0210_193linstab.nb

Orr-Sommerfeld Equation

A good reference for this section is R. L. Panton, Incompressible flow, Wiley, 1984

Here we derive the Orr-Sommerfeld equation which is a 4th order ODE that describes the growth on infinitesimal periodic distrubances that are governed by the Navier-Stokes equations.

The linearized x-momentum equations for a nearly-parallel flow are
x direction (first in traditional form)

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y direction (first in traditional form)

The continuity equation is

We need to expand these in terms of normal modes where the variable will be assumed to have the form ξ[x,y,t]=[y] Exp[I(α x -α c t)] based on the idea that we need functions that can describe spatially and temporally varying disturbances

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Which is the result for the x equation

which is the result for the y equation

which is the continuity equation

Now we use the continuity equation and the disturbance stream function definition
which is uh[y]= and vh[y]=-I αψ[y].

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Now combine the x and y equations to eliminate pressure,

This, with little effort, is the Orr-Sommerfeld equation.

Converted by Mathematica December 22, 1999