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**Some numerical methods for the computation of capillary free boundaries governed by the Navier-Stokes equations.**
*(English)*
Zbl 0706.76027

Summary: In § 1, the trial method, Newton’s method, and the total linearizaion method are applied to a simple one-dimensional free boundary (FB) problem for which the solution is known. Attention will be focused on convergence speed and computer implementation facilities of these methods. Section 2 discusses static FB problems in two- and three-dimensional containers. These FBs are governed by the Laplace-Young equation. In § 3, two stationary FB problems governed by the Navier-Stokes equations are formulated and these problems are solved using the three methods introduced in § 1. A thermocapillary FB problem governed by the Navier-Stokes equations coupled with the heat equation is studied in § 4. In § 5, a moving FB problem governed by the Navier-Stokes equations will be considered within the context of perturbation theory. Finally, in § 6, some remarks are made on present and future research in the field of capillary FB problems governed by the Navier-Stokes equations.

### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

35R35 | Free boundary problems for PDEs |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |