# A study of Taylor instability of superposed fluids

TL;DR: In this article, a solution of hydrodynamic equations for the case of (Taylor) instability of the interface of two fluids of different but constant densities when accelerated from the heavier fluid to the lighter fluid is sought.

Abstract: A solution of hydrodynamic equations for the case of (Taylor) instability of the interface of two fluids of different but constant densities when accelerated from the heavier fluid to the lighter fluid is sought. After identifying the problem as one of singular perturbation, it is solved by the method of strained coordinates. A third order theory is presented. It is found that the ratio of the densities of the fluids has a significant effect on the stability or growth of the interface. Growth of the interface is found to depend also on the wave number and amplitude of initial disturbance. Stability criterion is calculated for different values of density ratio.

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...A long-wave analysis for this case (Appendix B) shows that the growth in the deforming regions again becomes catastrophic due to finite-amplitude effects, as in the Newtonian case. A companion work (Canright & Morris 1993) considers a different model for the lithosphere : as a thermal boundary layer growing under a suddenly cooled horizontal boundary, where the fluid viscosity depends strongly on temperature. The long-wave analysis shows that, again, the nonlinear effects of finite amplitude give catastrophic growth, yielding sheets in finite time. For that case the force balance and the resulting growth of peaks is essentially the same as that considered here, because thermal diffusion becomes unimportant where the layer is thick. The Rayleigh-Taylor problem considered here is the simplest example of this dynamic balance. The instability of a dense fluid supported by a lighter one has been studied extensively, beginning with the analysis of Rayleigh (1883). Taylor (1950) noted that acceleration would give the same effect as gravity, in agreement with the experiments of Lewis (1950). A thorough introduction to the linear theory is given by Chandrasekhar (1961, chap....

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...A long-wave analysis for this case (Appendix B) shows that the growth in the deforming regions again becomes catastrophic due to finite-amplitude effects, as in the Newtonian case. A companion work (Canright & Morris 1993) considers a different model for the lithosphere : as a thermal boundary layer growing under a suddenly cooled horizontal boundary, where the fluid viscosity depends strongly on temperature. The long-wave analysis shows that, again, the nonlinear effects of finite amplitude give catastrophic growth, yielding sheets in finite time. For that case the force balance and the resulting growth of peaks is essentially the same as that considered here, because thermal diffusion becomes unimportant where the layer is thick. The Rayleigh-Taylor problem considered here is the simplest example of this dynamic balance. The instability of a dense fluid supported by a lighter one has been studied extensively, beginning with the analysis of Rayleigh (1883). Taylor (1950) noted that acceleration would give the same effect as gravity, in agreement with the experiments of Lewis (1950)....

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...A long-wave analysis for this case (Appendix B) shows that the growth in the deforming regions again becomes catastrophic due to finite-amplitude effects, as in the Newtonian case. A companion work (Canright & Morris 1993) considers a different model for the lithosphere : as a thermal boundary layer growing under a suddenly cooled horizontal boundary, where the fluid viscosity depends strongly on temperature. The long-wave analysis shows that, again, the nonlinear effects of finite amplitude give catastrophic growth, yielding sheets in finite time. For that case the force balance and the resulting growth of peaks is essentially the same as that considered here, because thermal diffusion becomes unimportant where the layer is thick. The Rayleigh-Taylor problem considered here is the simplest example of this dynamic balance. The instability of a dense fluid supported by a lighter one has been studied extensively, beginning with the analysis of Rayleigh (1883). Taylor (1950) noted that acceleration would give the same effect as gravity, in agreement with the experiments of Lewis (1950). A thorough introduction to the linear theory is given by Chandrasekhar (1961, chap. X), and recent surveys of previous work are given by Sharp (1984) and Kull(l991). Many of the analyses (e.g. Bellman & Pennington 1954; and Menikoff et al. 1978) have focused on the initial small-amplitude growth, where linearized equations are appropriate. One approach for examining the nonlinear effects of large amplitude is direct numerical simulation, e.g. Harlow & Welch (1966) and Tryggvason (1988), or through boundary-integral computations, e....

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...A long-wave analysis for this case (Appendix B) shows that the growth in the deforming regions again becomes catastrophic due to finite-amplitude effects, as in the Newtonian case. A companion work (Canright & Morris 1993) considers a different model for the lithosphere : as a thermal boundary layer growing under a suddenly cooled horizontal boundary, where the fluid viscosity depends strongly on temperature. The long-wave analysis shows that, again, the nonlinear effects of finite amplitude give catastrophic growth, yielding sheets in finite time. For that case the force balance and the resulting growth of peaks is essentially the same as that considered here, because thermal diffusion becomes unimportant where the layer is thick. The Rayleigh-Taylor problem considered here is the simplest example of this dynamic balance. The instability of a dense fluid supported by a lighter one has been studied extensively, beginning with the analysis of Rayleigh (1883). Taylor (1950) noted that acceleration would give the same effect as gravity, in agreement with the experiments of Lewis (1950). A thorough introduction to the linear theory is given by Chandrasekhar (1961, chap. X), and recent surveys of previous work are given by Sharp (1984) and Kull(l991). Many of the analyses (e.g. Bellman & Pennington 1954; and Menikoff et al. 1978) have focused on the initial small-amplitude growth, where linearized equations are appropriate. One approach for examining the nonlinear effects of large amplitude is direct numerical simulation, e.g. Harlow & Welch (1966) and Tryggvason (1988), or through boundary-integral computations, e.g. Baker & Meiron (1984), and Newhouse & Pozrikidis (1990). To investigate the nonlinear effects analytically, the most common approach has been the perturbation method, where various quantities are expressed as power series in the small initial amplitude or slope, giving linearized equations for each order (e.g. Emmons, Chang & Watson 1960). However, a wide variety of other methods have been employed (primarily for inviscid fluids) e.g. least-squares approximation (Kull 1986), averaging (Drazin 1969), multiple timescales (Nayfeh 1969), strained coordinates (Amaranath & Rajappa 1976), generalized coordinates (Dienes 1978), Lagrangian formulations (Ott 1972), and heuristic models (Baker & Freeman 1981; Aref & Tryggvason 1989). Also, Dussan V. (1975) used energy methods to determine the stability of disturbances of arbitrary amplitude. In the above studies, inertia is important. The situation we consider, of a thin viscous layer in creeping flow, has been examined primarily in the geophysics literature, e.g. the experiments and weakly nonlinear theory of Whitehead & Luther (1975). Our analysis is equivalent to a perturbation expansion in the interfacial slope, which for long waves remains small even for large amplitudes....

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...A long-wave analysis for this case (Appendix B) shows that the growth in the deforming regions again becomes catastrophic due to finite-amplitude effects, as in the Newtonian case. A companion work (Canright & Morris 1993) considers a different model for the lithosphere : as a thermal boundary layer growing under a suddenly cooled horizontal boundary, where the fluid viscosity depends strongly on temperature. The long-wave analysis shows that, again, the nonlinear effects of finite amplitude give catastrophic growth, yielding sheets in finite time. For that case the force balance and the resulting growth of peaks is essentially the same as that considered here, because thermal diffusion becomes unimportant where the layer is thick. The Rayleigh-Taylor problem considered here is the simplest example of this dynamic balance. The instability of a dense fluid supported by a lighter one has been studied extensively, beginning with the analysis of Rayleigh (1883). Taylor (1950) noted that acceleration would give the same effect as gravity, in agreement with the experiments of Lewis (1950). A thorough introduction to the linear theory is given by Chandrasekhar (1961, chap. X), and recent surveys of previous work are given by Sharp (1984) and Kull(l991). Many of the analyses (e.g. Bellman & Pennington 1954; and Menikoff et al. 1978) have focused on the initial small-amplitude growth, where linearized equations are appropriate. One approach for examining the nonlinear effects of large amplitude is direct numerical simulation, e.g. Harlow & Welch (1966) and Tryggvason (1988), or through boundary-integral computations, e.g. Baker & Meiron (1984), and Newhouse & Pozrikidis (1990). To investigate the nonlinear effects analytically, the most common approach has been the perturbation method, where various quantities are expressed as power series in the small initial amplitude or slope, giving linearized equations for each order (e....

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