We study the steady flow of a second grade fluid down an open inclined channel. We formulate the mathematical problem, a mixed boundary value problem for the Laplacian with an unknown free boundary described by a nonlinear second-order ODE, and prove existence of a unique solution for small data using a contraction argument.
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José Miguel Urbano, Juha H. Videman, Unidirectional steady flow of a viscoelastic fluid with a free surface. Interfaces Free Bound. 3 (2001), no. 4, pp. 447–463DOI 10.4171/IFB/48