258:
perturbation retains its sinusoidal shape. However, after the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid). The growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models, resulting in a growth rate that is approximately constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored. The spikes and bubbles then begin to interact with one another in the third stage. Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Also, bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated. This eventually develops into a region of turbulent mixing, which is the fourth and final stage in the evolution. It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the
Reynolds number is sufficiently large.
6164:
6288:. This in turn creates additional vorticity, leading to further misalignment. This concept is depicted in the figure, where it is observed that the two counter-rotating vortices have velocity fields that sum at the peak and trough of the perturbed interface. In the stable configuration, the vorticity, and thus the induced velocity field, will be in a direction that decreases the misalignment and therefore stabilizes the system.
31:
249:
72:
267:
6064:
880:
603:
158:, as the denser material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh. The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are
174:
equations are required to describe fluid motions. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities
252:
This figure represents the evolution of the
Rayleigh–Taylor instability from small wavelength perturbations at the interface (a) which grow into the ubiquitous mushroom shaped spikes (fluid structures of heavy into light fluid) and bubbles (fluid structures of light into heavy fluid) (b) and these
165:
As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely
257:
The evolution of the RTI follows four main stages. In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth. In the early portion of this stage, a sinusoidal initial
5809:
5365:
5081:
4501:
153:
or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of
229:(also known as Rayleigh instability) of a liquid jet. This instability, sometimes called the hosepipe (or firehose) instability, occurs due to surface tension, which acts to break a cylindrical jet into a stream of droplets having the same total volume but higher surface area.
4316:
5847:
4002:
3002:
1288:
3895:
5219:
5480:
681:
461:
2196:
253:
fluid structures interact due to bubble merging and competition (c) eventually developing into a mixing region (d). Here ρ2 represents the heavy fluid and ρ1 represents the light fluid. Gravity is acting downward and the system is RT unstable.
6299:
The analysis in the previous section breaks down when the amplitude of the perturbation is large. The growth then becomes non-linear as the spikes and bubbles of the instability tangle and roll up into vortices. Then, as in the figure,
278:
two-dimensional
Rayleigh–Taylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the simple nature of the base state. Consider a base state in which there is an interface, located at
1848:
2047:
2552:
6268:
3475:
3298:
5692:
5232:
1604:
4935:
4808:
4355:
2852:
1039:
3069:
6167:
Visualization of an unstable
Rayleigh–Taylor instability configuration where baroclinic torque at the interface creates vorticity and induces a velocity field that increases the baroclinic torque. Here ω is vorticity,
3765:
2747:
4177:
4122:
5090:
4669:
179:, A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.
3906:
2913:
1405:
6281:
When in the unstable configuration, for a particular harmonic component of the initial perturbation, the torque on the interface creates vorticity that will tend to increase the misalignment of the
5374:
3774:
3370:
1900:
1153:
1704:
453:
3638:
4706:
3108:
2077:
6059:{\displaystyle \Psi \left(x,z,t\right)=Ae^{-\alpha |z|}\exp \left=A\exp \left(\alpha {\sqrt {\frac {g{\tilde {\mathcal {A}}}}{\alpha }}}t\right)\exp \left(i\alpha x-\alpha |z|\right)\,}
875:{\displaystyle \mathbf {v} ={\overline {\mathbf {v} }}+{\hat {\mathbf {v} }}(z)e^{ikx+\sigma t},\quad p={\overline {p}}+{\hat {p}}(z)e^{ikx+\sigma t},\quad f={\hat {f}}e^{ikx+\sigma t}}
2906:
2679:
2600:
598:{\displaystyle {\overline {\mathbf {v} }}=\mathbf {0} ,\quad {\overline {p}}={\begin{cases}-\rho _{1}gz\quad {\text{for }}z<0,\\-\rho _{2}gz\quad {\text{for }}z>0,\end{cases}}}
5609:
5527:
6147:
When the two layers of the fluid are allowed to have a relative velocity, the instability is generalized to the Kelvin–Helmholtz–Rayleigh–Taylor instability, which includes both the
4170:
1457:
1122:
1082:
4851:
4565:
2234:
1937:
6869:
Roberts, M.S.; Jacobs, J.W. (2015). "The effects of forced small-wavelength, finite-bandwidth initial perturbations and miscibility on the turbulent
Rayleigh Taylor instability".
2267:
1951:
2476:
5836:
4928:
2338:
6103:
3379:
2442:
2375:
1330:
4348:
3165:
2305:
5685:
5643:
5561:
3212:
2405:
2469:
4713:
2783:
2070:
383:
330:
6141:
4038:
673:
4524:
3201:
2632:
1145:
943:
170:, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and
4592:
409:
356:
4878:
3135:
2776:
1715:
632:
303:
65:
3655:
2686:
923:
903:
6201:
6151:
and the
Rayleigh–Taylor instability as special cases. It was recently discovered that the fluid equations governing the linear dynamics of the system admit a
3011:
162:, with the less dense fluid accelerating into the denser fluid. This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.
6627:
Betti, R.; Goncharov, V.N.; McCrory, R.L.; Verdon, C.P. (1998). "Growth rates of the ablative
Rayleigh–Taylor instability in inertial confinement fusion".
1465:
6163:
951:
6824:
Singh, Chamkor; Das, Arup K.; Das, Prasanta K. (2016), "Single-mode instability of a ferrofluid-mercury interface under a nonuniform magnetic field",
134:
explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.
4045:
7180:
7156:
7130:
6567:
5804:{\displaystyle c=\pm i{\sqrt {\frac {g{\mathcal {A}}}{\alpha }}},\qquad {\mathcal {A}}={\frac {\rho _{G}-\rho _{L}}{\rho _{G}+\rho _{L}}},\,}
5360:{\displaystyle c^{2}={\frac {g}{\alpha }}{\frac {\rho _{L}-\rho _{G}}{\rho _{L}+\rho _{G}}}+{\frac {\sigma \alpha }{\rho _{L}+\rho _{G}}}.\,}
1407:; that is to say, surface tension stabilises large wavenumbers or small length scales. Then the maximum growth rate occurs at the wavenumber
3319:
7232:
5076:{\displaystyle c^{2}\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\Psi \left(\rho _{G}-\rho _{L}\right)-\sigma \alpha ^{2}\Psi .\,}
4496:{\displaystyle c^{2}\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\Psi \left(\rho _{G}-\rho _{L}\right)-\sigma \alpha ^{2}\Psi ,\,}
6785:"Numerical Simulations of the Magnetic Rayleigh–Taylor Instability in the Kippenhahn-Schlüter Prominence Model. I. Formation of Upflows"
3484:
6358:
6318:
2269:. To specify the solution fully, it is necessary to fix conditions at the boundaries and interface. This determines the wave speed
237:
114:
which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the
6333:
6323:
6148:
226:
38:
6191:
created by the misalignment of the pressure and density gradients at the perturbed interface, as described by the two-dimensional
4604:
4311:{\displaystyle c\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\eta \left(\rho _{G}-\rho _{L}\right)+\sigma \eta _{xx}.\,}
3900:
6663:
6155:, and the Kelvin–Helmholtz–Rayleigh–Taylor instability occurs when and only when the parity-time symmetry breaks spontaneously.
945:
is the growth rate of the perturbation. Then the linear stability analysis based on the inviscid governing equations shows that
7034:
6313:
1335:
6383:
1283:{\displaystyle \sigma ^{2}={\frac {\rho _{2}-\rho _{1}}{\rho _{2}+\rho _{1}}}gk-{\frac {\gamma k^{3}}{\rho _{2}+\rho _{1}}},}
3899:
The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised
7237:
6987:
Experiments and
Simulations on the Incompressible, Rayleigh–Taylor Instability with Small Wavelength Initial Perturbations
5369:
To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then,
7197:
6301:
5214:{\displaystyle Ac^{2}\alpha \left(-\rho _{G}-\rho _{L}\right)=Ag\left(\rho _{G}-\rho _{L}\right)-\sigma \alpha ^{2}A.\,}
414:
1863:
4674:
7015:
20.^ A. R. Piriz, O. D. Cortazar, J. J. López Cela, and N. A. Tahir, "The
Rayleigh-Taylor instability", Am. J. Phys.
3997:{\displaystyle {\frac {\partial u_{i}'}{\partial t}}=-{\frac {1}{\rho _{i}}}{\frac {\partial p_{i}'}{\partial x}}\,}
248:
7202:
7148:
7114:
6348:
2997:{\displaystyle {\frac {\partial \eta }{\partial t}}+u'{\frac {\partial \eta }{\partial x}}=w'\left(\eta \right).\,}
7060:
214:
explosion 1000 years ago. The RT instability has also recently been discovered in the Sun's outer atmosphere, or
5475:{\displaystyle c^{2}={\frac {g}{\alpha }}{\frac {\rho _{L}-\rho _{G}}{\rho _{L}+\rho _{G}}},\qquad \sigma =0,\,}
3890:{\displaystyle p'_{G}-p'_{L}=g\eta \left(\rho _{G}-\rho _{L}\right)+\sigma \eta _{xx},\qquad {\text{on }}z=0.\,}
2865:
2559:
1618:
The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,
67:), as well as the formation of a "mushroom cap" at a later stage in the third and fourth frame in the sequence.
5573:
5491:
1621:
6554:
Hillebrandt, W.; Höflich, P. (1992). "Supernova 1987a in the Large
Magellanic Cloud". In R. J. Tayler (ed.).
6467:
4601:
Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation
1410:
7227:
7070:(1950). "The instability of liquid surfaces when accelerated in a direction perpendicular to their planes".
6932:
Qin, H.; et al. (2019). "Kelvin–Helmholtz instability is the result of parity-time symmetry breaking".
6732:"Quiescent Prominence Dynamics Observed with the Hinode Solar Optical Telescope. I. Turbulent Upflow Plumes"
3076:
1087:
1047:
6291:
A much simpler explanation of the basic physics of the Rayleigh-Taylor instability can be found in Ref.20.
4815:
4529:
2203:
1905:
7067:
6487:
6343:
2239:
146:
1856:. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays
455:. The velocity field and pressure field in this equilibrium state, denoted with an overbar, are given by
222:
overlies a less dense plasma bubble. This latter case resembles magnetically modulated RT instabilities.
7039:"Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density"
6584:
Chen, H. B.; Hilko, B.; Panarella, E. (1994). "The Rayleigh–Taylor instability in the spherical pinch".
187:
103:
6501:
Wang, C.-Y. & Chevalier R. A. (2000). "Instabilities and Clumping in Type Ia Supernova Remnants".
5816:
4883:
2310:
7079:
6990:
6951:
6878:
6833:
6796:
6743:
6704:
6636:
6593:
6520:
6421:
6353:
6152:
6075:
2637:
1296:
195:
4323:
3140:
2191:{\displaystyle \left(D^{2}-\alpha ^{2}\right)\Psi _{j}=0,\,\,\,\ D={\frac {d}{dz}},\,\,\,\ j=L,G.\,}
507:
4129:
3647:
2276:
The first of these conditions is provided by details at the boundary. The perturbation velocities
1853:
150:
138:
5656:
5614:
5532:
2380:
145:
fluid, the denser fluid on top of the less dense one and both subject to the Earth's gravity. The
7122:
7095:
6967:
6941:
6902:
6609:
6536:
6510:
2447:
675:. Correspondingly, the base state is also slightly perturbed. In the linear theory, we can write
203:
3204:
2410:
2343:
2856:
where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition.
2054:
361:
308:
7176:
7152:
7126:
6894:
6849:
6563:
6108:
4007:
2279:
637:
137:
Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be
127:
123:
4508:
3180:
2611:
1130:
928:
7169:
7087:
7050:
6998:
6959:
6886:
6841:
6804:
6751:
6712:
6708:
6644:
6601:
6528:
6429:
6409:
6285:
4570:
388:
335:
219:
155:
115:
6716:
4856:
3113:
2754:
2042:{\displaystyle \psi \left(x,z,t\right)=e^{i\alpha \left(x-ct\right)}\Psi \left(z\right),\,}
17:
3174:
2547:{\displaystyle \Psi _{L}\left(-\infty \right)=0,\qquad \Psi _{G}\left(\infty \right)=0.\,}
1125:
167:
37:
simulation of a single "finger" of the Rayleigh–Taylor instability. Note the formation of
6180:
is gravity. The thick circular arrows represent the velocity field created by the vortex.
2200:
The domain of the problem is the following: the fluid with label 'L' lives in the region
611:
282:
44:
7083:
7072:
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
6994:
6955:
6882:
6837:
6800:
6747:
6730:
Berger, Thomas E.; Slater, Gregory; Hurlburt, Neal; Shine, Richard; et al. (2010).
6640:
6597:
6524:
6425:
6379:
6263:{\displaystyle {\frac {D\omega }{Dt}}={\frac {1}{\rho ^{2}}}\nabla \rho \times \nabla p}
3470:{\displaystyle p_{G}\left(z=\eta \right)-p_{L}\left(z=\eta \right)=\sigma \eta _{xx}.\,}
6756:
6731:
6338:
6328:
1707:
908:
888:
119:
6809:
6784:
2307:
should satisfy a no-flux condition, so that fluid does not leak out at the boundaries
7221:
7140:
6906:
6670:
6613:
6453:
6433:
5839:
3293:{\displaystyle p_{G}\left(z=\eta \right)-p_{L}\left(z=\eta \right)=\sigma \kappa ,\,}
176:
91:
7099:
6971:
6540:
7213:
plasma Rayleigh–Taylor instability experiment at California Institute of Technology
6472:
1857:
1599:{\displaystyle \sigma _{m}^{2}={\frac {2\gamma }{\rho _{2}-\rho _{1}}}\left^{3/2}.}
215:
191:
159:
95:
34:
4803:{\displaystyle \Psi _{L}=A_{L}e^{\alpha z},\qquad \Psi _{G}=A_{G}e^{-\alpha z}.\,}
2847:{\displaystyle \Psi _{L}\left(0\right)=\Psi _{G}\left(0\right)+{\text{H.O.T.}},\,}
30:
1034:{\displaystyle \sigma ^{2}={\frac {\rho _{2}-\rho _{1}}{\rho _{2}+\rho _{1}}}gk.}
7212:
6275:
207:
199:
99:
76:
7054:
6845:
3137:. Using the normal-mode and streamfunction representations, this condition is
236:, although some might claim this is more accurately described as an example of
41:, in the second and later snapshots shown (starting initially around the level
7061:
https://www.irphe.fr/~clanet/otherpaperfile/articles/Rayleigh/rayleigh1883.pdf
6185:
1843:{\displaystyle {\textbf {u}}'=(u'(x,z,t),w'(x,z,t))=(\psi _{z},-\psi _{x}),\,}
171:
142:
6559:
6195:
3479:
However, this condition refers to the total pressure (base+perturbed), thus
3311:
2072:
is a spatial wavenumber. Thus, the problem reduces to solving the equation
1293:
which indicates that the instability occurs only for a range of wavenumbers
634:. Let this interface be slightly perturbed, so that it assumes the position
233:
211:
183:
131:
111:
7091:
6853:
6515:
3760:{\displaystyle P_{L}=-\rho _{L}gz+p_{0},\qquad P_{G}=-\rho _{G}gz+p_{0},\,}
5842:. By taking the positive solution, we see that the solution has the form
2742:{\displaystyle \Psi _{L}\left(\eta \right)=\Psi _{G}\left(\eta \right).\,}
1610:
Details of the linear stability analysis. A similar derivation appears in
6890:
6282:
6192:
1706:
Because the fluid is assumed incompressible, this velocity field has the
275:
7003:
3064:{\displaystyle {\frac {\partial \eta }{\partial t}}=w'\left(0\right),\,}
270:
Base state of the Rayleigh–Taylor instability. Gravity points downwards.
240:
due to the active heating of the fluid layer at the bottom of the lamp.
71:
7208:
Experiments on Rayleigh–Taylor instability at the University of Arizona
6783:
Hillier, A.; Berger, Thomas; Isobe, Hiroaki; Shibata, Kazunari (2012).
6605:
3642:(As usual, The perturbed quantities can be linearized onto the surface
6963:
6898:
5649:
is purely imaginary. This happens when the heavier fluid sits on top.
6695:
Hester, J. Jeff (2008). "The Crab Nebula: an Astrophysical Chimera".
6648:
6188:
1944:
7038:
2556:
The other three conditions are provided by details at the interface
266:
182:
This process is evident not only in many terrestrial examples, from
6946:
6664:"EVALUATION OF VARIOUS THEORETICAL MODELS FOR UNDERWATER EXPLOSION"
6532:
2273:, which in turn determines the stability properties of the system.
2236:, while the fluid with the label 'G' lives in the upper half-plane
1939:
Next, because of the translational invariance of the system in the
6162:
265:
247:
244:
Stages of development and eventual evolution into turbulent mixing
107:
70:
29:
27:
Unstable behavior of two contacting fluids of different densities
6274:
is the pressure. In this case the dominant pressure gradient is
4350:
and using the normal-mode representation, this relation becomes
608:
where the reference location for the pressure is taken to be at
7207:
232:
Many people have witnessed the RT instability by looking at a
1124:, it is unstable for all wavenumbers. If the interface has a
5986:
5822:
5735:
5717:
5085:
Plugging the solution into this equation gives the relation
4664:{\displaystyle \left(D^{2}-\alpha ^{2}\right)\Psi _{i}=0,\,}
411:. The gravitatioanl acceleration is described by the vector
225:
Note that the RT instability is not to be confused with the
4117:{\displaystyle p_{i}'=\rho _{i}cD\Psi _{i},\qquad i=L,G.\,}
591:
5567:
is real. This happens when the lighter fluid sits on top;
198:. For example, RT instability structure is evident in the
6380:"Parallel AMR Code for Compressible MHD or HD Equations"
6304:
of the full problem is required to describe the system.
2681:. Using the stream function representation, this gives
1400:{\displaystyle k_{c}^{2}=(\rho _{2}-\rho _{1})g/\gamma }
7175:(2nd ed.). Cambridge: Cambridge University Press.
3365:{\displaystyle \kappa =\nabla ^{2}\eta =\eta _{xx}.\,}
6204:
6111:
6078:
5850:
5819:
5695:
5659:
5617:
5576:
5535:
5494:
5377:
5235:
5093:
4938:
4886:
4859:
4818:
4716:
4677:
4607:
4573:
4532:
4511:
4358:
4326:
4180:
4132:
4126:
Putting this last equation and the jump condition on
4048:
4010:
3909:
3777:
3658:
3487:
3382:
3322:
3314:
of the interface, which in a linear approximation is
3215:
3183:
3143:
3116:
3079:
3014:
2916:
2868:
2786:
2757:
2689:
2640:
2614:
2562:
2479:
2450:
2413:
2383:
2346:
2313:
2282:
2242:
2206:
2080:
2057:
1954:
1908:
1866:
1718:
1624:
1468:
1413:
1338:
1299:
1156:
1133:
1090:
1050:
954:
931:
911:
891:
684:
640:
614:
464:
417:
391:
364:
338:
311:
305:
that separates fluid media with different densities,
285:
47:
6989:(PhD thesis). University of Arizona Dissertations.
7198:Java demonstration of the RT instability in fluids
7168:
6262:
6135:
6097:
6058:
5830:
5803:
5679:
5637:
5603:
5555:
5521:
5474:
5359:
5213:
5075:
4922:
4872:
4845:
4802:
4700:
4663:
4586:
4559:
4518:
4495:
4342:
4310:
4164:
4116:
4032:
3996:
3889:
3759:
3632:
3469:
3364:
3292:
3195:
3159:
3129:
3102:
3063:
2996:
2900:
2846:
2770:
2741:
2673:
2626:
2594:
2546:
2463:
2436:
2399:
2369:
2332:
2299:
2261:
2228:
2190:
2064:
2041:
1931:
1894:
1842:
1698:
1598:
1451:
1399:
1324:
1282:
1139:
1116:
1076:
1033:
937:
917:
897:
874:
667:
626:
597:
447:
403:
377:
350:
324:
297:
59:
6068:and this is associated to the interface position
448:{\displaystyle \mathbf {g} =-g\,\mathbf {e} _{z}}
6184:The RT instability can be seen as the result of
3633:{\displaystyle \left-\left=\sigma \eta _{xx}.\,}
3177:, the pressure difference over the interface at
1895:{\displaystyle \nabla \times {\textbf {u}}'=0\,}
6918:
6916:
4701:{\displaystyle \Psi \left(\pm \infty \right)\,}
7043:Proceedings of the London Mathematical Society
6864:
6862:
6449:
6447:
6445:
6443:
4320:Substituting the second interfacial condition
6770:
6669:. U.S. Government. p. 86. Archived from
5227:cancels from both sides and we are left with
1611:
8:
6410:"An Overview of Rayleigh–Taylor Instability"
4930:The third interfacial condition states that
4812:The first interfacial condition states that
6697:Annual Review of Astronomy and Astrophysics
2471:. In terms of the streamfunction, this is
141:by two completely plane-parallel layers of
6488:"Why Nuclear Bombs Create Mushroom Clouds"
7002:
6945:
6808:
6755:
6514:
6403:
6401:
6399:
6237:
6228:
6205:
6203:
6132:
6121:
6110:
6094:
6077:
6055:
6045:
6037:
5985:
5983:
5982:
5975:
5904:
5896:
5889:
5849:
5827:
5821:
5820:
5818:
5800:
5788:
5775:
5763:
5750:
5743:
5734:
5733:
5716:
5715:
5708:
5694:
5676:
5664:
5658:
5653:Now, when the heavier fluid sits on top,
5634:
5622:
5616:
5600:
5594:
5581:
5575:
5552:
5540:
5534:
5518:
5512:
5499:
5493:
5471:
5446:
5433:
5421:
5408:
5401:
5391:
5382:
5376:
5356:
5344:
5331:
5316:
5304:
5291:
5279:
5266:
5259:
5249:
5240:
5234:
5210:
5198:
5177:
5164:
5135:
5122:
5101:
5092:
5072:
5060:
5039:
5026:
4997:
4984:
4971:
4958:
4943:
4937:
4919:
4904:
4891:
4885:
4869:
4858:
4842:
4836:
4823:
4817:
4799:
4784:
4774:
4761:
4744:
4734:
4721:
4715:
4697:
4676:
4660:
4645:
4630:
4617:
4606:
4583:
4572:
4556:
4550:
4537:
4531:
4515:
4510:
4492:
4480:
4459:
4446:
4417:
4404:
4391:
4378:
4363:
4357:
4339:
4325:
4307:
4295:
4274:
4261:
4232:
4219:
4206:
4193:
4179:
4153:
4137:
4131:
4113:
4085:
4069:
4053:
4047:
4029:
4009:
3993:
3973:
3963:
3955:
3946:
3920:
3910:
3908:
3886:
3872:
3859:
3838:
3825:
3798:
3782:
3776:
3756:
3747:
3728:
3712:
3698:
3679:
3663:
3657:
3629:
3617:
3582:
3558:
3521:
3497:
3486:
3466:
3454:
3419:
3387:
3381:
3361:
3349:
3333:
3321:
3289:
3252:
3220:
3214:
3182:
3156:
3142:
3126:
3115:
3099:
3078:
3060:
3015:
3013:
2993:
2948:
2917:
2915:
2901:{\displaystyle z=\eta \left(x,t\right)\,}
2897:
2867:
2843:
2835:
2815:
2791:
2785:
2767:
2756:
2738:
2718:
2694:
2688:
2670:
2661:
2645:
2639:
2613:
2595:{\displaystyle z=\eta \left(x,t\right)\,}
2591:
2561:
2543:
2520:
2484:
2478:
2460:
2449:
2433:
2418:
2412:
2396:
2382:
2366:
2351:
2345:
2329:
2312:
2296:
2287:
2281:
2258:
2241:
2225:
2205:
2187:
2165:
2164:
2163:
2145:
2135:
2134:
2133:
2118:
2103:
2090:
2079:
2061:
2056:
2038:
1990:
1953:
1928:
1913:
1907:
1902:. In the streamfunction representation,
1891:
1875:
1874:
1865:
1839:
1827:
1811:
1721:
1720:
1717:
1695:
1623:
1583:
1579:
1552:
1539:
1529:
1515:
1502:
1487:
1478:
1473:
1467:
1442:
1437:
1431:
1418:
1412:
1389:
1377:
1364:
1348:
1343:
1337:
1316:
1298:
1268:
1255:
1243:
1233:
1215:
1202:
1190:
1177:
1170:
1161:
1155:
1132:
1108:
1095:
1089:
1068:
1055:
1049:
1013:
1000:
988:
975:
968:
959:
953:
930:
910:
890:
851:
836:
835:
804:
780:
779:
766:
735:
711:
709:
708:
695:
693:
685:
683:
639:
613:
571:
558:
530:
517:
502:
489:
480:
467:
465:
463:
439:
434:
432:
418:
416:
390:
369:
363:
337:
316:
310:
284:
210:is sweeping up ejected material from the
46:
7119:Hydrodynamic and Hydromagnetic Stability
6468:"Rayleigh–Taylor instability and mixing"
5604:{\displaystyle \rho _{G}>\rho _{L}\,}
5522:{\displaystyle \rho _{G}<\rho _{L}\,}
1699:{\displaystyle (u'(x,z,t),w'(x,z,t)).\,}
1084:, the base state is stable and while if
6370:
3171:Pressure relation across the interface:
1943:-direction, it is possible to make the
1452:{\displaystyle k_{m}=k_{c}/{\sqrt {3}}}
1147:, then the dispersion relation becomes
175:divided by their sum is defined as the
7203:Actual images and videos of RT fingers
7145:Introduction to hydrodynamic stability
6717:10.1146/annurev.astro.45.051806.110608
6270:, where ω is vorticity, ρ density and
3103:{\displaystyle w'\left(\eta \right)\,}
1117:{\displaystyle \rho _{2}>\rho _{1}}
1077:{\displaystyle \rho _{2}<\rho _{1}}
75:RT instability fingers evident in the
4846:{\displaystyle \Psi _{L}=\Psi _{G}\,}
4560:{\displaystyle \Psi _{L}=\Psi _{G}\,}
2229:{\displaystyle -\infty <z\leq 0\,}
1932:{\displaystyle \nabla ^{2}\psi =0.\,}
7:
7035:Rayleigh, Lord (John William Strutt)
3167:, the second interfacial condition.
2262:{\displaystyle 0\leq z<\infty \,}
7167:Drazin, P. G.; Reid, W. H. (2004).
6278:, resulting from the acceleration.
1876:
1722:
6254:
6245:
6088:
5851:
5066:
5014:
4994:
4968:
4833:
4820:
4758:
4718:
4689:
4678:
4642:
4547:
4534:
4512:
4486:
4434:
4414:
4388:
4336:
4229:
4203:
4082:
3984:
3966:
3931:
3913:
3330:
3153:
3026:
3018:
2959:
2951:
2928:
2920:
2812:
2788:
2715:
2691:
2530:
2517:
2498:
2481:
2457:
2393:
2323:
2255:
2210:
2115:
2021:
1910:
1867:
25:
7059:(Original paper is available at:
2908:, the kinematic condition holds:
2634:, the vertical velocities match,
5831:{\displaystyle {\mathcal {A}}\,}
4923:{\displaystyle A_{L}=A_{G}=A.\,}
4505:where there is no need to label
3110:is linearized on to the surface
2606:Continuity of vertical velocity:
2333:{\displaystyle z=\pm \infty .\,}
712:
696:
686:
481:
468:
435:
419:
6098:{\displaystyle c\eta =\Psi .\,}
5732:
5458:
4756:
4526:(only its derivatives) because
4094:
3871:
3707:
2674:{\displaystyle w'_{L}=w'_{G}\,}
2515:
1325:{\displaystyle 0<k<k_{c}}
828:
759:
570:
529:
488:
6384:Los Alamos National Laboratory
6046:
6038:
5990:
5905:
5897:
4343:{\displaystyle c\eta =\Psi \,}
3160:{\displaystyle c\eta =\Psi \,}
1852:where the subscripts indicate
1833:
1804:
1798:
1795:
1777:
1763:
1745:
1734:
1689:
1686:
1668:
1654:
1636:
1625:
1558:
1532:
1383:
1357:
905:is the real wavenumber in the
841:
797:
791:
785:
728:
722:
716:
662:
650:
39:Kelvin–Helmholtz instabilities
1:
6319:Richtmyer–Meshkov instability
4671:with the boundary conditions
4165:{\displaystyle p'_{G}-p'_{L}}
6757:10.1088/0004-637X/716/2/1288
6434:10.1016/0167-2789(84)90510-4
6334:Plateau–Rayleigh instability
6324:Kelvin–Helmholtz instability
6149:Kelvin–Helmholtz instability
5680:{\displaystyle c^{2}<0\,}
5638:{\displaystyle c^{2}<0\,}
5556:{\displaystyle c^{2}>0\,}
3006:Linearizing, this is simply
2400:{\displaystyle z=-\infty \,}
771:
700:
494:
472:
227:Plateau–Rayleigh instability
7233:Fluid dynamic instabilities
7115:Chandrasekhar, Subrahmanyan
7068:Taylor, Sir Geoffrey Ingram
6810:10.1088/0004-637X/746/2/120
6378:Li, Shengtai & Hui Li.
6172:is pressure, ρ is density,
3306:is the surface tension and
2860:The free-surface condition:
2464:{\displaystyle z=\infty \,}
84:Rayleigh–Taylor instability
18:Rayleigh-Taylor instability
7254:
7149:Cambridge University Press
6871:Journal of Fluid Mechanics
6846:10.1103/PhysRevE.94.012803
6359:Rayleigh–Bénard convection
6314:Saffman–Taylor instability
2437:{\displaystyle w_{G}'=0\,}
2370:{\displaystyle w_{L}'=0\,}
238:Rayleigh–Bénard convection
218:, when a relatively dense
6789:The Astrophysical Journal
6736:The Astrophysical Journal
6503:The Astrophysical Journal
2065:{\displaystyle \alpha \,}
378:{\displaystyle \rho _{2}}
325:{\displaystyle \rho _{1}}
262:Linear stability analysis
202:, in which the expanding
7055:10.1112/plms/s1-14.1.170
7028:Original research papers
6922:Drazin (2002) pp. 48–52.
6586:Journal of Fusion Energy
6136:{\displaystyle B=A/c.\,}
4033:{\displaystyle i=L,G,\,}
2300:{\displaystyle w'_{i}\,}
668:{\displaystyle z=f(x,t)}
149:here is unstable to any
6709:2008ARA&A..46..127H
6662:John Pritchett (1971).
4519:{\displaystyle \Psi \,}
3903:for the perturbations,
3196:{\displaystyle z=\eta }
2627:{\displaystyle z=\eta }
1140:{\displaystyle \gamma }
938:{\displaystyle \sigma }
7171:Hydrodynamic stability
7092:10.1098/rspa.1950.0052
6985:Roberts, M.S. (2012).
6344:Hydrodynamic stability
6264:
6181:
6137:
6099:
6060:
5832:
5805:
5681:
5639:
5605:
5557:
5523:
5476:
5361:
5215:
5077:
4924:
4874:
4847:
4804:
4702:
4665:
4588:
4587:{\displaystyle z=0.\,}
4561:
4520:
4497:
4344:
4312:
4166:
4118:
4034:
3998:
3891:
3761:
3634:
3471:
3366:
3294:
3197:
3161:
3131:
3104:
3065:
2998:
2902:
2848:
2772:
2743:
2675:
2628:
2596:
2548:
2465:
2438:
2401:
2371:
2334:
2301:
2263:
2230:
2192:
2066:
2043:
1933:
1896:
1844:
1700:
1600:
1453:
1401:
1326:
1284:
1141:
1118:
1078:
1035:
939:
919:
899:
876:
669:
628:
599:
449:
405:
404:{\displaystyle z>0}
379:
352:
351:{\displaystyle z<0}
326:
299:
271:
254:
79:
68:
61:
6265:
6166:
6159:Vorticity explanation
6138:
6100:
6061:
5833:
5806:
5682:
5640:
5606:
5558:
5524:
5477:
5362:
5216:
5078:
4925:
4875:
4873:{\displaystyle z=0\,}
4848:
4805:
4703:
4666:
4589:
4562:
4521:
4498:
4345:
4313:
4167:
4119:
4035:
3999:
3892:
3762:
3635:
3472:
3367:
3295:
3198:
3162:
3132:
3130:{\displaystyle z=0\,}
3105:
3066:
2999:
2903:
2849:
2773:
2771:{\displaystyle z=0\,}
2744:
2676:
2629:
2597:
2549:
2466:
2439:
2402:
2372:
2335:
2302:
2264:
2231:
2193:
2067:
2044:
1934:
1897:
1845:
1701:
1601:
1454:
1402:
1327:
1285:
1142:
1119:
1079:
1036:
940:
920:
900:
877:
670:
629:
600:
450:
406:
380:
353:
327:
300:
269:
251:
74:
62:
33:
7238:Plasma instabilities
6891:10.1017/jfm.2015.599
6562:. pp. 249–302.
6556:Stellar Astrophysics
6466:David Youngs (ed.).
6408:Sharp, D.H. (1984).
6354:Fluid thread breakup
6349:Kármán vortex street
6302:numerical simulation
6202:
6153:parity-time symmetry
6109:
6076:
5848:
5817:
5693:
5657:
5615:
5574:
5533:
5492:
5375:
5233:
5091:
4936:
4884:
4857:
4816:
4714:
4675:
4605:
4571:
4530:
4509:
4356:
4324:
4178:
4130:
4046:
4008:
3907:
3775:
3656:
3485:
3380:
3320:
3213:
3181:
3141:
3114:
3077:
3012:
2914:
2866:
2862:At the free surface
2784:
2755:
2687:
2638:
2612:
2560:
2477:
2448:
2411:
2381:
2344:
2311:
2280:
2240:
2204:
2078:
2055:
1952:
1906:
1864:
1716:
1622:
1466:
1411:
1336:
1297:
1154:
1131:
1088:
1048:
952:
929:
909:
889:
682:
638:
612:
462:
415:
389:
362:
336:
309:
283:
196:electrohydrodynamics
45:
7084:1950RSPSA.201..192T
6995:2012PhDT.......222R
6956:2019PhPl...26c2102Q
6883:2016JFM...787...50R
6838:2016PhRvE..94a2803S
6801:2012ApJ...746..120H
6748:2010ApJ...716.1288B
6676:on October 18, 2012
6641:1998PhPl....5.1446B
6598:1994JFuE...13..275C
6525:2001ApJ...549.1119W
6490:. 20 November 2013.
6426:1984PhyD...12....3S
6295:Late-time behaviour
4161:
4145:
4061:
3981:
3928:
3806:
3790:
3648:hydrostatic balance
3590:
3529:
3073:where the velocity
2669:
2653:
2426:
2359:
2295:
1854:partial derivatives
1614:, §92, pp. 433–435.
1483:
1353:
627:{\displaystyle z=0}
298:{\displaystyle z=0}
60:{\displaystyle y=0}
7123:Dover Publications
6934:Physics of Plasmas
6771:Chandrasekhar 1981
6629:Physics of Plasmas
6606:10.1007/BF02215847
6516:astro-ph/0005105v1
6260:
6182:
6133:
6095:
6056:
5828:
5801:
5677:
5635:
5601:
5553:
5519:
5472:
5357:
5211:
5073:
4920:
4870:
4843:
4800:
4698:
4661:
4584:
4557:
4516:
4493:
4340:
4308:
4162:
4149:
4133:
4114:
4049:
4030:
3994:
3969:
3916:
3887:
3794:
3778:
3757:
3630:
3578:
3517:
3467:
3362:
3290:
3193:
3173:For the case with
3157:
3127:
3100:
3061:
2994:
2898:
2844:
2768:
2739:
2671:
2657:
2641:
2624:
2592:
2544:
2461:
2434:
2414:
2397:
2367:
2347:
2330:
2297:
2283:
2259:
2226:
2188:
2062:
2039:
1929:
1892:
1840:
1696:
1612:Chandrasekhar 1981
1596:
1469:
1459:and its value is
1449:
1397:
1339:
1322:
1280:
1137:
1114:
1074:
1031:
935:
915:
895:
872:
665:
624:
595:
590:
445:
401:
375:
348:
322:
295:
272:
255:
204:pulsar wind nebula
188:weather inversions
128:nuclear explosions
124:volcanic eruptions
80:
69:
57:
7182:978-0-521-52541-1
7158:978-0-521-00965-2
7132:978-0-486-64071-6
7078:(1065): 192–196.
6964:10.1063/1.5088498
6826:Physical Review E
6569:978-0-7503-0200-5
6456:(2002) pp. 50–51.
6243:
6223:
6001:
6000:
5993:
5795:
5727:
5726:
5453:
5399:
5351:
5311:
5257:
4708:has the solution
3991:
3961:
3938:
3875:
3033:
2966:
2935:
2838:
2168:
2158:
2138:
1878:
1724:
1573:
1522:
1447:
1275:
1222:
1020:
918:{\displaystyle x}
898:{\displaystyle k}
844:
788:
774:
719:
703:
574:
533:
497:
475:
16:(Redirected from
7245:
7186:
7174:
7162:
7136:
7103:
7058:
7009:
7008:
7006:
6982:
6976:
6975:
6949:
6929:
6923:
6920:
6911:
6910:
6866:
6857:
6856:
6821:
6815:
6814:
6812:
6780:
6774:
6768:
6762:
6761:
6759:
6742:(2): 1288–1307.
6727:
6721:
6720:
6692:
6686:
6685:
6683:
6681:
6675:
6668:
6659:
6653:
6652:
6649:10.1063/1.872802
6635:(5): 1446–1454.
6624:
6618:
6617:
6581:
6575:
6573:
6551:
6545:
6544:
6518:
6509:(2): 1119–1134.
6498:
6492:
6491:
6484:
6478:
6477:
6463:
6457:
6451:
6438:
6437:
6405:
6394:
6393:
6391:
6390:
6375:
6269:
6267:
6266:
6261:
6244:
6242:
6241:
6229:
6224:
6222:
6214:
6206:
6176:is velocity and
6142:
6140:
6139:
6134:
6125:
6104:
6102:
6101:
6096:
6065:
6063:
6062:
6057:
6054:
6050:
6049:
6041:
6010:
6006:
6002:
5996:
5995:
5994:
5989:
5984:
5977:
5976:
5954:
5950:
5949:
5945:
5910:
5909:
5908:
5900:
5878:
5874:
5837:
5835:
5834:
5829:
5826:
5825:
5810:
5808:
5807:
5802:
5796:
5794:
5793:
5792:
5780:
5779:
5769:
5768:
5767:
5755:
5754:
5744:
5739:
5738:
5728:
5722:
5721:
5720:
5710:
5709:
5686:
5684:
5683:
5678:
5669:
5668:
5644:
5642:
5641:
5636:
5627:
5626:
5610:
5608:
5607:
5602:
5599:
5598:
5586:
5585:
5562:
5560:
5559:
5554:
5545:
5544:
5528:
5526:
5525:
5520:
5517:
5516:
5504:
5503:
5481:
5479:
5478:
5473:
5454:
5452:
5451:
5450:
5438:
5437:
5427:
5426:
5425:
5413:
5412:
5402:
5400:
5392:
5387:
5386:
5366:
5364:
5363:
5358:
5352:
5350:
5349:
5348:
5336:
5335:
5325:
5317:
5312:
5310:
5309:
5308:
5296:
5295:
5285:
5284:
5283:
5271:
5270:
5260:
5258:
5250:
5245:
5244:
5220:
5218:
5217:
5212:
5203:
5202:
5187:
5183:
5182:
5181:
5169:
5168:
5145:
5141:
5140:
5139:
5127:
5126:
5106:
5105:
5082:
5080:
5079:
5074:
5065:
5064:
5049:
5045:
5044:
5043:
5031:
5030:
5007:
5003:
5002:
5001:
4989:
4988:
4976:
4975:
4963:
4962:
4948:
4947:
4929:
4927:
4926:
4921:
4909:
4908:
4896:
4895:
4879:
4877:
4876:
4871:
4852:
4850:
4849:
4844:
4841:
4840:
4828:
4827:
4809:
4807:
4806:
4801:
4795:
4794:
4779:
4778:
4766:
4765:
4752:
4751:
4739:
4738:
4726:
4725:
4707:
4705:
4704:
4699:
4696:
4692:
4670:
4668:
4667:
4662:
4650:
4649:
4640:
4636:
4635:
4634:
4622:
4621:
4593:
4591:
4590:
4585:
4566:
4564:
4563:
4558:
4555:
4554:
4542:
4541:
4525:
4523:
4522:
4517:
4502:
4500:
4499:
4494:
4485:
4484:
4469:
4465:
4464:
4463:
4451:
4450:
4427:
4423:
4422:
4421:
4409:
4408:
4396:
4395:
4383:
4382:
4368:
4367:
4349:
4347:
4346:
4341:
4317:
4315:
4314:
4309:
4303:
4302:
4284:
4280:
4279:
4278:
4266:
4265:
4242:
4238:
4237:
4236:
4224:
4223:
4211:
4210:
4198:
4197:
4171:
4169:
4168:
4163:
4157:
4141:
4123:
4121:
4120:
4115:
4090:
4089:
4074:
4073:
4057:
4039:
4037:
4036:
4031:
4003:
4001:
4000:
3995:
3992:
3990:
3982:
3977:
3964:
3962:
3960:
3959:
3947:
3939:
3937:
3929:
3924:
3911:
3896:
3894:
3893:
3888:
3876:
3873:
3867:
3866:
3848:
3844:
3843:
3842:
3830:
3829:
3802:
3786:
3766:
3764:
3763:
3758:
3752:
3751:
3733:
3732:
3717:
3716:
3703:
3702:
3684:
3683:
3668:
3667:
3639:
3637:
3636:
3631:
3625:
3624:
3606:
3602:
3601:
3586:
3574:
3563:
3562:
3545:
3541:
3540:
3525:
3513:
3502:
3501:
3476:
3474:
3473:
3468:
3462:
3461:
3443:
3439:
3424:
3423:
3411:
3407:
3392:
3391:
3371:
3369:
3368:
3363:
3357:
3356:
3338:
3337:
3299:
3297:
3296:
3291:
3276:
3272:
3257:
3256:
3244:
3240:
3225:
3224:
3203:is given by the
3202:
3200:
3199:
3194:
3166:
3164:
3163:
3158:
3136:
3134:
3133:
3128:
3109:
3107:
3106:
3101:
3098:
3087:
3070:
3068:
3067:
3062:
3056:
3045:
3034:
3032:
3024:
3016:
3003:
3001:
3000:
2995:
2989:
2978:
2967:
2965:
2957:
2949:
2947:
2936:
2934:
2926:
2918:
2907:
2905:
2904:
2899:
2896:
2892:
2853:
2851:
2850:
2845:
2839:
2836:
2831:
2820:
2819:
2807:
2796:
2795:
2777:
2775:
2774:
2769:
2751:Expanding about
2748:
2746:
2745:
2740:
2734:
2723:
2722:
2710:
2699:
2698:
2680:
2678:
2677:
2672:
2665:
2649:
2633:
2631:
2630:
2625:
2601:
2599:
2598:
2593:
2590:
2586:
2553:
2551:
2550:
2545:
2536:
2525:
2524:
2505:
2501:
2489:
2488:
2470:
2468:
2467:
2462:
2443:
2441:
2440:
2435:
2422:
2406:
2404:
2403:
2398:
2376:
2374:
2373:
2368:
2355:
2339:
2337:
2336:
2331:
2306:
2304:
2303:
2298:
2291:
2268:
2266:
2265:
2260:
2235:
2233:
2232:
2227:
2197:
2195:
2194:
2189:
2166:
2159:
2157:
2146:
2136:
2123:
2122:
2113:
2109:
2108:
2107:
2095:
2094:
2071:
2069:
2068:
2063:
2048:
2046:
2045:
2040:
2034:
2020:
2019:
2018:
2014:
1982:
1978:
1938:
1936:
1935:
1930:
1918:
1917:
1901:
1899:
1898:
1893:
1884:
1880:
1879:
1849:
1847:
1846:
1841:
1832:
1831:
1816:
1815:
1776:
1744:
1730:
1726:
1725:
1705:
1703:
1702:
1697:
1667:
1635:
1605:
1603:
1602:
1597:
1592:
1591:
1587:
1578:
1574:
1572:
1564:
1557:
1556:
1544:
1543:
1530:
1523:
1521:
1520:
1519:
1507:
1506:
1496:
1488:
1482:
1477:
1458:
1456:
1455:
1450:
1448:
1443:
1441:
1436:
1435:
1423:
1422:
1406:
1404:
1403:
1398:
1393:
1382:
1381:
1369:
1368:
1352:
1347:
1331:
1329:
1328:
1323:
1321:
1320:
1289:
1287:
1286:
1281:
1276:
1274:
1273:
1272:
1260:
1259:
1249:
1248:
1247:
1234:
1223:
1221:
1220:
1219:
1207:
1206:
1196:
1195:
1194:
1182:
1181:
1171:
1166:
1165:
1146:
1144:
1143:
1138:
1123:
1121:
1120:
1115:
1113:
1112:
1100:
1099:
1083:
1081:
1080:
1075:
1073:
1072:
1060:
1059:
1040:
1038:
1037:
1032:
1021:
1019:
1018:
1017:
1005:
1004:
994:
993:
992:
980:
979:
969:
964:
963:
944:
942:
941:
936:
924:
922:
921:
916:
904:
902:
901:
896:
881:
879:
878:
873:
871:
870:
846:
845:
837:
824:
823:
790:
789:
781:
775:
767:
755:
754:
721:
720:
715:
710:
704:
699:
694:
689:
674:
672:
671:
666:
633:
631:
630:
625:
604:
602:
601:
596:
594:
593:
575:
572:
563:
562:
534:
531:
522:
521:
498:
490:
484:
476:
471:
466:
454:
452:
451:
446:
444:
443:
438:
422:
410:
408:
407:
402:
384:
382:
381:
376:
374:
373:
357:
355:
354:
349:
331:
329:
328:
323:
321:
320:
304:
302:
301:
296:
220:solar prominence
168:linear equations
166:approximated by
156:potential energy
126:and atmospheric
122:like those from
116:gravity of Earth
66:
64:
63:
58:
21:
7253:
7252:
7248:
7247:
7246:
7244:
7243:
7242:
7218:
7217:
7194:
7183:
7166:
7163:xvii+238 pages.
7159:
7139:
7133:
7113:
7110:
7066:
7033:
7030:
7025:
7013:
7012:
6984:
6983:
6979:
6931:
6930:
6926:
6921:
6914:
6868:
6867:
6860:
6823:
6822:
6818:
6782:
6781:
6777:
6769:
6765:
6729:
6728:
6724:
6694:
6693:
6689:
6679:
6677:
6673:
6666:
6661:
6660:
6656:
6626:
6625:
6621:
6583:
6582:
6578:
6574:. See page 274.
6570:
6553:
6552:
6548:
6500:
6499:
6495:
6486:
6485:
6481:
6465:
6464:
6460:
6452:
6441:
6407:
6406:
6397:
6388:
6386:
6377:
6376:
6372:
6367:
6310:
6297:
6233:
6215:
6207:
6200:
6199:
6161:
6145:
6144:
6107:
6106:
6074:
6073:
6021:
6017:
5978:
5971:
5967:
5932:
5928:
5921:
5917:
5885:
5858:
5854:
5846:
5845:
5815:
5814:
5784:
5771:
5770:
5759:
5746:
5745:
5711:
5691:
5690:
5660:
5655:
5654:
5618:
5613:
5612:
5590:
5577:
5572:
5571:
5536:
5531:
5530:
5508:
5495:
5490:
5489:
5442:
5429:
5428:
5417:
5404:
5403:
5378:
5373:
5372:
5340:
5327:
5326:
5318:
5300:
5287:
5286:
5275:
5262:
5261:
5236:
5231:
5230:
5194:
5173:
5160:
5159:
5155:
5131:
5118:
5114:
5110:
5097:
5089:
5088:
5056:
5035:
5022:
5021:
5017:
4993:
4980:
4967:
4954:
4953:
4949:
4939:
4934:
4933:
4900:
4887:
4882:
4881:
4880:, which forces
4855:
4854:
4832:
4819:
4814:
4813:
4780:
4770:
4757:
4740:
4730:
4717:
4712:
4711:
4685:
4681:
4673:
4672:
4641:
4626:
4613:
4612:
4608:
4603:
4602:
4569:
4568:
4546:
4533:
4528:
4527:
4507:
4506:
4476:
4455:
4442:
4441:
4437:
4413:
4400:
4387:
4374:
4373:
4369:
4359:
4354:
4353:
4322:
4321:
4291:
4270:
4257:
4256:
4252:
4228:
4215:
4202:
4189:
4188:
4184:
4176:
4175:
4128:
4127:
4081:
4065:
4044:
4043:
4006:
4005:
3983:
3965:
3951:
3930:
3912:
3905:
3904:
3901:Euler equations
3855:
3834:
3821:
3820:
3816:
3773:
3772:
3743:
3724:
3708:
3694:
3675:
3659:
3654:
3653:
3613:
3591:
3564:
3554:
3553:
3549:
3530:
3503:
3493:
3492:
3488:
3483:
3482:
3450:
3429:
3425:
3415:
3397:
3393:
3383:
3378:
3377:
3345:
3329:
3318:
3317:
3262:
3258:
3248:
3230:
3226:
3216:
3211:
3210:
3179:
3178:
3175:surface tension
3139:
3138:
3112:
3111:
3088:
3080:
3075:
3074:
3046:
3038:
3025:
3017:
3010:
3009:
2979:
2971:
2958:
2950:
2940:
2927:
2919:
2912:
2911:
2882:
2878:
2864:
2863:
2821:
2811:
2797:
2787:
2782:
2781:
2753:
2752:
2724:
2714:
2700:
2690:
2685:
2684:
2636:
2635:
2610:
2609:
2576:
2572:
2558:
2557:
2526:
2516:
2494:
2490:
2480:
2475:
2474:
2446:
2445:
2409:
2408:
2379:
2378:
2342:
2341:
2309:
2308:
2278:
2277:
2238:
2237:
2202:
2201:
2150:
2114:
2099:
2086:
2085:
2081:
2076:
2075:
2053:
2052:
2024:
2001:
1997:
1986:
1962:
1958:
1950:
1949:
1909:
1904:
1903:
1873:
1862:
1861:
1823:
1807:
1769:
1737:
1719:
1714:
1713:
1710:representation
1660:
1628:
1620:
1619:
1615:
1565:
1548:
1535:
1531:
1525:
1524:
1511:
1498:
1497:
1489:
1464:
1463:
1427:
1414:
1409:
1408:
1373:
1360:
1334:
1333:
1312:
1295:
1294:
1264:
1251:
1250:
1239:
1235:
1211:
1198:
1197:
1186:
1173:
1172:
1157:
1152:
1151:
1129:
1128:
1126:surface tension
1104:
1091:
1086:
1085:
1064:
1051:
1046:
1045:
1009:
996:
995:
984:
971:
970:
955:
950:
949:
927:
926:
925:-direction and
907:
906:
887:
886:
847:
800:
731:
680:
679:
636:
635:
610:
609:
589:
588:
554:
548:
547:
513:
503:
460:
459:
433:
413:
412:
387:
386:
365:
360:
359:
334:
333:
312:
307:
306:
281:
280:
264:
246:
206:powered by the
120:mushroom clouds
43:
42:
28:
23:
22:
15:
12:
11:
5:
7251:
7249:
7241:
7240:
7235:
7230:
7228:Fluid dynamics
7220:
7219:
7216:
7215:
7210:
7205:
7200:
7193:
7192:External links
7190:
7189:
7188:
7181:
7164:
7157:
7137:
7131:
7109:
7106:
7105:
7104:
7064:
7029:
7026:
7024:
7021:
7011:
7010:
6977:
6924:
6912:
6858:
6816:
6795:(2): 120–133.
6775:
6763:
6722:
6687:
6654:
6619:
6592:(4): 275–280.
6576:
6568:
6546:
6533:10.1086/319439
6493:
6479:
6458:
6439:
6395:
6369:
6368:
6366:
6363:
6362:
6361:
6356:
6351:
6346:
6341:
6339:Salt fingering
6336:
6331:
6329:Mushroom cloud
6326:
6321:
6316:
6309:
6306:
6296:
6293:
6259:
6256:
6253:
6250:
6247:
6240:
6236:
6232:
6227:
6221:
6218:
6213:
6210:
6160:
6157:
6131:
6128:
6124:
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6093:
6090:
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6084:
6081:
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6048:
6044:
6040:
6036:
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6027:
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6020:
6016:
6013:
6009:
6005:
5999:
5992:
5988:
5981:
5974:
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5966:
5963:
5960:
5957:
5953:
5948:
5944:
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5938:
5935:
5931:
5927:
5924:
5920:
5916:
5913:
5907:
5903:
5899:
5895:
5892:
5888:
5884:
5881:
5877:
5873:
5870:
5867:
5864:
5861:
5857:
5853:
5824:
5799:
5791:
5787:
5783:
5778:
5774:
5766:
5762:
5758:
5753:
5749:
5742:
5737:
5731:
5725:
5719:
5714:
5707:
5704:
5701:
5698:
5675:
5672:
5667:
5663:
5651:
5650:
5633:
5630:
5625:
5621:
5597:
5593:
5589:
5584:
5580:
5568:
5551:
5548:
5543:
5539:
5515:
5511:
5507:
5502:
5498:
5470:
5467:
5464:
5461:
5457:
5449:
5445:
5441:
5436:
5432:
5424:
5420:
5416:
5411:
5407:
5398:
5395:
5390:
5385:
5381:
5355:
5347:
5343:
5339:
5334:
5330:
5324:
5321:
5315:
5307:
5303:
5299:
5294:
5290:
5282:
5278:
5274:
5269:
5265:
5256:
5253:
5248:
5243:
5239:
5209:
5206:
5201:
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5193:
5190:
5186:
5180:
5176:
5172:
5167:
5163:
5158:
5154:
5151:
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5144:
5138:
5134:
5130:
5125:
5121:
5117:
5113:
5109:
5104:
5100:
5096:
5071:
5068:
5063:
5059:
5055:
5052:
5048:
5042:
5038:
5034:
5029:
5025:
5020:
5016:
5013:
5010:
5006:
5000:
4996:
4992:
4987:
4983:
4979:
4974:
4970:
4966:
4961:
4957:
4952:
4946:
4942:
4918:
4915:
4912:
4907:
4903:
4899:
4894:
4890:
4868:
4865:
4862:
4839:
4835:
4831:
4826:
4822:
4798:
4793:
4790:
4787:
4783:
4777:
4773:
4769:
4764:
4760:
4755:
4750:
4747:
4743:
4737:
4733:
4729:
4724:
4720:
4695:
4691:
4688:
4684:
4680:
4659:
4656:
4653:
4648:
4644:
4639:
4633:
4629:
4625:
4620:
4616:
4611:
4599:
4598:
4582:
4579:
4576:
4553:
4549:
4545:
4540:
4536:
4514:
4491:
4488:
4483:
4479:
4475:
4472:
4468:
4462:
4458:
4454:
4449:
4445:
4440:
4436:
4433:
4430:
4426:
4420:
4416:
4412:
4407:
4403:
4399:
4394:
4390:
4386:
4381:
4377:
4372:
4366:
4362:
4338:
4335:
4332:
4329:
4306:
4301:
4298:
4294:
4290:
4287:
4283:
4277:
4273:
4269:
4264:
4260:
4255:
4251:
4248:
4245:
4241:
4235:
4231:
4227:
4222:
4218:
4214:
4209:
4205:
4201:
4196:
4192:
4187:
4183:
4160:
4156:
4152:
4148:
4144:
4140:
4136:
4112:
4109:
4106:
4103:
4100:
4097:
4093:
4088:
4084:
4080:
4077:
4072:
4068:
4064:
4060:
4056:
4052:
4028:
4025:
4022:
4019:
4016:
4013:
3989:
3986:
3980:
3976:
3972:
3968:
3958:
3954:
3950:
3945:
3942:
3936:
3933:
3927:
3923:
3919:
3915:
3885:
3882:
3879:
3870:
3865:
3862:
3858:
3854:
3851:
3847:
3841:
3837:
3833:
3828:
3824:
3819:
3815:
3812:
3809:
3805:
3801:
3797:
3793:
3789:
3785:
3781:
3755:
3750:
3746:
3742:
3739:
3736:
3731:
3727:
3723:
3720:
3715:
3711:
3706:
3701:
3697:
3693:
3690:
3687:
3682:
3678:
3674:
3671:
3666:
3662:
3650:, in the form
3628:
3623:
3620:
3616:
3612:
3609:
3605:
3600:
3597:
3594:
3589:
3585:
3581:
3577:
3573:
3570:
3567:
3561:
3557:
3552:
3548:
3544:
3539:
3536:
3533:
3528:
3524:
3520:
3516:
3512:
3509:
3506:
3500:
3496:
3491:
3465:
3460:
3457:
3453:
3449:
3446:
3442:
3438:
3435:
3432:
3428:
3422:
3418:
3414:
3410:
3406:
3403:
3400:
3396:
3390:
3386:
3360:
3355:
3352:
3348:
3344:
3341:
3336:
3332:
3328:
3325:
3288:
3285:
3282:
3279:
3275:
3271:
3268:
3265:
3261:
3255:
3251:
3247:
3243:
3239:
3236:
3233:
3229:
3223:
3219:
3192:
3189:
3186:
3155:
3152:
3149:
3146:
3125:
3122:
3119:
3097:
3094:
3091:
3086:
3083:
3059:
3055:
3052:
3049:
3044:
3041:
3037:
3031:
3028:
3023:
3020:
2992:
2988:
2985:
2982:
2977:
2974:
2970:
2964:
2961:
2956:
2953:
2946:
2943:
2939:
2933:
2930:
2925:
2922:
2895:
2891:
2888:
2885:
2881:
2877:
2874:
2871:
2842:
2834:
2830:
2827:
2824:
2818:
2814:
2810:
2806:
2803:
2800:
2794:
2790:
2766:
2763:
2760:
2737:
2733:
2730:
2727:
2721:
2717:
2713:
2709:
2706:
2703:
2697:
2693:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2623:
2620:
2617:
2589:
2585:
2582:
2579:
2575:
2571:
2568:
2565:
2542:
2539:
2535:
2532:
2529:
2523:
2519:
2514:
2511:
2508:
2504:
2500:
2497:
2493:
2487:
2483:
2459:
2456:
2453:
2432:
2429:
2425:
2421:
2417:
2395:
2392:
2389:
2386:
2365:
2362:
2358:
2354:
2350:
2328:
2325:
2322:
2319:
2316:
2294:
2290:
2286:
2257:
2254:
2251:
2248:
2245:
2224:
2221:
2218:
2215:
2212:
2209:
2186:
2183:
2180:
2177:
2174:
2171:
2162:
2156:
2153:
2149:
2144:
2141:
2132:
2129:
2126:
2121:
2117:
2112:
2106:
2102:
2098:
2093:
2089:
2084:
2060:
2037:
2033:
2030:
2027:
2023:
2017:
2013:
2010:
2007:
2004:
2000:
1996:
1993:
1989:
1985:
1981:
1977:
1974:
1971:
1968:
1965:
1961:
1957:
1927:
1924:
1921:
1916:
1912:
1890:
1887:
1883:
1872:
1869:
1838:
1835:
1830:
1826:
1822:
1819:
1814:
1810:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1775:
1772:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1743:
1740:
1736:
1733:
1729:
1708:streamfunction
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1666:
1663:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1634:
1631:
1627:
1616:
1609:
1608:
1607:
1606:
1595:
1590:
1586:
1582:
1577:
1571:
1568:
1563:
1560:
1555:
1551:
1547:
1542:
1538:
1534:
1528:
1518:
1514:
1510:
1505:
1501:
1495:
1492:
1486:
1481:
1476:
1472:
1446:
1440:
1434:
1430:
1426:
1421:
1417:
1396:
1392:
1388:
1385:
1380:
1376:
1372:
1367:
1363:
1359:
1356:
1351:
1346:
1342:
1319:
1315:
1311:
1308:
1305:
1302:
1291:
1290:
1279:
1271:
1267:
1263:
1258:
1254:
1246:
1242:
1238:
1232:
1229:
1226:
1218:
1214:
1210:
1205:
1201:
1193:
1189:
1185:
1180:
1176:
1169:
1164:
1160:
1136:
1111:
1107:
1103:
1098:
1094:
1071:
1067:
1063:
1058:
1054:
1042:
1041:
1030:
1027:
1024:
1016:
1012:
1008:
1003:
999:
991:
987:
983:
978:
974:
967:
962:
958:
934:
914:
894:
883:
882:
869:
866:
863:
860:
857:
854:
850:
843:
840:
834:
831:
827:
822:
819:
816:
813:
810:
807:
803:
799:
796:
793:
787:
784:
778:
773:
770:
765:
762:
758:
753:
750:
747:
744:
741:
738:
734:
730:
727:
724:
718:
714:
707:
702:
698:
692:
688:
664:
661:
658:
655:
652:
649:
646:
643:
623:
620:
617:
606:
605:
592:
587:
584:
581:
578:
569:
566:
561:
557:
553:
550:
549:
546:
543:
540:
537:
528:
525:
520:
516:
512:
509:
508:
506:
501:
496:
493:
487:
483:
479:
474:
470:
442:
437:
431:
428:
425:
421:
400:
397:
394:
372:
368:
347:
344:
341:
319:
315:
294:
291:
288:
263:
260:
245:
242:
190:, but also in
88:RT instability
56:
53:
50:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7250:
7239:
7236:
7234:
7231:
7229:
7226:
7225:
7223:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7195:
7191:
7184:
7178:
7173:
7172:
7165:
7160:
7154:
7150:
7146:
7142:
7141:Drazin, P. G.
7138:
7134:
7128:
7124:
7120:
7116:
7112:
7111:
7107:
7101:
7097:
7093:
7089:
7085:
7081:
7077:
7073:
7069:
7065:
7062:
7056:
7052:
7048:
7044:
7040:
7036:
7032:
7031:
7027:
7022:
7020:
7019:, 1095(2006)
7018:
7005:
7000:
6996:
6992:
6988:
6981:
6978:
6973:
6969:
6965:
6961:
6957:
6953:
6948:
6943:
6940:(3): 032102.
6939:
6935:
6928:
6925:
6919:
6917:
6913:
6908:
6904:
6900:
6896:
6892:
6888:
6884:
6880:
6876:
6872:
6865:
6863:
6859:
6855:
6851:
6847:
6843:
6839:
6835:
6832:(1): 012803,
6831:
6827:
6820:
6817:
6811:
6806:
6802:
6798:
6794:
6790:
6786:
6779:
6776:
6772:
6767:
6764:
6758:
6753:
6749:
6745:
6741:
6737:
6733:
6726:
6723:
6718:
6714:
6710:
6706:
6702:
6698:
6691:
6688:
6672:
6665:
6658:
6655:
6650:
6646:
6642:
6638:
6634:
6630:
6623:
6620:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6587:
6580:
6577:
6571:
6565:
6561:
6557:
6550:
6547:
6542:
6538:
6534:
6530:
6526:
6522:
6517:
6512:
6508:
6504:
6497:
6494:
6489:
6483:
6480:
6475:
6474:
6469:
6462:
6459:
6455:
6450:
6448:
6446:
6444:
6440:
6435:
6431:
6427:
6423:
6419:
6415:
6411:
6404:
6402:
6400:
6396:
6385:
6381:
6374:
6371:
6364:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6315:
6312:
6311:
6307:
6305:
6303:
6294:
6292:
6289:
6287:
6284:
6279:
6277:
6273:
6257:
6251:
6248:
6238:
6234:
6230:
6225:
6219:
6216:
6211:
6208:
6197:
6194:
6190:
6187:
6179:
6175:
6171:
6165:
6158:
6156:
6154:
6150:
6143:
6129:
6126:
6122:
6118:
6115:
6112:
6091:
6085:
6082:
6079:
6071:
6066:
6051:
6042:
6034:
6031:
6028:
6025:
6022:
6018:
6014:
6011:
6007:
6003:
5997:
5979:
5972:
5968:
5964:
5961:
5958:
5955:
5951:
5946:
5942:
5939:
5936:
5933:
5929:
5925:
5922:
5918:
5914:
5911:
5901:
5893:
5890:
5886:
5882:
5879:
5875:
5871:
5868:
5865:
5862:
5859:
5855:
5843:
5841:
5840:Atwood number
5811:
5797:
5789:
5785:
5781:
5776:
5772:
5764:
5760:
5756:
5751:
5747:
5740:
5729:
5723:
5712:
5705:
5702:
5699:
5696:
5688:
5673:
5670:
5665:
5661:
5648:
5631:
5628:
5623:
5619:
5595:
5591:
5587:
5582:
5578:
5569:
5566:
5549:
5546:
5541:
5537:
5513:
5509:
5505:
5500:
5496:
5487:
5486:
5485:
5482:
5468:
5465:
5462:
5459:
5455:
5447:
5443:
5439:
5434:
5430:
5422:
5418:
5414:
5409:
5405:
5396:
5393:
5388:
5383:
5379:
5370:
5367:
5353:
5345:
5341:
5337:
5332:
5328:
5322:
5319:
5313:
5305:
5301:
5297:
5292:
5288:
5280:
5276:
5272:
5267:
5263:
5254:
5251:
5246:
5241:
5237:
5228:
5226:
5221:
5207:
5204:
5199:
5195:
5191:
5188:
5184:
5178:
5174:
5170:
5165:
5161:
5156:
5152:
5149:
5146:
5142:
5136:
5132:
5128:
5123:
5119:
5115:
5111:
5107:
5102:
5098:
5094:
5086:
5083:
5069:
5061:
5057:
5053:
5050:
5046:
5040:
5036:
5032:
5027:
5023:
5018:
5011:
5008:
5004:
4998:
4990:
4985:
4981:
4977:
4972:
4964:
4959:
4955:
4950:
4944:
4940:
4931:
4916:
4913:
4910:
4905:
4901:
4897:
4892:
4888:
4866:
4863:
4860:
4837:
4829:
4824:
4810:
4796:
4791:
4788:
4785:
4781:
4775:
4771:
4767:
4762:
4753:
4748:
4745:
4741:
4735:
4731:
4727:
4722:
4709:
4693:
4686:
4682:
4657:
4654:
4651:
4646:
4637:
4631:
4627:
4623:
4618:
4614:
4609:
4596:
4595:
4594:
4580:
4577:
4574:
4551:
4543:
4538:
4503:
4489:
4481:
4477:
4473:
4470:
4466:
4460:
4456:
4452:
4447:
4443:
4438:
4431:
4428:
4424:
4418:
4410:
4405:
4401:
4397:
4392:
4384:
4379:
4375:
4370:
4364:
4360:
4351:
4333:
4330:
4327:
4318:
4304:
4299:
4296:
4292:
4288:
4285:
4281:
4275:
4271:
4267:
4262:
4258:
4253:
4249:
4246:
4243:
4239:
4233:
4225:
4220:
4216:
4212:
4207:
4199:
4194:
4190:
4185:
4181:
4173:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4124:
4110:
4107:
4104:
4101:
4098:
4095:
4091:
4086:
4078:
4075:
4070:
4066:
4062:
4058:
4054:
4050:
4041:
4026:
4023:
4020:
4017:
4014:
4011:
3987:
3978:
3974:
3970:
3956:
3952:
3948:
3943:
3940:
3934:
3925:
3921:
3917:
3902:
3897:
3883:
3880:
3877:
3868:
3863:
3860:
3856:
3852:
3849:
3845:
3839:
3835:
3831:
3826:
3822:
3817:
3813:
3810:
3807:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3770:
3769:this becomes
3767:
3753:
3748:
3744:
3740:
3737:
3734:
3729:
3725:
3721:
3718:
3713:
3709:
3704:
3699:
3695:
3691:
3688:
3685:
3680:
3676:
3672:
3669:
3664:
3660:
3651:
3649:
3645:
3640:
3626:
3621:
3618:
3614:
3610:
3607:
3603:
3598:
3595:
3592:
3587:
3583:
3579:
3575:
3571:
3568:
3565:
3559:
3555:
3550:
3546:
3542:
3537:
3534:
3531:
3526:
3522:
3518:
3514:
3510:
3507:
3504:
3498:
3494:
3489:
3480:
3477:
3463:
3458:
3455:
3451:
3447:
3444:
3440:
3436:
3433:
3430:
3426:
3420:
3416:
3412:
3408:
3404:
3401:
3398:
3394:
3388:
3384:
3375:
3372:
3358:
3353:
3350:
3346:
3342:
3339:
3334:
3326:
3323:
3315:
3313:
3309:
3305:
3300:
3286:
3283:
3280:
3277:
3273:
3269:
3266:
3263:
3259:
3253:
3249:
3245:
3241:
3237:
3234:
3231:
3227:
3221:
3217:
3208:
3206:
3205:Young–Laplace
3190:
3187:
3184:
3176:
3172:
3168:
3150:
3147:
3144:
3123:
3120:
3117:
3095:
3092:
3089:
3084:
3081:
3071:
3057:
3053:
3050:
3047:
3042:
3039:
3035:
3029:
3021:
3007:
3004:
2990:
2986:
2983:
2980:
2975:
2972:
2968:
2962:
2954:
2944:
2941:
2937:
2931:
2923:
2909:
2893:
2889:
2886:
2883:
2879:
2875:
2872:
2869:
2861:
2857:
2854:
2840:
2832:
2828:
2825:
2822:
2816:
2808:
2804:
2801:
2798:
2792:
2779:
2764:
2761:
2758:
2749:
2735:
2731:
2728:
2725:
2719:
2711:
2707:
2704:
2701:
2695:
2682:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2621:
2618:
2615:
2607:
2603:
2587:
2583:
2580:
2577:
2573:
2569:
2566:
2563:
2554:
2540:
2537:
2533:
2527:
2521:
2512:
2509:
2506:
2502:
2495:
2491:
2485:
2472:
2454:
2451:
2430:
2427:
2423:
2419:
2415:
2390:
2387:
2384:
2363:
2360:
2356:
2352:
2348:
2326:
2320:
2317:
2314:
2292:
2288:
2284:
2274:
2272:
2252:
2249:
2246:
2243:
2222:
2219:
2216:
2213:
2207:
2198:
2184:
2181:
2178:
2175:
2172:
2169:
2160:
2154:
2151:
2147:
2142:
2139:
2130:
2127:
2124:
2119:
2110:
2104:
2100:
2096:
2091:
2087:
2082:
2073:
2058:
2049:
2035:
2031:
2028:
2025:
2015:
2011:
2008:
2005:
2002:
1998:
1994:
1991:
1987:
1983:
1979:
1975:
1972:
1969:
1966:
1963:
1959:
1955:
1947:
1946:
1942:
1925:
1922:
1919:
1914:
1888:
1885:
1881:
1870:
1859:
1855:
1850:
1836:
1828:
1824:
1820:
1817:
1812:
1808:
1801:
1792:
1789:
1786:
1783:
1780:
1773:
1770:
1766:
1760:
1757:
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1390:
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189:
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178:
177:Atwood number
173:
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163:
161:
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152:
151:perturbations
148:
144:
140:
135:
133:
129:
125:
121:
117:
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110:of different
109:
105:
101:
97:
93:
92:Lord Rayleigh
89:
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35:Hydrodynamics
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6678:. Retrieved
6671:the original
6657:
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6628:
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6579:
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6473:Scholarpedia
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6373:
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6280:
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5484:and clearly
5483:
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1858:irrotational
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224:
216:solar corona
192:astrophysics
181:
164:
136:
106:between two
96:G. I. Taylor
87:
83:
81:
7049:: 170–177.
6703:: 127–155.
6420:(1): 3–18.
6276:hydrostatic
6105:Now define
208:Crab pulsar
200:Crab Nebula
160:accelerated
147:equilibrium
100:instability
77:Crab Nebula
7222:Categories
7187:626 pages.
7023:References
6947:1810.11460
6773:, Chap. X.
6680:October 9,
6389:2006-09-05
6198:equation,
6186:baroclinic
4172:together,
3646:.) Using
3207:equation:
184:salt domes
172:non-linear
143:immiscible
6907:126143908
6877:: 50–83.
6614:122223176
6560:CRC Press
6414:Physica D
6255:∇
6252:×
6249:ρ
6246:∇
6235:ρ
6212:ω
6196:vorticity
6089:Ψ
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2247:≤
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2208:−
2116:Ψ
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2022:Ψ
2006:−
1995:α
1956:ψ
1920:ψ
1911:∇
1871:×
1868:∇
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1821:−
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1471:σ
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1371:−
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1200:ρ
1188:ρ
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1093:ρ
1066:ρ
1053:ρ
1044:Thus, if
1011:ρ
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865:σ
842:^
818:σ
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717:^
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573:for
556:ρ
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532:for
515:ρ
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367:ρ
314:ρ
234:lava lamp
212:supernova
132:supernova
112:densities
104:interface
98:), is an
7143:(2002).
7117:(1981).
7100:98831861
7037:(1883).
6972:53658729
6854:27575198
6541:15244583
6308:See also
6283:gradient
6193:inviscid
4597:Solution
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1860:, hence
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276:inviscid
7080:Bibcode
6991:Bibcode
6952:Bibcode
6899:1436483
6879:Bibcode
6834:Bibcode
6797:Bibcode
6744:Bibcode
6705:Bibcode
6637:Bibcode
6594:Bibcode
6521:Bibcode
6422:Bibcode
6286:vectors
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139:modeled
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6189:torque
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3374:Thus,
3302:where
2837:H.O.T.
2778:gives
2407:, and
2340:Thus,
2167:
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2051:where
1945:ansatz
1332:where
885:where
108:fluids
102:of an
7108:Other
7096:S2CID
6968:S2CID
6942:arXiv
6903:S2CID
6674:(PDF)
6667:(PDF)
6610:S2CID
6537:S2CID
6511:arXiv
6365:Notes
4004:with
86:, or
7177:ISBN
7153:ISBN
7127:ISBN
6895:OSTI
6850:PMID
6682:2012
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2385:z
2364:0
2361:=
2353:L
2349:w
2327:.
2318:=
2315:z
2289:i
2285:w
2271:c
2250:z
2244:0
2223:0
2217:z
2185:.
2182:G
2179:,
2176:L
2173:=
2170:j
2161:,
2155:z
2152:d
2148:d
2143:=
2140:D
2131:,
2128:0
2125:=
2120:j
2111:)
2105:2
2092:2
2088:D
2083:(
2036:,
2032:)
2029:z
2026:(
2016:)
2012:t
2009:c
2003:x
1999:(
1992:i
1988:e
1984:=
1980:)
1976:t
1973:,
1970:z
1967:,
1964:x
1960:(
1941:x
1923:=
1915:2
1889:0
1886:=
1877:u
1837:,
1834:)
1829:x
1818:,
1813:z
1805:(
1802:=
1799:)
1796:)
1793:t
1790:,
1787:z
1784:,
1781:x
1778:(
1771:w
1767:,
1764:)
1761:t
1758:,
1755:z
1752:,
1749:x
1746:(
1739:u
1735:(
1732:=
1723:u
1693:.
1690:)
1687:)
1684:t
1681:,
1678:z
1675:,
1672:x
1669:(
1662:w
1658:,
1655:)
1652:t
1649:,
1646:z
1643:,
1640:x
1637:(
1630:u
1626:(
1594:.
1589:2
1585:/
1581:3
1576:]
1567:3
1562:g
1559:)
1554:1
1541:2
1533:(
1527:[
1517:1
1504:2
1491:2
1485:=
1480:2
1475:m
1445:3
1439:/
1433:c
1429:k
1425:=
1420:m
1416:k
1391:/
1387:g
1384:)
1379:1
1366:2
1358:(
1355:=
1350:2
1345:c
1341:k
1318:c
1314:k
1307:k
1301:0
1278:,
1270:1
1262:+
1257:2
1245:3
1241:k
1228:k
1225:g
1217:1
1209:+
1204:2
1192:1
1179:2
1168:=
1163:2
1110:1
1097:2
1070:1
1057:2
1029:.
1026:k
1023:g
1015:1
1007:+
1002:2
990:1
977:2
966:=
961:2
913:x
893:k
868:t
862:+
859:x
856:k
853:i
849:e
839:f
833:=
830:f
826:,
821:t
815:+
812:x
809:k
806:i
802:e
798:)
795:z
792:(
783:p
777:+
769:p
764:=
761:p
757:,
752:t
746:+
743:x
740:k
737:i
733:e
729:)
726:z
723:(
713:v
706:+
697:v
691:=
687:v
663:)
660:t
657:,
654:x
651:(
648:f
645:=
642:z
622:0
619:=
616:z
586:,
583:0
577:z
568:z
565:g
560:2
545:,
542:0
536:z
527:z
524:g
519:1
505:{
500:=
492:p
486:,
482:0
478:=
469:v
441:z
436:e
430:g
424:=
420:g
399:0
393:z
371:2
346:0
340:z
318:1
293:0
290:=
287:z
55:0
52:=
49:y
20:)
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