Non-linear liquid motion in conical containers

Summary The non-linear behavior of an incompressible and frictionless liquid with a free surface in an annular conical sector frustrum container has been determined. All special cases of conical tanks may be obtained from the results. Some configurations have been evaluated numerically. It was found that the oscillating liquid system exhibits softening behavior, showing for increased free surface amplitudes decreased natural frequencies as compared to the linearized frequencies. This non-linear behavior is more pronounced for larger vertex angles and smaller frustrum heights.Notation a, b radius (describing the free surface,r=a the container bottomr=b) - k=b/a ratio of radii - g gravity constant or acceleration along the axis of the container - h liquid height - P ^m ,Q ^m associated Legendre functions of first and second kind and orderm - r, , spherical coordinates - p liquid pressure - t time - 2, 2 outer- and inner vertex angle resp - 2 sector angle of container - (r, , ,t) velocity potential - liquid density - zeros of associated Legendre function or cross product Legendre functions with respect to the degree - non-linear frequency - _mn ⁽⁰⁾ linearized natural frequencies of liquid - free surface elevation With 7 Figures