{"id":{"repo_id":"ncsu","oai_identifier":"oai:repository.lib.ncsu.edu:1840.16/3917"},"canonical_url":"https://search.dev.ndltd.org/etd/ncsu/oai:repository.lib.ncsu.edu:1840.16/3917","repository":{"repo_id":"ncsu","name":"North Carolina State University","base_url":"https://repository.lib.ncsu.edu/server/oai/request"},"display":{"title":"Partial Differential Equations of Thin Liquid Films: Analysis and Numerical Simulation","abstract":"We consider four problems related to Marangoni-driven thin liquid films. The first compares two models for the motion of a contact line: the precursor model and the Navier slip model. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. The range of effective contact slopes and parameter values are explained with the aid of Poincaré sections of the phase diagram of the third order ODE. In the second problem, we use theory from hyperbolic conservation laws to map classical shocks, nonclassical shock waves (known as undercompressive shocks) and rarefactions that arise as solutions to the Cauchy problem. To create such a 'Riemann map', we employ a kinetic relation that describes admissible nonclassical shock waves, and a nucleation condition that determines when a nonclassical solution is selected. The hyperbolic theory captures features observed in thin film flow, such as multiple long-time solutions for the same initial upstream and downstream states. The third problem incorporates localized heating by an infrared (IR) laser to the model of a Marangoni-driven thin film from the previous problems. We analyze two types of steady state solutions, using a dynamical systems approach to explain homoclinic solutions and PDE simulations to explain heteroclinic solutions. We discuss several methods for controlling the downstream height and the strength of forcing required to create homoclinic solutions from uniform or monotonic initial data. The fourth problem explores a model for a different physical scenario, in which a thin film is driven down a solid substrate by gravity and surfactant. The model couples the thin film PDE for the height of the film with an equation for the transport of surfactant. Solutions of the parabolic-hyperbolic system include a complicated [em double wave solution], with discontinuities in the height and surfactant concentration gradient. Agreement of analytical solutions with data from numerical simulations indicates that we have successfully modeled long-time wave structures.","abstract_html":"We consider four problems related to Marangoni-driven thin liquid films. The first compares two models for the motion of a contact line: the precursor model and the Navier slip model. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. The range of effective contact slopes and parameter values are explained with the aid of Poincaré sections of the phase diagram of the third order ODE. In the second problem, we use theory from hyperbolic conservation laws to map classical shocks, nonclassical shock waves (known as undercompressive shocks) and rarefactions that arise as solutions to the Cauchy problem. To create such a 'Riemann map', we employ a kinetic relation that describes admissible nonclassical shock waves, and a nucleation condition that determines when a nonclassical solution is selected. The hyperbolic theory captures features observed in thin film flow, such as multiple long-time solutions for the same initial upstream and downstream states. The third problem incorporates localized heating by an infrared (IR) laser to the model of a Marangoni-driven thin film from the previous problems. We analyze two types of steady state solutions, using a dynamical systems approach to explain homoclinic solutions and PDE simulations to explain heteroclinic solutions. We discuss several methods for controlling the downstream height and the strength of forcing required to create homoclinic solutions from uniform or monotonic initial data. The fourth problem explores a model for a different physical scenario, in which a thin film is driven down a solid substrate by gravity and surfactant. The model couples the thin film PDE for the height of the film with an equation for the transport of surfactant. Solutions of the parabolic-hyperbolic system include a complicated [em double wave solution], with discontinuities in the height and surfactant concentration gradient. Agreement of analytical solutions with data from numerical simulations indicates that we have successfully modeled long-time wave structures.","abstract_has_math":false,"creators":["Levy, Rachel"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Dr. Tom Witelski, Committee Member","Dr. Mette Olufsen, Committee Member","Dr. Michael Shearer, Committee Chair","Dr. Alina Chertock, Committee Member"],"committee_chairs":[],"committee_members":[],"year":2005,"date_issued":"2005-07-26","date_published":"2005-07-26","updated_at":"2026-08-21T22:21:56Z","subjects":["thin liquid films","partial differential equations","shock","contact line","nucleation","surfactant"],"languages":[],"rights":["I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04222005-162444"],"render_values":[{"text":"etd-04222005-162444","href":null,"code":true}]}]},"links":{"outbound_url":"http://www.lib.ncsu.edu/resolver/1840.16/3917","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"source_record":{"url":"https://repository.lib.ncsu.edu/server/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Arepository.lib.ncsu.edu%3A1840.16%2F3917","prefix":"dim"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Dr. Tom Witelski, Committee Member","Dr. Mette Olufsen, Committee Member","Dr. Michael Shearer, Committee Chair","Dr. Alina Chertock, Committee Member"]},{"key":"dc:creator","label":"Author","values":["Levy, Rachel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2010-04-02T18:40:15Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2010-04-02T18:40:15Z"]},{"key":"dc:date.issued","label":"Date","values":["2005-07-26"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["thin liquid films","partial differential equations","shock","contact line","nucleation","surfactant"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04222005-162444"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://www.lib.ncsu.edu/resolver/1840.16/3917"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["North Carolina State University Theses Mathematics."]},{"key":"dc:description.abstract","label":"Abstract","values":["We consider four problems related to Marangoni-driven thin liquid films. The first compares two models for the motion of a contact line: the precursor model and the Navier slip model. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. The range of effective contact slopes and parameter values are explained with the aid of Poincaré sections of the phase diagram of the third order ODE. In the second problem, we use theory from hyperbolic conservation laws to map classical shocks, nonclassical shock waves (known as undercompressive shocks) and rarefactions that arise as solutions to the Cauchy problem. To create such a 'Riemann map', we employ a kinetic relation that describes admissible nonclassical shock waves, and a nucleation condition that determines when a nonclassical solution is selected. The hyperbolic theory captures features observed in thin film flow, such as multiple long-time solutions for the same initial upstream and downstream states. The third problem incorporates localized heating by an infrared (IR) laser to the model of a Marangoni-driven thin film from the previous problems. We analyze two types of steady state solutions, using a dynamical systems approach to explain homoclinic solutions and PDE simulations to explain heteroclinic solutions. We discuss several methods for controlling the downstream height and the strength of forcing required to create homoclinic solutions from uniform or monotonic initial data. The fourth problem explores a model for a different physical scenario, in which a thin film is driven down a solid substrate by gravity and surfactant. The model couples the thin film PDE for the height of the film with an equation for the transport of surfactant. Solutions of the parabolic-hyperbolic system include a complicated [em double wave solution], with discontinuities in the height and surfactant concentration gradient. Agreement of analytical solutions with data from numerical simulations indicates that we have successfully modeled long-time wave structures."]},{"key":"dc:format","label":"Dc Format","values":["Thesis (Ph.D.)--North Carolina State University."]},{"key":"dc:title","label":"Title","values":["Partial Differential Equations of Thin Liquid Films: Analysis and Numerical Simulation"]}]}],"canonical_facts":{"dc:contributor.advisor":["Dr. Tom Witelski, Committee Member","Dr. Mette Olufsen, Committee Member","Dr. Michael Shearer, Committee Chair","Dr. Alina Chertock, Committee Member"],"dc:creator":["Levy, Rachel"],"dc:date.accessioned":["2010-04-02T18:40:15Z"],"dc:date.available":["2010-04-02T18:40:15Z"],"dc:date.issued":["2005-07-26"],"dc:description":["North Carolina State University Theses Mathematics."],"dc:description.abstract":["We consider four problems related to Marangoni-driven thin liquid films. The first compares two models for the motion of a contact line: the precursor model and the Navier slip model. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. The range of effective contact slopes and parameter values are explained with the aid of Poincaré sections of the phase diagram of the third order ODE. In the second problem, we use theory from hyperbolic conservation laws to map classical shocks, nonclassical shock waves (known as undercompressive shocks) and rarefactions that arise as solutions to the Cauchy problem. To create such a 'Riemann map', we employ a kinetic relation that describes admissible nonclassical shock waves, and a nucleation condition that determines when a nonclassical solution is selected. The hyperbolic theory captures features observed in thin film flow, such as multiple long-time solutions for the same initial upstream and downstream states. The third problem incorporates localized heating by an infrared (IR) laser to the model of a Marangoni-driven thin film from the previous problems. We analyze two types of steady state solutions, using a dynamical systems approach to explain homoclinic solutions and PDE simulations to explain heteroclinic solutions. We discuss several methods for controlling the downstream height and the strength of forcing required to create homoclinic solutions from uniform or monotonic initial data. The fourth problem explores a model for a different physical scenario, in which a thin film is driven down a solid substrate by gravity and surfactant. The model couples the thin film PDE for the height of the film with an equation for the transport of surfactant. Solutions of the parabolic-hyperbolic system include a complicated [em double wave solution], with discontinuities in the height and surfactant concentration gradient. Agreement of analytical solutions with data from numerical simulations indicates that we have successfully modeled long-time wave structures."],"dc:format":["Thesis (Ph.D.)--North Carolina State University."],"dc:identifier.other":["etd-04222005-162444"],"dc:identifier.uri":["http://www.lib.ncsu.edu/resolver/1840.16/3917"],"dc:rights":["I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report."],"dc:subject":["thin liquid films","partial differential equations","shock","contact line","nucleation","surfactant"],"dc:title":["Partial Differential Equations of Thin Liquid Films: Analysis and Numerical Simulation"]},"updated_at":"2026-08-21T22:21:56Z"}