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Steady vortex flows past a circular cylinder are obtained numerically as solutions of the partial differential equation Δψ = f(ψ), f(ψ) = ω(1 − H(ω − α)), where H is the Heaviside function. Only symmetric solutions are considered so the flow may be thought of as that past a semicircular bump in a half-plane. The flow is transplanted by the complex logarithm to a semi-infinite strip. This strip is truncated at a finite height, a numerical boundary condition is used on the top, and the difference equations resulting from the five-point discretization for the Laplacian on a uniform grid are solved using Fourier methods and an iteration for the nonlinear equation. If the area of the vortex region is prescribed the magnitude of the vorticity ω is adjusted in an inner iteration to satisfy this area constraint.Three types of solutions are discussed: vortices attached to the cylinder, vortex patches standing off from the cylinder and strips of vorticity extending to infinity. Three families of each type of solution have been found. Equilibrium positions for point vortices, including the Föppl pair, are related to these families by continuation.

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Abstract. If a drop of fluid of density ρ 1 rests on the surface of a fluid of density ρ 2 below a fluid of density ρ 0 , ρ 0 < ρ 1 < ρ 2 , the surface of the drop is made up of a sessile drop and an inverted sessile drop which match an external capillary surface. Solutions of this problem are constructed by matching solutions of the axisymmetric capillary surface equation. For general values of the surface tensions at the common boundaries of the three fluids the surfaces need not be graphs and the profiles of these axisymmetric surfaces are parametrized by their tangent angles. The solutions are obtained by finding the value of the tangent angle for which the three surfaces match. In addition the asymptotic form of the solution is found for small drops. Mathematics Subject Classification (2000). 76B45, 35A15, 35R35, 49J05.

We prove the uniqueness of the determination of a surface crack from one special boundary measurement of an electrical or elastic field. Then we suggest and test a numerical algorithm for identification of a polygonal plane crack based on the Schwarz-Christoffel formula. The numerical experiments with cracks consisting of one or two intervals show the high stability and precision of this algorithm.