An alternative numerical solution for the Orr–Sommerfeld problem

A simple tutorial for the numerical solution of the Orr-Sommerfeld problem is presented. The aim is the determination of the neutral stability threshold to the linear instability for the Poiseuille flow within a plane parallel channel. Such an instability is hydrodynamic in nature as it is driven by the advection term in the local momentum balance equation. The numerical solution is achieved by a spectral method. The basis functions are chosen as Chebyshev polynomials multiplied by a suitable overall factor meant to allow the fulfilment of the boundary conditions. The details of the numerical code implemented via the Mathematica software environment are also provided. A comparison with other numerical solutions of the same problem implemented via the tau-method is presented.