= Orr-Sommerfeld equation
{c}
{title2=$[(-i\omega+i\alpha U)(D^2-\kappa^2)-i\alpha U''-Re^{-1}(D^2-\kappa^2)^2]\widehat v=0$}
The Orr-Sommerfeld equation governs the wall-normal velocity of <normal mode> disturbances to an incompressible viscous parallel <shear flow>. Here $D=d/dy$, $\kappa^2=\alpha^2+\beta^2$, and $Re$ is the <Reynolds number>. It follows by eliminating pressure from the linearized <Navier-Stokes equations>. At a rigid no-slip wall, $\widehat v=D\widehat v=0$. In the inviscid limit it reduces to the <Rayleigh equation for inviscid shear flow>.
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