Jets rising and falling under gravity

Steady two-dimensional jets of inviscid incompressible fluid, rising and falling under gravity, are calculated numkrically. The shape of each jet depends upon a single parameter, the Froude number λ = qr c(Qg)-1/3 , which ranges from zero to infinity. Here q c is the velocity at the crest of the jet, i.e. the highest point of the upper surface, Q is the flux in the jet. and g is the acceleration of gravity. For λ = ∞ the jet is slender and parabolic. It becomes thicker as λ decreases, and reaches a limiting form at λ = 0. Then there is a stagnation point at the crest, where the surface makes a 120° angle with itself. This angle is predicted by the same argument Stokes used in his study of water waves. The problem is formulated as an integro-differential equation for the two free surfaces of the jet, This equation is dlscretized to yield a set of nonlinear equations, which are solved numerically by Newton's method. In addition, asymptotic results for large λ are obtained analytically. Graphs of the results are presented.