, under certain conditions, the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Numerical tests are carried out to verify the accuracy and stability of the scheme. The behavior of the solution near the contact line is examined. It is verified that, when the interface intersects with the boundary, the consistent splitting scheme and for the Navier Stokes equations has the better accuracy for pressure. In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1,2,4]. The nonlinear version of the scheme is semi-implicit in time and is based on a convex splitting of the Cahn–Hilliard free energy (including the boundary energy) together with a projection method for the Navier–Stokes equations. We show, under certain conditions, the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Numerical tests are carried out to verify the accuracy and stability of the scheme. The behavior of the solution near the contact line is examined. It is verified that, when the interface intersects with the boundary, the consistent splitting scheme [21,22] for the Navier Stokes equations has the better accuracy for pressure.
Author	Gao, Min Wang, Xiao-Ping
Author_xml	– sequence: 1 givenname: Min surname: Gao fullname: Gao, Min email: gaomin@ust.hk – sequence: 2 givenname: Xiao-Ping surname: Wang fullname: Wang, Xiao-Ping email: mawang@ust.hk
BackLink	http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=25929503$$DView record in Pascal Francis
BookMark	eNp9kcFO3DAQhq2KSl0oD8DNFyQu2c4kjhOLE0LQVlqpF3q2HHvCeuUki22QeHu8XcShB072jL9_Rvp8yk7mZSbGLhDWCCh_7NY7u1_XgFjqNWD7ha0QFFR1h_KErQBqrJRS-I2dprQDgL4V_Yo93PDHaJynOfOUzRCIJ7ulifi4RG74fmtSuXsKjk-Lo_Cvn7dUqhc_P3K7zNnYzIOfie_jUiZM39nX0YRE5-_nGft7f_dw-6va_Pn5-_ZmU9lGQq5GdJY6i41A20oSSogBhQXVu8EMBtpBtoMQUjhlZN0LN4ixQ2ylEH15HJszdnWcW_Y-PVPKevLJUghmpuU5aZQdNl0HnSro5TtqkjVhjGa2Pul99JOJr7puVa1aaAqHR87GJaVI4weCoA-m9U4X0_pg-tAqpkum-y9jfTbZFzPR-PBp8vqYpGLpxVPUyZavsOR8JJu1W_wn6TdkZpkn
CODEN	JCTPAH
CitedBy_id	crossref_primary_10_1007_s10444_023_10031_5 crossref_primary_10_1137_23M1546816 crossref_primary_10_1016_j_jcp_2019_109179 crossref_primary_10_1155_2014_871021 crossref_primary_10_1016_j_cnsns_2015_06_012 crossref_primary_10_1137_17M1125005 crossref_primary_10_1002_fld_4349 crossref_primary_10_1137_20M1317773 crossref_primary_10_1002_adfm_202213621 crossref_primary_10_1017_jfm_2022_33 crossref_primary_10_1016_j_jcp_2014_07_038 crossref_primary_10_1051_m2an_2025039 crossref_primary_10_1002_num_22341 crossref_primary_10_1007_s00205_013_0713_x crossref_primary_10_1016_j_jcp_2019_109170 crossref_primary_10_1016_j_jcp_2022_111149 crossref_primary_10_1017_jfm_2014_696 crossref_primary_10_1017_jfm_2018_428 crossref_primary_10_1137_130940992 crossref_primary_10_1016_j_jcp_2014_02_043 crossref_primary_10_1137_18M1223459 crossref_primary_10_1080_00036811_2015_1102893 crossref_primary_10_1016_j_cpc_2013_08_016 crossref_primary_10_1016_j_cma_2017_08_011 crossref_primary_10_1016_j_jcp_2017_10_002 crossref_primary_10_1016_j_jcp_2022_110968 crossref_primary_10_3390_ma16175932 crossref_primary_10_1017_jfm_2019_664 crossref_primary_10_1016_j_jcp_2012_07_027 crossref_primary_10_1007_s10494_015_9655_8 crossref_primary_10_3934_ipi_2013_7_947 crossref_primary_10_1007_s10444_020_09764_4 crossref_primary_10_1007_s10915_017_0448_1 crossref_primary_10_3934_dcdsb_2023163 crossref_primary_10_1002_fld_4200 crossref_primary_10_1007_s10915_019_00934_1 crossref_primary_10_1007_s10915_022_01863_2 crossref_primary_10_1007_s13160_014_0151_7 crossref_primary_10_1016_j_jcp_2017_01_026 crossref_primary_10_1016_j_ijmultiphaseflow_2017_12_016 crossref_primary_10_1007_s00205_019_01383_8 crossref_primary_10_1016_j_ijmultiphaseflow_2017_04_008 crossref_primary_10_1063_5_0220227 crossref_primary_10_1016_j_cpc_2019_106870 crossref_primary_10_1016_j_jcp_2019_109006 crossref_primary_10_1007_s10915_017_0391_1 crossref_primary_10_1016_j_jmaa_2018_06_075 crossref_primary_10_1016_j_jcp_2016_05_016 crossref_primary_10_1002_fld_5100 crossref_primary_10_1016_j_jcp_2019_108959 crossref_primary_10_1016_j_jcp_2014_12_046 crossref_primary_10_1016_j_cma_2016_09_003 crossref_primary_10_1016_j_jcp_2014_04_054 crossref_primary_10_1016_j_jcp_2020_109521 crossref_primary_10_1007_s00526_016_0992_9 crossref_primary_10_1016_j_apm_2020_02_022 crossref_primary_10_1016_j_jcp_2019_04_037 crossref_primary_10_1016_j_apm_2018_12_017
Cites_doi	10.1016/j.jcp.2003.07.035 10.1017/S0022112008001456 10.1017/S0022112006001935 10.1006/jcph.2001.6715 10.1103/PhysRevE.68.016306 10.1017/S0022112004000370 10.1016/j.cma.2005.10.010 10.1007/PL00005429 10.3934/dcds.2010.28.1669 10.1137/080738143 10.1090/S0025-5718-06-01915-6 10.1016/j.physa.2004.08.076 10.1016/j.jcp.2004.02.009 10.1016/j.jcp.2009.01.009 10.1016/j.jcp.2009.04.020 10.1016/j.physa.2009.01.026 10.1016/j.jcp.2007.06.028 10.1016/j.jcp.2003.07.009 10.1103/PhysRevE.68.066703 10.1016/j.jcp.2011.03.022 10.1016/j.jcp.2006.02.021 10.1137/0728069
ContentType	Journal Article
Copyright	2011 Elsevier Inc. 2015 INIST-CNRS
Copyright_xml	– notice: 2011 Elsevier Inc. – notice: 2015 INIST-CNRS
DBID	AAYXX CITATION IQODW 7SC 7SP 7U5 8FD JQ2 L7M L~C L~D
DOI	10.1016/j.jcp.2011.10.015
DatabaseName	CrossRef Pascal-Francis Computer and Information Systems Abstracts Electronics & Communications Abstracts Solid State and Superconductivity Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional
DatabaseTitle	CrossRef Technology Research Database Computer and Information Systems Abstracts – Academic Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Solid State and Superconductivity Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional
DatabaseTitleList	Technology Research Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Applied Sciences Physics
EISSN	1090-2716
EndPage	1386
ExternalDocumentID	25929503 10_1016_j_jcp_2011_10_015 S0021999111006152
GroupedDBID	--K --M -~X .~1 0R~ 1B1 1RT 1~. 1~5 29K 4.4 457 4G. 5GY 5VS 6OB 6TJ 7-5 71M 8P~ 8WZ 9JN A6W AABNK AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO AAYFN ABBOA ABFNM ABFRF ABJNI ABMAC ABNEU ABTAH ABXDB ABYKQ ACBEA ACDAQ ACFVG ACGFO ACGFS ACNCT ACNNM ACRLP ACZNC ADBBV ADEZE ADFGL ADIYS ADJOM ADMUD AEBSH AEFWE AEKER AENEX AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHZHX AIALX AIEXJ AIKHN AITUG AIVDX AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ASPBG AVWKF AXJTR AZFZN BBWZM BKOJK BLXMC CAG COF CS3 D-I DM4 DU5 EBS EFBJH EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA GBOLZ HLZ HME HMV HVGLF HZ~ IHE J1W K-O KOM LG5 LX9 LZ4 M37 M41 MO0 N9A NDZJH O-L O9- OAUVE OGIMB OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SBC SDF SDG SDP SES SEW SHN SPC SPCBC SPD SPG SSQ SSV SSZ T5K T9H TN5 UPT UQL WUQ XFK YQT ZMT ZU3 ZY4 ~02 ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD AFXIZ AGCQF AGRNS BNPGV IQODW SSH 7SC 7SP 7U5 8FD JQ2 L7M L~C L~D
ID	FETCH-LOGICAL-c360t-f1dce7c1341c56e4944b14c098dbaba05b65b4464d9a6284db4f7115644805bf3
ISICitedReferencesCount	81
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000300462100016&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0021-9991
IngestDate	Sun Nov 09 13:51:54 EST 2025 Mon Jul 21 09:15:00 EDT 2025 Sat Nov 29 06:46:04 EST 2025 Tue Nov 18 20:57:59 EST 2025 Fri Feb 23 02:18:52 EST 2024
IsPeerReviewed	true
IsScholarly	true
Issue	4
Keywords	Navier–Stokes equations Moving contact line Convex splitting Cahn–Hilliard equation Phase field Cahn-Hilliard equation Total energy Cahn Hilliard equation Boundary conditions Models Calculation Free energy Navier-Stokes equations Calculation methods
Language	English
License	https://www.elsevier.com/tdm/userlicense/1.0 CC BY 4.0
LinkModel	OpenURL
MergedId	FETCHMERGED-LOGICAL-c360t-f1dce7c1341c56e4944b14c098dbaba05b65b4464d9a6284db4f7115644805bf3
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23
PQID	1671377079
PQPubID	23500
PageCount	15
ParticipantIDs	proquest_miscellaneous_1671377079 pascalfrancis_primary_25929503 crossref_primary_10_1016_j_jcp_2011_10_015 crossref_citationtrail_10_1016_j_jcp_2011_10_015 elsevier_sciencedirect_doi_10_1016_j_jcp_2011_10_015
PublicationCentury	2000
PublicationDate	2012-02-20
PublicationDateYYYYMMDD	2012-02-20
PublicationDate_xml	– month: 02 year: 2012 text: 2012-02-20 day: 20
PublicationDecade	2010
PublicationPlace	Kidlington
PublicationPlace_xml	– name: Kidlington
PublicationTitle	Journal of computational physics
PublicationYear	2012
Publisher	Elsevier Inc Elsevier
Publisher_xml	– name: Elsevier Inc – name: Elsevier
References	Qian, Wang, Sheng (b0015) 2006; 1 Yang, Feng, Liu, Shen (b0070) 2006; 218 Choi, Lee, Jeong, Kim (b0040) 2009; 388 He, Glowinski, Wang (b0125) 2011; 230 Shen, Yang (b0080) 2010; 28 Mello, Filho (b0055) 2005; 347 Wise, Wang, Lowengrub (b0100) 2009; 47 Guermond, Shen (b0130) 2003; 192 Feng, He, Liu (b0085) 2007; 76 Guermond, Minev, Shen (b0110) 2006; 195 Kim, Kang, Lowengrub (b0090) 2004; 193 Wang, Qian, Sheng (b0010) 2008; 605 Yue, Feng, Liu, Shen (b0065) 2004; 515 Du, Nicolaides (b0030) 1991; 28 Furihata (b0035) 2001; 87 Brown, Cortez, Minion (b0115) 2001; 168 Qian, Wang, Sheng (b0005) 2003; 68 Shen, Yang (b0075) 2009; 228 Johnston, Liu (b0105) 2004; 199 Qian, Wang, Sheng (b0020) 2006; 564 D.J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, June 1998. Ding, Spelt, Shu (b0120) 2007; 266 Vollmayr-Lee, Rutenberg (b0060) 2003; 68 Sun (b0025) 1995; 64 Hu, Wise, Wang, Lowengrub (b0095) 2009; 228 Eyre (b0050) 1998 Eyre (10.1016/j.jcp.2011.10.015_b0050) 1998 Wise (10.1016/j.jcp.2011.10.015_b0100) 2009; 47 Qian (10.1016/j.jcp.2011.10.015_b0005) 2003; 68 Guermond (10.1016/j.jcp.2011.10.015_b0110) 2006; 195 Yang (10.1016/j.jcp.2011.10.015_b0070) 2006; 218 Mello (10.1016/j.jcp.2011.10.015_b0055) 2005; 347 Qian (10.1016/j.jcp.2011.10.015_b0015) 2006; 1 Furihata (10.1016/j.jcp.2011.10.015_b0035) 2001; 87 Kim (10.1016/j.jcp.2011.10.015_b0090) 2004; 193 10.1016/j.jcp.2011.10.015_b0045 Guermond (10.1016/j.jcp.2011.10.015_b0130) 2003; 192 Hu (10.1016/j.jcp.2011.10.015_b0095) 2009; 228 Du (10.1016/j.jcp.2011.10.015_b0030) 1991; 28 Wang (10.1016/j.jcp.2011.10.015_b0010) 2008; 605 Qian (10.1016/j.jcp.2011.10.015_b0020) 2006; 564 Feng (10.1016/j.jcp.2011.10.015_b0085) 2007; 76 Johnston (10.1016/j.jcp.2011.10.015_b0105) 2004; 199 Shen (10.1016/j.jcp.2011.10.015_b0080) 2010; 28 Ding (10.1016/j.jcp.2011.10.015_b0120) 2007; 266 Vollmayr-Lee (10.1016/j.jcp.2011.10.015_b0060) 2003; 68 Shen (10.1016/j.jcp.2011.10.015_b0075) 2009; 228 Sun (10.1016/j.jcp.2011.10.015_b0025) 1995; 64 Choi (10.1016/j.jcp.2011.10.015_b0040) 2009; 388 Brown (10.1016/j.jcp.2011.10.015_b0115) 2001; 168 He (10.1016/j.jcp.2011.10.015_b0125) 2011; 230 Yue (10.1016/j.jcp.2011.10.015_b0065) 2004; 515
References_xml	– volume: 266 start-page: 2078 year: 2007 end-page: 2095 ident: b0120 article-title: Diffuse interface model for incompressible two-phase flows with large density ratios publication-title: J. Comput. Phys. – volume: 605 start-page: 59 year: 2008 end-page: 78 ident: b0010 article-title: Moving contact line on chemically patterned surfaces publication-title: J. Fluid Mech. – volume: 230 start-page: 4991 year: 2011 end-page: 5009 ident: b0125 article-title: A least-squares/finite element method for the numerical solution of the Navier–Stokes–Cahn–Hilliard system modeling the motion of the contact line publication-title: J. Comput. Phys. – volume: 199 start-page: 221 year: 2004 end-page: 259 ident: b0105 article-title: Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term publication-title: J. Comput. Phys. – volume: 168 start-page: 464 year: 2001 end-page: 499 ident: b0115 article-title: Accurate projection methods for the incompressible Navier–Stokes equations publication-title: J. Comput. Phys. – volume: 347 start-page: 429 year: 2005 end-page: 443 ident: b0055 article-title: Numerical study of the Cahn–Hilliard equations in one, two and three dimensions publication-title: Phys. A – volume: 76 start-page: 539 year: 2007 end-page: 571 ident: b0085 article-title: Analysis of finite element approximations of a phase field model for two-phase fluids publication-title: Math. Comput. – start-page: 39 year: 1998 end-page: 46 ident: b0050 publication-title: Computational and Mathematical Models of Microstructural Evolution – volume: 87 start-page: 675 year: 2001 end-page: 699 ident: b0035 article-title: A stable and conservative finite difference scheme for the Cahn–Hilliard equation publication-title: Numer. Math. – volume: 68 start-page: 066703 year: 2003 ident: b0060 article-title: Fast and accurate coarsening simulation with an unconditionally stable time step publication-title: Phys. Rev. E – volume: 28 start-page: 1310 year: 1991 end-page: 1322 ident: b0030 article-title: Numerical analysis of a continuum model of phase transition publication-title: SIAM J. Numer. Anal. – volume: 228 start-page: 5323 year: 2009 end-page: 5339 ident: b0095 article-title: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase filed crystal equation publication-title: J. Comput. Phys. – volume: 47 start-page: 2269 year: 2009 end-page: 2288 ident: b0100 article-title: An energy-stable and convergent finite-difference scheme for the phase field crystal equation publication-title: SIAM J. Numer. Anal. – volume: 68 start-page: 016306 year: 2003 ident: b0005 article-title: Molecular scale contact line hydrodynamics of immiscible flows publication-title: Phys. Rev. E – volume: 64 start-page: 1463 year: 1995 ident: b0025 article-title: A second-order accurate linearized difference scheme for the two-dimensional Cahn–Hilliard equation publication-title: Math. Comput. – volume: 218 start-page: 417 year: 2006 end-page: 428 ident: b0070 article-title: Numerical simulations of jet pinching-off and drop formation using energetic variational phase-field method publication-title: J. Comput. Phys. – volume: 388 start-page: 1791 year: 2009 end-page: 1803 ident: b0040 article-title: An unconditionally gradient stable numerical method for solving the Allen–Cahn equation publication-title: Phys. A – reference: D.J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, June 1998. – volume: 195 start-page: 6011 year: 2006 end-page: 6045 ident: b0110 article-title: An overview of projection methods for incompressible flows publication-title: Comput. Methods Appl. Mech. Eng. – volume: 515 start-page: 293 year: 2004 end-page: 317 ident: b0065 article-title: A diffuse-interface method for simulating two-phase flows of complex fluids publication-title: J. Fluid Mech. – volume: 192 start-page: 262 year: 2003 end-page: 276 ident: b0130 article-title: A new class of truly consistent splitting schemes for incompressible flows publication-title: J. Comput. Phys. – volume: 1 start-page: 1 year: 2006 end-page: 52 ident: b0015 article-title: Molecular hydrodynamics of the moving contact line in two-phase immiscible flows publication-title: Commun. Comput. Phys. – volume: 564 start-page: 306 year: 2006 end-page: 333 ident: b0020 article-title: A variational approach to the moving contact line hydrodynamics publication-title: J. Fluid Mech. – volume: 228 start-page: 2978 year: 2009 end-page: 2992 ident: b0075 article-title: An efficient moving mesh spectral method for the phase-field model of two-phase flows publication-title: J. Comput. Phys. – volume: 193 start-page: 511 year: 2004 end-page: 543 ident: b0090 article-title: Conservative multigrid methods for Cahn–Hilliard fluids publication-title: J. Comput. Phys. – volume: 28 start-page: 1669 year: 2010 end-page: 1691 ident: b0080 article-title: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations publication-title: DCDS-A – volume: 193 start-page: 511 year: 2004 ident: 10.1016/j.jcp.2011.10.015_b0090 article-title: Conservative multigrid methods for Cahn–Hilliard fluids publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2003.07.035 – volume: 605 start-page: 59 year: 2008 ident: 10.1016/j.jcp.2011.10.015_b0010 article-title: Moving contact line on chemically patterned surfaces publication-title: J. Fluid Mech. doi: 10.1017/S0022112008001456 – volume: 564 start-page: 306 year: 2006 ident: 10.1016/j.jcp.2011.10.015_b0020 article-title: A variational approach to the moving contact line hydrodynamics publication-title: J. Fluid Mech. doi: 10.1017/S0022112006001935 – volume: 168 start-page: 464 issue: 2 year: 2001 ident: 10.1016/j.jcp.2011.10.015_b0115 article-title: Accurate projection methods for the incompressible Navier–Stokes equations publication-title: J. Comput. Phys. doi: 10.1006/jcph.2001.6715 – volume: 68 start-page: 016306 year: 2003 ident: 10.1016/j.jcp.2011.10.015_b0005 article-title: Molecular scale contact line hydrodynamics of immiscible flows publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.68.016306 – volume: 1 start-page: 1 year: 2006 ident: 10.1016/j.jcp.2011.10.015_b0015 article-title: Molecular hydrodynamics of the moving contact line in two-phase immiscible flows publication-title: Commun. Comput. Phys. – volume: 515 start-page: 293 year: 2004 ident: 10.1016/j.jcp.2011.10.015_b0065 article-title: A diffuse-interface method for simulating two-phase flows of complex fluids publication-title: J. Fluid Mech. doi: 10.1017/S0022112004000370 – volume: 195 start-page: 6011 year: 2006 ident: 10.1016/j.jcp.2011.10.015_b0110 article-title: An overview of projection methods for incompressible flows publication-title: Comput. Methods Appl. Mech. Eng. doi: 10.1016/j.cma.2005.10.010 – ident: 10.1016/j.jcp.2011.10.015_b0045 – volume: 87 start-page: 675 issue: 4 year: 2001 ident: 10.1016/j.jcp.2011.10.015_b0035 article-title: A stable and conservative finite difference scheme for the Cahn–Hilliard equation publication-title: Numer. Math. doi: 10.1007/PL00005429 – start-page: 39 year: 1998 ident: 10.1016/j.jcp.2011.10.015_b0050 – volume: 28 start-page: 1669 issue: 4 year: 2010 ident: 10.1016/j.jcp.2011.10.015_b0080 article-title: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations publication-title: DCDS-A doi: 10.3934/dcds.2010.28.1669 – volume: 47 start-page: 2269 issue: 3 year: 2009 ident: 10.1016/j.jcp.2011.10.015_b0100 article-title: An energy-stable and convergent finite-difference scheme for the phase field crystal equation publication-title: SIAM J. Numer. Anal. doi: 10.1137/080738143 – volume: 76 start-page: 539 issue: 258 year: 2007 ident: 10.1016/j.jcp.2011.10.015_b0085 article-title: Analysis of finite element approximations of a phase field model for two-phase fluids publication-title: Math. Comput. doi: 10.1090/S0025-5718-06-01915-6 – volume: 347 start-page: 429 year: 2005 ident: 10.1016/j.jcp.2011.10.015_b0055 article-title: Numerical study of the Cahn–Hilliard equations in one, two and three dimensions publication-title: Phys. A doi: 10.1016/j.physa.2004.08.076 – volume: 199 start-page: 221 year: 2004 ident: 10.1016/j.jcp.2011.10.015_b0105 article-title: Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2004.02.009 – volume: 228 start-page: 2978 year: 2009 ident: 10.1016/j.jcp.2011.10.015_b0075 article-title: An efficient moving mesh spectral method for the phase-field model of two-phase flows publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2009.01.009 – volume: 228 start-page: 5323 year: 2009 ident: 10.1016/j.jcp.2011.10.015_b0095 article-title: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase filed crystal equation publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2009.04.020 – volume: 64 start-page: 1463 issue: 212 year: 1995 ident: 10.1016/j.jcp.2011.10.015_b0025 article-title: A second-order accurate linearized difference scheme for the two-dimensional Cahn–Hilliard equation publication-title: Math. Comput. – volume: 388 start-page: 1791 year: 2009 ident: 10.1016/j.jcp.2011.10.015_b0040 article-title: An unconditionally gradient stable numerical method for solving the Allen–Cahn equation publication-title: Phys. A doi: 10.1016/j.physa.2009.01.026 – volume: 266 start-page: 2078 year: 2007 ident: 10.1016/j.jcp.2011.10.015_b0120 article-title: Diffuse interface model for incompressible two-phase flows with large density ratios publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2007.06.028 – volume: 192 start-page: 262 year: 2003 ident: 10.1016/j.jcp.2011.10.015_b0130 article-title: A new class of truly consistent splitting schemes for incompressible flows publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2003.07.009 – volume: 68 start-page: 066703 issue: 8 year: 2003 ident: 10.1016/j.jcp.2011.10.015_b0060 article-title: Fast and accurate coarsening simulation with an unconditionally stable time step publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.68.066703 – volume: 230 start-page: 4991 issue: 12 year: 2011 ident: 10.1016/j.jcp.2011.10.015_b0125 article-title: A least-squares/finite element method for the numerical solution of the Navier–Stokes–Cahn–Hilliard system modeling the motion of the contact line publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2011.03.022 – volume: 218 start-page: 417 year: 2006 ident: 10.1016/j.jcp.2011.10.015_b0070 article-title: Numerical simulations of jet pinching-off and drop formation using energetic variational phase-field method publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2006.02.021 – volume: 28 start-page: 1310 issue: 5 year: 1991 ident: 10.1016/j.jcp.2011.10.015_b0030 article-title: Numerical analysis of a continuum model of phase transition publication-title: SIAM J. Numer. Anal. doi: 10.1137/0728069
SSID	ssj0008548
Score	2.3814569
Snippet	In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the...
SourceID	proquest pascalfrancis crossref elsevier
SourceType	Aggregation Database Index Database Enrichment Source Publisher
StartPage	1372
SubjectTerms	Accuracy Boundaries Cahn–Hilliard equation Computational techniques Contact Convex splitting Exact sciences and technology Free energy Mathematical analysis Mathematical methods in physics Mathematical models Moving contact line Navier-Stokes equations Phase field Physics Splitting
Title	A gradient stable scheme for a phase field model for the moving contact line problem
URI	https://dx.doi.org/10.1016/j.jcp.2011.10.015 https://www.proquest.com/docview/1671377079
Volume	231
WOSCitedRecordID	wos000300462100016&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1090-2716 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0008548 issn: 0021-9991 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1bb9MwFLaqjgckxB1RBpOReKIKshMnjh8rNG5C0x4K9AnLcZzRakujtpv283d8C7uoE3vgJWrTxkl7vpxzfPL5Owi946SuIa0ViSnrOrH9kJKKcppY5ZPGaEor1w7o53d-cFDOZuJwMPgd18KcHfO2Lc_PRfdfTQ37wNh26ewdzN0PCjvgNRgdtmB22P6T4Sfjo5XjcW1sncAujIIJrDnx4t5q3P2BuDV2xDXfBqcnGp744oIlr9uFky7_DP1mtqSw2rWEiOVEXyTpc_TPaul5-T38foXa9GyulslhjJmh5GC5G2mSkstu1PI6hG-zFd1oGrz5_HKRwDlFmvnuPCHA0syLX99w3r6OsPiw0J3XVrW0O7_Y86pQ9rUA1tMKI2NtIWEIaYeA95JYDYKdlOeiHKKdydf92bc-Vpc587E6_KD43NsxAK9dx7bM5UGn1nA_Nb4Ryo2Y7hKV6WP0MJgHTzwynqCBaZ-iR2G2gYMvXz9D0wmOQMEeKNgDBQMgsMIOKNgBBTuguP0AFOyBggNQsAUKDkB5jn582p9-_JKEHhuJzgqySRpaa8O1lfXTeWGYYKyiTBNR1pWqFMmrIq8YK1gtVAGpTF2xhlOrMMRK-LDJXqBhu2zNS4RzZZguBWk048zq_Jm6MRkkuFobmDXzESLx_5M6CNDbPijHcqvdRuh9f0jn1Vdu-zKLRpEhffRpoQSA3XbY3hUD9idKc5g65CQbobfRohJ8r32gplqzPF1LWnAr2Em4eHWXC91F9__eVa_RcLM6NW_QPX22ma9XewGiF1u7pXg
linkProvider	Elsevier
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+gradient+stable+scheme+for+a+phase+field+model+for+the+moving+contact+line+problem&rft.jtitle=Journal+of+computational+physics&rft.au=Gao%2C+Min&rft.au=Wang%2C+Xiao-Ping&rft.date=2012-02-20&rft.issn=0021-9991&rft.volume=231&rft.issue=4&rft.spage=1372&rft.epage=1386&rft_id=info:doi/10.1016%2Fj.jcp.2011.10.015&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_jcp_2011_10_015
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0021-9991&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0021-9991&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0021-9991&client=summon