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.
Reference:
Le Roux, C. 1991. Mixed variational problems associated with stationary viscous incompressible free boundary flows. University of Cape Town.
Le Roux, C. (1991). Mixed variational problems associated with stationary viscous incompressible free boundary flows. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/15965
Le Roux, Christiaan. "Mixed variational problems associated with stationary viscous incompressible free boundary flows." Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1991. http://hdl.handle.net/11427/15965
Le Roux C. Mixed variational problems associated with stationary viscous incompressible free boundary flows. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1991 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/15965