{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/15965"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/15965","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"Mixed variational problems associated with stationary viscous incompressible free boundary flows","abstract":"A strategy that is often used in the study of capillary free boundary (FB) problems for viscous incompressible flows is the following: (1) Ignore one of the boundary conditions at the FB and prove that for every chosen position of the FB the resultant problem, here called the auxiliary problem (AP), is well posed. (2) Establish regularity results for the solution of the AP. (3) Using (2) and the remaining boundary condition, determine the position of the FB. We study the existence and uniqueness of the weak solution(s) to the AP, i.e., step (1), under minimal regularity constraints on the data and domain. The analysis is carried out for stationary two-dimensional flows, governed by either the Stokes or Navier-Stokes equations, in the context of four standard examples. A Green's formula is derived which allows the AP to be formulated as a mixed variational problem in which the pressure and normal stress appear as Lagrange multipliers. Existence and uniqueness results are obtained by using the Ladyzhenskaya-Babuska-Brezzi theory for mixed problems. By analogy with step (3), the dependence of the normal stress on the position of the FB is investigated.","abstract_html":"A strategy that is often used in the study of capillary free boundary (FB) problems for viscous incompressible flows is the following: (1) Ignore one of the boundary conditions at the FB and prove that for every chosen position of the FB the resultant problem, here called the auxiliary problem (AP), is well posed. (2) Establish regularity results for the solution of the AP. (3) Using (2) and the remaining boundary condition, determine the position of the FB. We study the existence and uniqueness of the weak solution(s) to the AP, i.e., step (1), under minimal regularity constraints on the data and domain. The analysis is carried out for stationary two-dimensional flows, governed by either the Stokes or Navier-Stokes equations, in the context of four standard examples. A Green's formula is derived which allows the AP to be formulated as a mixed variational problem in which the pressure and normal stress appear as Lagrange multipliers. Existence and uniqueness results are obtained by using the Ladyzhenskaya-Babuska-Brezzi theory for mixed problems. By analogy with step (3), the dependence of the normal stress on the position of the FB is investigated.","abstract_has_math":false,"creators":["Le Roux, Christiaan"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Reddy, B Daya"],"committee_chairs":[],"committee_members":[],"year":1991,"date_issued":"1991","date_published":"1991","updated_at":"2026-07-22T22:23:16Z","subjects":[],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/15965","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Reddy, B Daya"]},{"key":"dc:creator","label":"Author","values":["Le Roux, Christiaan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-12-28T06:03:37Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-12-28T06:03:37Z"]},{"key":"dc:date.issued","label":"Date","values":["1991"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cape Town"]},{"key":"dc:type","label":"Dc Type","values":["Master Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Masters"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["MSc"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/15965"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Bibliography: pages 93-97."]},{"key":"dc:description.abstract","label":"Abstract","values":["A strategy that is often used in the study of capillary free boundary (FB) problems for viscous incompressible flows is the following: (1) Ignore one of the boundary conditions at the FB and prove that for every chosen position of the FB the resultant problem, here called the auxiliary problem (AP), is well posed. (2) Establish regularity results for the solution of the AP. (3) Using (2) and the remaining boundary condition, determine the position of the FB. We study the existence and uniqueness of the weak solution(s) to the AP, i.e., step (1), under minimal regularity constraints on the data and domain. The analysis is carried out for stationary two-dimensional flows, governed by either the Stokes or Navier-Stokes equations, in the context of four standard examples. A Green's formula is derived which allows the AP to be formulated as a mixed variational problem in which the pressure and normal stress appear as Lagrange multipliers. Existence and uniqueness results are obtained by using the Ladyzhenskaya-Babuska-Brezzi theory for mixed problems. By analogy with step (3), the dependence of the normal stress on the position of the FB is investigated."]},{"key":"dc:title","label":"Title","values":["Mixed variational problems associated with stationary viscous incompressible free boundary flows"]}]}],"canonical_facts":{"dc:contributor.advisor":["Reddy, B Daya"],"dc:creator":["Le Roux, Christiaan"],"dc:date.accessioned":["2015-12-28T06:03:37Z"],"dc:date.available":["2015-12-28T06:03:37Z"],"dc:date.issued":["1991"],"dc:description":["Bibliography: pages 93-97."],"dc:description.abstract":["A strategy that is often used in the study of capillary free boundary (FB) problems for viscous incompressible flows is the following: (1) Ignore one of the boundary conditions at the FB and prove that for every chosen position of the FB the resultant problem, here called the auxiliary problem (AP), is well posed. (2) Establish regularity results for the solution of the AP. (3) Using (2) and the remaining boundary condition, determine the position of the FB. We study the existence and uniqueness of the weak solution(s) to the AP, i.e., step (1), under minimal regularity constraints on the data and domain. The analysis is carried out for stationary two-dimensional flows, governed by either the Stokes or Navier-Stokes equations, in the context of four standard examples. A Green's formula is derived which allows the AP to be formulated as a mixed variational problem in which the pressure and normal stress appear as Lagrange multipliers. Existence and uniqueness results are obtained by using the Ladyzhenskaya-Babuska-Brezzi theory for mixed problems. By analogy with step (3), the dependence of the normal stress on the position of the FB is investigated."],"dc:identifier.uri":["http://hdl.handle.net/11427/15965"],"dc:language.iso":["eng"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:publisher.institution":["University of Cape Town"],"dc:title":["Mixed variational problems associated with stationary viscous incompressible free boundary flows"],"dc:type":["Master Thesis"],"dc:type.qualificationlevel":["Masters"],"dc:type.qualificationname":["MSc"]},"updated_at":"2026-07-22T22:23:16Z"}