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Jets rising and falling under gravity

Published online by Cambridge University Press:  20 April 2006

Jean-Marc Vanden-Broeck  and
Joseph B. Keller
Jean-Marc Vanden-Broeck
Affiliation:
Department of Mathematics, Stanford University, Stanford. CA 94305
Joseph B. Keller
Affiliation:
Department of Mathematics, Stanford University, Stanford. CA 94305
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Abstract

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 $\lambda = q_{r\m c}(Qg)^{-\frac{1}{3}}$, which ranges from zero to infinity. Here qc 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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 124 , November 1982 , pp. 335 - 345
Copyright
© 1982 Cambridge University Press

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References

Grant, M. A. 1973 J. Fluid Mech. 59, 257.
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Keller, J. B. & Weitz, M. L. 1957 In Proc. 9th Int. Congr. Appl. Mech., Brussels, vol. 1, p. 316.
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Vanden-Broeck, J.-M. & Keller, J. B. 1980 J. Fluid Mech. 98, 161.
Vanden-Broeck, J.-M. & Schwartz, L. W. 1979 Phys. Fluids 22, 1868.
Stokes, G. G. 1880 In Mathematical and Physical Papers, vol. 1, p. 225.