The OrrSommerfeld problem is a classic problem from the field of hydrodynamic stability. In the simplest case it models the flow of a Newtonian fluid between two infinite plates, governed by the NavierStokes equations. The Reynolds number $Re$ is a nondimensional parameter corresponding roughly to velocity divided by viscosity. For any value of $Re$ there is a smooth ("laminar") solution to the NavierStokes equations, but the stability, and hence the observability, of this solution is a delicate and muchstudied question. In practice one observes instability and transition to turbulence when $Re$ is in the thousands or higher.
The OrrSommerfeld operator is the linear operator $L$ that maps infinitesimal perturbations on the laminar flow to their infinitesimal growth rates. Classically speaking we expect the flow to be stable if all the eigenvalues of $L$ lie in the left halfplane, and unstable if there are any eigenvalues in the right halfplane.
The following Chebfun code computes the rightmost $50$ eigenvalues of $L$ for $Re=2000$. This mathematical formulation, due to Reddy and Henningson and described in Appendix A of [1], involves a fourth order complex generalized eigenvalue problem.
Re = 2000; % Reynolds number alph = 1; % longitudinal Fourier parameter A = chebop(1,1); A.op = @(x,u) (diff(u,4)2*alph^2*diff(u,2)+alph^4*u)/Re  ... 2i*alph*u  1i*alph*(1x.^2).*(diff(u,2)alph^2*u); B = chebop(1,1); B.op = @(x,u) diff(u,2)  u; A.lbc = [0; 0]; A.rbc = [0; 0]; e = eigs(A,B,50,'LR'); FS = 'fontsize'; MS = 'markersize'; maxe = max(real(e)); plot(e,'.r',MS,16), grid on, axis([.9 .1 1 0]), axis square title(sprintf('Re = %8.2f \\lambda_r = %7.5f',Re,maxe),FS,16)
(The eigenvalues on the lowerright branch are neardegenerate pairs.) Here is the same computation for $Re = 5772.22$, the critical value at which an eigenvalue first crosses into the right halfplane:
Re = 5772.22; alph = 1.02; A.op = @(x,u) (diff(u,4)2*alph^2*diff(u,2)+alph^4*u)/Re  ... 2i*alph*u  1i*alph*(1x.^2).*(diff(u,2)alph^2*u); e = eigs(A,B,50,'LR'); maxe = max(real(e)); plot(e,'.r',MS,16), grid on, axis([.9 .1 1 0]), axis square title(['Re = ' sprintf('%5d',Re) ... ', \lambda_r = ' sprintf('%7.5f',maxe)],FS,16)
Although the OrrSommerfeld equation is very famous, this eigenvalue analysis actually has little to do with what makes fluid flows unstable in practice, and it is difficult to see the number $5772.22$ in the laboratory [2].
References

P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Springer, 2001.

L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton U. Press, 2005.