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I'm trying to bolster my understanding of the Rayleigh-Taylor instability, and I've gotten stuck on the point of which fluid (more or less dense) is being accelerated into the other. Cases of uniform acceleration make sense (e.g. plane-parallel fluids under gravity; or being accelerated by a piston), but my understanding falters in more exotic configurations. Consider the following:

A wind tunnel(-like) setup---without gravity---in which one fluid (density $\rho_1$) is at rest, while another ($\rho_2$) is blown into the first with a constant velocity. Because the two substances are colliding --- they'll both feel an acceleration. But because there is no acceleration between the reference frames of $\rho_1$ and $\rho_2$, how do you tell which configuration is stable, and which is not?

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  • $\begingroup$ Also, what do you mean by "configuration is stable." My understanding of colliding flows is that Vishniac instabilities will appear. $\endgroup$
    – Kyle Kanos
    Dec 13, 2013 at 3:34
  • $\begingroup$ @KyleKanos I had never heard of that effect, it seems to be exclusively relevant to radiative transfer, specifically anisotropy formation by scattering... $\endgroup$
    – DilithiumMatrix
    Dec 13, 2013 at 3:50
  • $\begingroup$ It's related to the thermal pressure of a shock matching the ram pressure of an ambient medium, see this short paper for a review. As soon as an inhomogeneity arises in the ambient (which happens in colliding flows), the Vishniac instability kicks in. $\endgroup$
    – Kyle Kanos
    Dec 13, 2013 at 3:58
  • $\begingroup$ @KyleKanos Interesting; I don't see how that's any different from RT however. $\endgroup$
    – DilithiumMatrix
    Dec 13, 2013 at 4:03
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    $\begingroup$ What you are describing sounds closer to an incompressible Richtmyer–Meshkov instability. In the absence of surface tension and gravity, I believe both situations, heavy-light and light-heavy, are linearly unstable. $\endgroup$
    – SimpleLikeAnEgg
    Jan 8, 2014 at 1:28

2 Answers 2

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The initial conditions you are describing are those for a Richtmyer-Meshkov instability (RMI). Experimentally speaking, Richtmyer-Meshkov instability is usually generated by passing a shock wave through a perturbed interface between fluids of differing density. Although at first it might not appear that way, the situation you have proposed is entirely equivalent to a shock wave leading to an impulsive acceleration and can be modeled as such. In fact experiments on "incompressible" fluids under such conditions have been performed. In the absence of surface tension and gravity, both situations, heavy-light and light-heavy, are linearly unstable

Drop tank experiment results from the University of Arizona

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  • $\begingroup$ Thanks for your reply @SimpleLikeAnEgg. This is a special case of the general RT instability. I'm looking for some sort of understanding or explanation for why/why-not the ordering of the fluids matter. $\endgroup$
    – DilithiumMatrix
    Jan 8, 2014 at 21:20
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    $\begingroup$ @zhermes In terms of intuition, the ordering of the fluids matters for RT because of buoyancy. In the absence of a body force like gravity there is no buoyancy force to be concerned with and the order doesn't matter. $\endgroup$
    – SimpleLikeAnEgg
    Jan 9, 2014 at 19:28
  • $\begingroup$ Do you have source indicating why both positions are unstable? $\endgroup$
    – Rick
    Feb 3, 2015 at 17:44
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Instability occurring when a heavy fluid is superposed on a lighter one was first studied by Lord Rayleigh in 1883. The nature of this instability does not change when the problem is posed as a lighter fluid accelerating against a heavier one. G I Taylor first investigated it in this latter sense. The second paper in this reply will answer your question in detail. Its abstract says - "It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated."

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