{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/137640"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/137640","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Similarity Solutions for the Two-Dimensional Diffusion–Advection Equation in Sharp-Corner Geometries","abstract":"Problems involving corner geometries arise frequently in science and engineering, including in fluid mechanics. Low-Reynolds-number flow patterns near sharp corners, including Moffatt eddies, have been studied extensively in the past. Here, we focus on the effect of Moffatt eddies on concentration profiles near sharp corners. The Lie symmetry method is applied to the 2D diffusion-advection equation to obtain a similarity ansatz, reducing the equation to a time-independent second-order PDE. By modifying the ansatz, the problem can be recast as an eigenvalue problem. Exploiting elliptic regularity and the strong maximum principle, we invoke the Krein-Rutman theorem to establish the existence and domain-size monotonicity of the principal eigenvalue. We then investigate how varying domain length and corner angle affect the magnitude of the principal eigenvalue, observing an inverse relationship in both cases. The behavior of the eigenfunction is sensitive to small changes in the domain shape and size, so we truncate the domain along separating streamlines. This study adds to our understanding of the low-Reynolds-number transport dynamics near sharp corners from a pattern formation perspective, and can be extended to other systems.","abstract_html":"Problems involving corner geometries arise frequently in science and engineering, including in fluid mechanics. Low-Reynolds-number flow patterns near sharp corners, including Moffatt eddies, have been studied extensively in the past. Here, we focus on the effect of Moffatt eddies on concentration profiles near sharp corners. The Lie symmetry method is applied to the 2D diffusion-advection equation to obtain a similarity ansatz, reducing the equation to a time-independent second-order PDE. By modifying the ansatz, the problem can be recast as an eigenvalue problem. Exploiting elliptic regularity and the strong maximum principle, we invoke the Krein-Rutman theorem to establish the existence and domain-size monotonicity of the principal eigenvalue. We then investigate how varying domain length and corner angle affect the magnitude of the principal eigenvalue, observing an inverse relationship in both cases. The behavior of the eigenfunction is sensitive to small changes in the domain shape and size, so we truncate the domain along separating streamlines. This study adds to our understanding of the low-Reynolds-number transport dynamics near sharp corners from a pattern formation perspective, and can be extended to other systems.","abstract_has_math":false,"creators":["Khattab, Mohamed Osama Mahmoud"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Engineering Mechanics","degree_department":"Mechanical Engineering","school":null,"contributors":[],"advisors":[],"committee_chairs":["Stremler, Mark A."],"committee_members":["Staples, Anne E.","Paul, Mark R."],"year":2025,"date_issued":"2025-08-01","date_published":"2025-08-01","updated_at":"2026-07-22T22:19:19Z","subjects":["Lie Symmetry","Diffusion-advection","Krein-Rutman theorem","Similarity solutions","Moffatt eddies"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10919/137640","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Stremler, Mark A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Staples, Anne E.","Paul, Mark R."]},{"key":"dc:contributor.department","label":"Department","values":["Mechanical Engineering"]},{"key":"dc:creator","label":"Author","values":["Khattab, Mohamed Osama Mahmoud"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-09-08T19:33:01Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-09-08T19:33:01Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-08-01"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering Mechanics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Lie Symmetry","Diffusion-advection","Krein-Rutman theorem","Similarity solutions","Moffatt eddies"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10919/137640"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Problems involving corner geometries arise frequently in science and engineering, including in fluid mechanics. Low-Reynolds-number flow patterns near sharp corners, including Moffatt eddies, have been studied extensively in the past. Here, we focus on the effect of Moffatt eddies on concentration profiles near sharp corners. The Lie symmetry method is applied to the 2D diffusion-advection equation to obtain a similarity ansatz, reducing the equation to a time-independent second-order PDE. By modifying the ansatz, the problem can be recast as an eigenvalue problem. Exploiting elliptic regularity and the strong maximum principle, we invoke the Krein-Rutman theorem to establish the existence and domain-size monotonicity of the principal eigenvalue. We then investigate how varying domain length and corner angle affect the magnitude of the principal eigenvalue, observing an inverse relationship in both cases. The behavior of the eigenfunction is sensitive to small changes in the domain shape and size, so we truncate the domain along separating streamlines. This study adds to our understanding of the low-Reynolds-number transport dynamics near sharp corners from a pattern formation perspective, and can be extended to other systems."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["Low-Reynolds-number flow near sharp corners has been studied extensively, and here, we investigate the relevant problem of chemical species transport in such geometries, which has applications in biology, geophysics, and microfluidics. In this work, mathematical tools are used to characterize concentration profiles and pinpoint how known flow patterns influence particle transport. The work provides a mathematical approach to a relatively understudied problem, and the methods used here can be applied to similar systems."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Similarity Solutions for the Two-Dimensional Diffusion–Advection Equation in Sharp-Corner Geometries"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Stremler, Mark A."],"dc:contributor.committeemember":["Staples, Anne E.","Paul, Mark R."],"dc:contributor.department":["Mechanical Engineering"],"dc:creator":["Khattab, Mohamed Osama Mahmoud"],"dc:date.accessioned":["2025-09-08T19:33:01Z"],"dc:date.available":["2025-09-08T19:33:01Z"],"dc:date.issued":["2025-08-01"],"dc:description.abstract":["Problems involving corner geometries arise frequently in science and engineering, including in fluid mechanics. 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