How to Determine the Stability Condition in Couette Flow?

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The discussion centers on determining the stability condition in Couette flow, specifically with two moving plates. The user seeks assistance in performing a stability analysis, mentioning the need to expand around the unperturbed solution and linearize the Navier-Stokes equations. There is a suggestion to utilize the Rayleigh equation due to the incompressible inviscid limit. The conversation highlights the technical aspects of stability analysis in fluid dynamics. Overall, the focus is on applying theoretical principles to solve the problem at hand.
URIA
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Homework Statement
Hi all,
I would like to solve the attached .
Relevant Equations
Can someone help with the attached?
Dear All,
I tried to solve the attached question. it's about Couette flow, where the 2 plates move.
2023-01-13 110841.png

in fact, I have to find the stability condition. is someone familiar with this and can help?
many thanks,
uria
 
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Do you have a general idea on how to do the stability analysis? Basically it's expanding around the un-perturbed given solution and linearizing the NS equations.
 
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HI,
I assume I have to use the rayleigh equation because of the incompressible inviscid limit.
 

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I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW
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