How to find the value of the eigen value parameter in linear stability(decay or growth) of the fluid mechanics problems, Please help me. Further I am using bvp4c, but I can't find the value of eigen paramater(gamma or alpha etc)
Since you are interested linear stability analysis of fluid mechanic problems, you will likely have to deal with solving an Orr-Sommerfeld type eigenvalue problem. This stability problem is notoriously difficult to solve especially if your eigenvalue problem has variable coefficients (due to a nontrivial base flow, e.g flow in a channel). In that case one has to resort to numerical methods.
I suggest you look at the following report I have prepared that shows how one can solve the eigenvalue problem using a spectral collocation method. You can download it from Reasearchgate: Here are some details:
Stability Analysis of Rectilinear Flow in a 2-D Channel:
This report describes how to use the symbolic capabilities Mathematica to derive the Orr-Sommerfeld Equation for linear stability of a 2-D flow. We also show how to solve the Orr-Sommerfeld equation using a spectral collocation method with Mathematica. The results agree with the original work of Orsag (1971)
Stability Analysis of Rectilinear Flow in a 2-D Channel. Available from: https://www.researchgate.net/publication/269097836_Stability_Analysis_of_Rectilinear_Flow_in_a_2-D_Channel [accessed Oct 23, 2016].
Technical Report Stability Analysis of Rectilinear Flow in a 2-D Channel
There are some simpler ways than solving an Orr-Sommerfeld equation, if you are willing to use potential flow theory, like the 1D Kevin-Helmholtz (KH) and shallow water theories and the 2D Rayleigh analysis. They work sometimes. I apply the 1D KH approach to two-phase flow in my book: http://www.springer.com/us/book/9783319449678
If you are using bvp4c, then you are solving an ordinary differential eigenvalue problem. Presumably the Reynolds number is the only eigenvalue?
If your system of equations has order n, i.e. it has n first order equations, then you could declare Re to be your variable n+1. One more boundary condition is then needed, so you could then set any of the unknown initial conditions to equal 1 as a normalisation condition. I have had trouble from a few reviewers about this (e.g. "How can a constant be a variable?" Answer: Re'=0 is the equation for the variable, Re, and the numerical code then produces Re=constant!), but I have had decades of trouble-free use of this trick....
D. Andrew S. Rees
Dear sir,
I am trying to solve a similar problem. But with different initial guesses of the eigenvalue, I am getting positive and negative eigenvalues. How can I solve this issue?
Dear Korimerla Sai Vishwak ,
I don't know enough about your particular problem to know whether this is to be expected. If your eigenvalue is an exponential growth rate then having positive and negative values is generally to be expected. On the other hand, if your problem is similar in nature to the Rayleigh-Benard problem, then one would expect all the eigenvalues (the Rayleigh number) to be positive. Then one is interested in the smallest of these.
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