The character of the growth rates of the normal modes for Rayleigh–Taylor instability of superposed incompressible, viscous fluids is analyzed in terms of appropriately scaled dimensionless parameters and a particularly simple representation of the Rayleigh–Taylor dispersion relation. The chief feature that emerges is that the scaled growth rate is remarkably insensitive to the values of fluid densities and viscosities. To within a few percent, the physical growth rate depends only on the surface tension, the density-weighted average viscosity, and the effective acceleration. Approximate formulae for the most unstable wavenumber and the corresponding maximum growth rate are given.
The initial value problem associated with the development of small amplitude disturbances in Rayleigh–Taylor unstable, viscous, incompressible fluids is studied. Solutions to the linearized equations of motion which satisfy general initial conditions are obtained in terms of Fourier–Laplace transforms of the hydrodynamic variables, without restriction on the density or viscosity of either fluid. When the two fluids have equal kinematic viscosities, these transforms can be inverted explicitly to express the fluid variables as integrals of Green’s functions multiplied by initial data. In addition to normal modes, a set of continuum modes, not treated explicitly in the literature, makes an important contribution to the development of the fluid motion.
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