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I know that due to fluid instability there are some perturbations ( picture below )

enter image description here

after all of these, my question is the reason of existence of these perturbations. Why exactly they appear? Shear stress or capillary pressure ?

if the parameters I mentioned are in play please explain me the effect of each of them.

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The surface wave formed in a Rayleigh-Taylor instability is caused mainly by surface tension. Like i mentioned before, a liquid tends to minimize its surface area and $n$ droplets of volume $V/n$ have more surface area than a liquid column of volume $V$. Initially, the film is uniform and surface tension will minimize the area by starting to form waves. The surface tension is related to the capillary pressure.

Old (slightly irrelevant) answer: When i wrote this answer i was considering a different type of pertubation than the ones you meant

Pertubations are part of any real system caused by asymmetries in the system, or changes in air pressure, or someone walking by or a train speeding by, etc. etc.

A perfectly undisturbed symmetric system as you describe in your first picture is very difficult to obtain experimentally. Such a system is therefore a purely theoretical situation.

An example of the influence of pertubations is in determining the transition from the laminar to the turbulent regime in a channel flow. Generally, we say this occurs at $\mathrm{Re}\approx2000$ but notice the approximation sign; due to pertubations caused by external factors, the transition may occur one day at $\mathrm{Re}\approx1900$ while another day at $\mathrm{Re}\approx2100$ for the same experiment.

Note that in the case of the Rayleigh-Taylor instability there is an assumed pertubation of the form: $$e=e_0+\delta\left(t\right)\cos\left(\kappa x\right)$$ This means that the theoretical treatment assumes the pertubations are already present, i.e. the pertubations do not grow spontaneously from a initially undisturbed uniform film. Rectification: Even in a perfectly symmetric system the pertubation will occur because of surface tensile forces albeit much more slowly than in the case the system is perturbed by some external factor.

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  • $\begingroup$ So what makes them? Shear Stress? Pressure (capillary)? can you name those factors please? $\endgroup$
    – David 2000
    Commented Sep 14, 2015 at 19:11
  • $\begingroup$ @David2000 - I have modified my answer to make it more relevant $\endgroup$
    – nluigi
    Commented Sep 15, 2015 at 8:32
  • $\begingroup$ So, may I ask u another question? see physics.stackexchange.com/questions/206205/… that you have answered. what does a surface and collision do that this happens? $\endgroup$
    – David 2000
    Commented Sep 15, 2015 at 8:43
  • $\begingroup$ @David2000 - considering the duplicate answers in that question, i am now less sure what you observe is indeed the Rayleigh-Plateau instability. Possibly you will have to start looking into impinging jets instead as suggested by this answer. sorry :) $\endgroup$
    – nluigi
    Commented Sep 15, 2015 at 8:51
  • $\begingroup$ so if you know about that, would you please edit your answer and tell me about that ? $\endgroup$
    – David 2000
    Commented Sep 15, 2015 at 8:54

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