{"id":"https://openalex.org/W2019005251","doi":"https://doi.org/10.1137/040618400","title":"Finite Element Approximation of Soluble Surfactant Spreading on a Thin Film","display_name":"Finite Element Approximation of Soluble Surfactant Spreading on a Thin Film","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W2019005251","doi":"https://doi.org/10.1137/040618400","mag":"2019005251"},"language":"en","primary_location":{"id":"doi:10.1137/040618400","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040618400","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071664437","display_name":"John W. Barrett","orcid":"https://orcid.org/0000-0002-4048-1692"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"John W. Barrett","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036622790","display_name":"Robert N\u00fcrnberg","orcid":"https://orcid.org/0000-0002-2489-2416"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Robert N\u00fcrnberg","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5019581441","display_name":"M. R. E. Warner","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mark R. E. Warner","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5071664437"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.8463,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.73788462,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"44","issue":"3","first_page":"1218","last_page":"1247"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12141","display_name":"Fluid Dynamics and Thin Films","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12141","display_name":"Fluid Dynamics and Thin Films","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11379","display_name":"Rheology and Fluid Dynamics Studies","score":0.993399977684021,"subfield":{"id":"https://openalex.org/subfields/1507","display_name":"Fluid Flow and Transfer Processes"},"field":{"id":"https://openalex.org/fields/15","display_name":"Chemical Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11471","display_name":"Block Copolymer Self-Assembly","score":0.9840999841690063,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/nabla-symbol","display_name":"Nabla symbol","score":0.9753038883209229},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.5749334692955017},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4848860204219818},{"id":"https://openalex.org/keywords/p\u00e9clet-number","display_name":"P\u00e9clet number","score":0.4626612961292267},{"id":"https://openalex.org/keywords/pulmonary-surfactant","display_name":"Pulmonary surfactant","score":0.45370808243751526},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.4355301260948181},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.3578764796257019},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.212246835231781},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.1876215934753418},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.13707661628723145},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.13492152094841003}],"concepts":[{"id":"https://openalex.org/C54207081","wikidata":"https://www.wikidata.org/wiki/Q937750","display_name":"Nabla symbol","level":3,"score":0.9753038883209229},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5749334692955017},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4848860204219818},{"id":"https://openalex.org/C205684552","wikidata":"https://www.wikidata.org/wiki/Q899769","display_name":"P\u00e9clet number","level":2,"score":0.4626612961292267},{"id":"https://openalex.org/C58226133","wikidata":"https://www.wikidata.org/wiki/Q1324199","display_name":"Pulmonary surfactant","level":2,"score":0.45370808243751526},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.4355301260948181},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3578764796257019},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.212246835231781},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.1876215934753418},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.13707661628723145},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.13492152094841003}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/040618400","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040618400","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:iris.unitn.it:11572/283457","is_oa":false,"landing_page_url":"http://hdl.handle.net/11572/283457","pdf_url":null,"source":{"id":"https://openalex.org/S4306401913","display_name":"Institutional Research Information System (Universit\u00e0 degli Studi di Trento)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I193223587","host_organization_name":"University of Trento","host_organization_lineage":["https://openalex.org/I193223587"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:spiral.imperial.ac.uk:10044/1/639","is_oa":false,"landing_page_url":"http://hdl.handle.net/10044/1/639","pdf_url":null,"source":{"id":"https://openalex.org/S4306401396","display_name":"Spiral (Imperial College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I47508984","host_organization_name":"Imperial College London","host_organization_lineage":["https://openalex.org/I47508984"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"1247","raw_type":"Journal Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1592302237","https://openalex.org/W1964461288","https://openalex.org/W1971723555","https://openalex.org/W1975766172","https://openalex.org/W1975968248","https://openalex.org/W1982606205","https://openalex.org/W1996883801","https://openalex.org/W1998903284","https://openalex.org/W2007148543","https://openalex.org/W2014116743","https://openalex.org/W2058436955","https://openalex.org/W2060052509","https://openalex.org/W2066908036","https://openalex.org/W2075085615","https://openalex.org/W2081517630","https://openalex.org/W2091412121","https://openalex.org/W2097826064","https://openalex.org/W2127556239","https://openalex.org/W2161508768","https://openalex.org/W4254216534"],"related_works":["https://openalex.org/W2229925025","https://openalex.org/W1539596337","https://openalex.org/W33469805","https://openalex.org/W1855007440","https://openalex.org/W2788247819","https://openalex.org/W2914699178","https://openalex.org/W4301243416","https://openalex.org/W2024985363","https://openalex.org/W3096849461","https://openalex.org/W2086903497"],"abstract_inverted_index":{"We":[0],"consider":[1],"a":[2,110,117,139,260],"fully":[3],"practical":[4],"finite":[5],"element":[6],"approximation":[7],"of":[8,12,136,160,167,176,209,241,259],"the":[9,122,134,158,161,165,168,174,177,181,191,207,210,214,219,236,239,295],"following":[10],"system":[11,267,299],"nonlinear":[13,264,297],"degenerate":[14,265],"parabolic":[15,266],"equations:\\begin{align*}":[16],"\\frac{\\partial":[17,48],"u}{\\partial":[18],"t}":[19,50,77],"+":[20,51,78,96],"\\frac12\\,\\nabla\\,{\\cdot}\\,(u^2":[21],"\\,\\nabla":[22,28,56,63],"[\\sigma(v)])":[23,57],"-":[24,35,40,58,68,71,83,89,100,188],"\\frac13\\,":[25,84],"\\nabla":[26,52,60,92],"\\,{\\cdot}\\,(u^3":[27],"w)":[29],"&=":[30,73,102],"0,":[31,74,141,198,201],"\\\\":[32,47,75],"w":[33,86],"=":[34],"c":[36,203],"\\,":[37,42,44],"\\Delta":[38],"u":[39,156],"\\delta":[41],"u^{-\\nu}+\\,a":[43],"u^{-3}\\,":[45],",\\hspace{0.1cm}&":[46],"v}{\\partial":[49],"\\,{\\cdot}\\,":[53,81,87,93],"(u":[54],"\\,v":[55,62],"\\frac12\\,":[59],"\\,{\\cdot}\\,(u^2":[61],"w)-":[64],"\\rho_s":[65],"\\,\\Delta":[66],"v":[67,163,189],"K":[69,229],"\\left(\\psi":[70],"v\\right)":[72],"\\frac{\\partial\\psi}{\\partial":[76],"\\frac12\\,u\\,\\nabla":[79],"[\\sigma(v)]":[80],"\\nabla\\psi":[82,88],"u^2\\,\\nabla":[85],"\\rho_b\\,":[90],"u^{-1}":[91],"\\left(u":[94],"\\nabla\\psi\\right)":[95],"\\beta\\,":[97],"K\\,":[98],"u^{-1}(\\psi":[99],"v)":[101],"0.":[103],"\\hspace{2cm}\\nonumber":[104],"\\end{align*}":[105],"The":[106,130],"above":[107],"equations":[108],"model":[109,131],"Marangoni":[111],"driven":[112],"thin":[113],"film":[114],"laden":[115],"with":[116,147],"soluble":[118],"surfactant,":[119],"in":[120,269,280,285],"which":[121],"bulk":[123,182,215],"surfactant":[124,170,178],"concentration":[125,166,175],"has":[126],"been":[127],"vertically":[128],"averaged.":[129],"accounts":[132],"for":[133,250,293],"presence":[135],"both":[137],"attractive,":[138],"\\geq":[140,197,288],"and":[142,184,202,218,228,238,256],"repulsive,":[143],"delta":[144],">":[145,149,200,204,226,230,275],"0":[146,205,227,231,276],"nu":[148,287],"3,":[150],"van":[151],"der":[152],"Waals":[153],"forces.":[154],"Here":[155],"is":[157,164,173,190],"height":[159],"film,":[162],"interfacial":[169,242],"species,":[171],"psi":[172],"species":[179],"within":[180],"phase,":[183],"sigma(v)":[185],":=":[186],"1":[187],"typical":[192],"surface":[193,211],"tension.":[194],"Moreover,":[195],"rhos":[196,274],"rhob":[199],"are":[206,232,300,306],"inverses":[208],"Peclet":[212,216],"number,":[213,217,222],"modified":[220],"capillary":[221],"respectively;":[223],"finally,":[224],"beta":[225],"parameters":[233],"that":[234],"characterize":[235],"solubility":[237],"rate":[240],"adsorption.":[243],"In":[244],"addition":[245],"to":[246,262],"showing":[247],"stability":[248],"bounds":[249],"our":[251],"approximation,":[252],"we":[253],"prove":[254],"convergence":[255],"hence":[257],"existence":[258],"solution":[261],"this":[263],"(i)":[268],"one":[270],"space":[271,282],"dimension":[272],"when":[273],"and,":[277],"moreover,":[278],"(ii)":[279],"two":[281],"dimensions":[283],"if,":[284],"addition,":[286],"7.":[289],"Furthermore,":[290],"iterative":[291],"schemes":[292],"solving":[294],"resulting":[296],"discrete":[298],"discussed.":[301],"Finally,":[302],"some":[303],"numerical":[304],"experiments":[305],"presented.":[307]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
