Orr Sommerfeld equation
 

My question is very fundamental.
why, at all do we need to have a separate equation that predicts the growth or decay of perturbations in a steady base flow..? If at all, there are any perturbations in the flow, why can't the resultant motion of fluid be predicted using the usual Navier Stokes equations??

Because Navier-Stokes equation is too complicated to be solved analytically. This is the main problem of all fluid dynamics. That is why we still know almost nothing exact about turbulence.

thanks a lot for replying... but as far as i know, even Orr - Sommerfeld equation is a fourth order linear differential equation which cannot be solved analytically for most cases, and various numerical methods have to be resorted to solve it... In that case would it not be better to bear with a little more complexity in terms of better numerical methods and solve the full N-S equations?

As you said Orr-Sommerfeld is a linear system, while Navier-Stokes is nonlinear and shows chaotic behavior (turbulence) which is difficult analyse even numerically (DNS is done on supercomputers).

In the case of Orr-Somerfeld you have a possibility of qualitative analysys of transition to turbulence. You can see how perturbations amplify etc.

thanks.. that was really helpful..