Abstract
Studying in this paper the stability of plane-parallel flows of an ordinary liquid can be naturally translated into the language of the theory of hydrodynamic resonances. Thus, resonant absorption of oscillations induces stability of the flows of an ideal liquid having a velocity profile without inflection points (Rayleigh theorem), while resonant emission leads to Rayleigh instability in the presence of an inflection point. The flow velocity profile has an inflection point. Thus, the presence of inflection points is a necessary condition for instability. If, however, the velocity profile has inflection points, the flow is stable (Rayleigh's theorem). Note that the sign of the jump depends on whether the neutral oscillations are regarded as the limiting cases of growing or damped oscillations.
Highlights
Rayleigh has established in 1880 (Rayleigh, 1880), that plane-parallel liquid flows (Barston, 1991) of an ideal liquid, with velocity profiles that have no inflection points, are stable Rayleigh's theorem (Rayleigh, 1880).Rayleigh instability (Khenner, et al, 1999; Wolf, 1970a, 1997b; Kumar, et al, 1994) occurs when a heavy fluid is supported by a lighter fluid
In experimental works by (Bezdeneznykh, et al, 1984; Piriz, et al, 2010) for a long horizontal reservoir filled with two immiscible viscous fluids, an interesting phenomenon was found at the interface: the horizontal vibrations lead to the formation of a steady relief
In this paper, based on the previous work (Dou, 2002; Kuznetsov, et al, 2011), it is demonstrated that the stability of plane-parallel flows of an ordinary liquid can be naturally translated into the language of the theory of hydrodynamic resonances
Summary
Rayleigh has established in 1880 (Rayleigh, 1880), that plane-parallel liquid flows (Barston, 1991) of an ideal liquid, with velocity profiles that have no inflection points, are stable Rayleigh's theorem (Rayleigh, 1880). In experimental works by (Bezdeneznykh, et al, 1984; Piriz, et al, 2010) for a long horizontal reservoir filled with two immiscible viscous fluids, an interesting phenomenon was found at the interface: the horizontal vibrations lead to the formation of a steady relief. This formation mechanism has a threshold nature; it is noteworthy that such a wavy relief appears on the interface only if the densities of the two fluids are close enough, i.e. it does not appear for the liquid/gas interface. In this paper, based on the previous work (Dou, 2002; Kuznetsov, et al, 2011), it is demonstrated that the stability of plane-parallel flows of an ordinary liquid can be naturally translated into the language of the theory of hydrodynamic resonances
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