Self-similar viscous gravity currents: Phase-plane formalism
A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy...
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1990
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221120_v210_n155_p155_Gratton http://hdl.handle.net/20.500.12110/paper_00221120_v210_n155_p155_Gratton |
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paper:paper_00221120_v210_n155_p155_Gratton2023-06-08T14:45:55Z Self-similar viscous gravity currents: Phase-plane formalism Gratton, Julio Minotti, Fernando Oscar Fluid Mechanics--Mathematical Models Gravitational Effects Lubrication--Theory Mathematical Techniques--Differential Equations Phase-Plane Formalism Scaling Laws Viscous Gravity Currents Flow of Fluids A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on. © 1990, Cambridge University Press. All rights reserved. Fil:Gratton, J. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Minotti, F. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 1990 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221120_v210_n155_p155_Gratton http://hdl.handle.net/20.500.12110/paper_00221120_v210_n155_p155_Gratton |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Fluid Mechanics--Mathematical Models Gravitational Effects Lubrication--Theory Mathematical Techniques--Differential Equations Phase-Plane Formalism Scaling Laws Viscous Gravity Currents Flow of Fluids |
| spellingShingle |
Fluid Mechanics--Mathematical Models Gravitational Effects Lubrication--Theory Mathematical Techniques--Differential Equations Phase-Plane Formalism Scaling Laws Viscous Gravity Currents Flow of Fluids Gratton, Julio Minotti, Fernando Oscar Self-similar viscous gravity currents: Phase-plane formalism |
| topic_facet |
Fluid Mechanics--Mathematical Models Gravitational Effects Lubrication--Theory Mathematical Techniques--Differential Equations Phase-Plane Formalism Scaling Laws Viscous Gravity Currents Flow of Fluids |
| description |
A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on. © 1990, Cambridge University Press. All rights reserved. |
| author |
Gratton, Julio Minotti, Fernando Oscar |
| author_facet |
Gratton, Julio Minotti, Fernando Oscar |
| author_sort |
Gratton, Julio |
| title |
Self-similar viscous gravity currents: Phase-plane formalism |
| title_short |
Self-similar viscous gravity currents: Phase-plane formalism |
| title_full |
Self-similar viscous gravity currents: Phase-plane formalism |
| title_fullStr |
Self-similar viscous gravity currents: Phase-plane formalism |
| title_full_unstemmed |
Self-similar viscous gravity currents: Phase-plane formalism |
| title_sort |
self-similar viscous gravity currents: phase-plane formalism |
| publishDate |
1990 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221120_v210_n155_p155_Gratton http://hdl.handle.net/20.500.12110/paper_00221120_v210_n155_p155_Gratton |
| work_keys_str_mv |
AT grattonjulio selfsimilarviscousgravitycurrentsphaseplaneformalism AT minottifernandooscar selfsimilarviscousgravitycurrentsphaseplaneformalism |
| _version_ |
1768545312708755456 |