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Free boundary problems for Stokes’ flows and finite element methods. (English) Zbl 0617.76033
Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 327-332 (1986).
[For the entire collection see Zbl 0595.00009.]
In two dimensions a Stokes’ flow is considered symmetric to the abscisse \(\eta =0\) and periodic with respect to \(\xi\). On the free boundary \(| \eta | =S(\xi)\) the conditions are: (i) the free boundary is a streamline, (ii) the tangential force vanishes, (iii) the normal force is proportional to the mean curvature of the boundary. By straightening the boundary, i.e. by introducing the variables \(x=\xi\), \(y=\eta /S(\xi)\), the problem is reduced to one in a fixed domain. The underlying differential equations are now highly nonlinear: They consist in an elliptic system coupled with an ordinary differential equation for S. The analytic properties of the solution as well as the convergence of the proposed finite element approximation are discussed.
MSC:
76D07 Stokes and related (Oseen, etc.) flows
35Q30 Navier-Stokes equations