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Self-similar solutions of the shallow-water equations representing gravity currents with variable inflow

Published online by Cambridge University Press:  21 April 2006

R. E. Grundy  and
James W. Rottmant
R. E. Grundy
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
James W. Rottmant
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Fluid Modelling Facility MD-81, Atmospheric Sciences Research Laboratory, United States Environmental Protection Agency, Research Triangle Park, NC 2711, USA.
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Abstract

A phase-plane method is used to study the existence of similarity solutions of the two-dimensional and axisymmetric shallow-water equations representing gravity currents with volumes proportional to tα, where α ≥ 0 and t is the time after the flow is initiated. Only currents for which there is a balance between the inertia of the current and the driving buoyancy force are considered. It is found that similarity solutions exist for the two-dimensional problem for all α ≥ 0, with some restrictions on the condition at the current front. It is shown that no similarity solutions satisfying the boundary conditions on the axis of symmetry exist for the axisymmetric problem except when α = 0.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 169 , August 1986 , pp. 337 - 351
Copyright
© 1986 Cambridge University Press

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