A vortex pair in a viscous, incompressible fluid rises vertically toward a deformable free surface. The mathematical, description of this flow situation is a time-dependent nonlinear free-surface problem that has been solved numerically for a two-dimensional laminar flow with the aid of the Navier-Stokes equations by using boundary-fitted coordinates. For a number of selected flow parameters, results are presented on the decay of the primary vortices and their paths, the generation of surface vorticity and secondary vortices, the development and final stage of the disturbed free surface, and the influence of surface tension. High and low Froude numbers represent the two extremes of free-surface yielding and stiffness, respectively. For an intermediate Froude number, a special rebounding due to the presence of secondary vortices has been observed: the path of the primary vortex centre portrays a complete loop.

The problem of a vortex pair, rising obliquely at an angle of 45° toward a deformable free surface in a viscous, incompressible fluid, is solved with the aid of the Navier—Stokes equations. The full nonlinear boundary conditions at the free surface are applied. The oblique interaction of the vortex pair with the free surface results in a number of novel features that have not been observed for the special case of a vertical rise, reported earlier. These features include the directional change of trajectories near the free surface and the occurrence of waves driven by the vortex pair. Moreover, surface tension can completely change the flow characteristics such as the direction of the trajectories and the generation of secondary vortices. Numerical solutions are presented for selected Reynolds, Froude, and Weber numbers.

The decay of two counter-rotating point vortices of equal strength in a viscous incompressible fluid is studied by solving the Navier–Stokes equations with a finite-difference technique. Solutions for Reynolds numbers 10, 50, and 100 are compared with a model presented by Cantwell and Rott. Equivorticity lines and the decay rate of translational velocity and circulation of the half-plane are presented, together with the diverging paths of the vortex pair.

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

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