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| On the Rayleigh-Taylor Instability for Two UniformViscous Incompressible Flows |
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Citation： |
Fei JIANG,Song JIANG,Weiwei WANG.On the Rayleigh-Taylor Instability for Two UniformViscous Incompressible Flows[J].Chinese Annals of Mathematics B,2014,35(6):907~940 |
| Page view： 1149
Net amount： 1203 |
Authors： |
Fei JIANG; Song JIANG; Weiwei WANG; |
Foundation： |
the National Natural Science Foundation of China (Nos. 11101044, 11271051,
11229101, 11301083, 11371065, 11471134), the Fujian Provincial Natural Science Foundation of China
(No. 2014J01011), the National Basic Research Program (No. 2011CB309705) and the Beijing Center for
Mathematics and Information Interdisciplinary Sciences. |
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| Abstract： |
The authors study the Rayleigh-Taylor instability for two incompressible immiscible
fluids with or without surface tension, evolving with a free interface in the presence of
a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead
of using the transformation to Lagrangian coordinates, the perturbed equations in Eulerian
coordinates are transformed to an integral form and the two-fluid flow is formulated
as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with
the jump conditions at the free interface. First, the linearized problem around the steady
state which describes a denser immiscible fluid lying above a light one with a free interface
separating the two fluids, both fluids being in (unstable) equilibrium is analyzed. By a
general method of studying a family of modes, the smooth (when restricted to each fluid
domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev
spaces are constructed, thus leading to a global instability result for the linearized problem.
Then, by using these pathological solutions, the global instability for the corresponding
nonlinear problem in an appropriate sense is demonstrated. |
Keywords： |
Rayleigh-Taylor instability, Viscous incompressible flows, Global instability |
Classification： |
76E17, 76D05 |
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