= Solution
Choose a length $\ell$, velocity $U_s$, time $\ell/U_s$, and pressure $\rho U_s^2$, giving <Reynolds number> $Re=U_s\ell/\nu$. Write perturbation velocity as $(u,v,w)$ and let $\mathcal L=\partial_t+U\partial_x-Re^{-1}\nabla^2$. Linearizing the incompressible <Navier-Stokes equations> gives
$$
\mathcal Lu+U'v=-p_x,\quad\mathcal Lv=-p_y,\quad\mathcal Lw=-p_z,\quad u_x+v_y+w_z=0.
$$
Taking the divergence gives $\nabla^2p=-2U'v_x$. Applying $\nabla^2$ to the wall-normal momentum equation and using
$$
\nabla^2(Uv_x)=U\nabla^2v_x+2U'v_{xy}+U''v_x
$$
then cancels the pressure derivatives and yields $\mathcal L\nabla^2v-U''v_x=0$. Define wall-normal <vorticity> $\eta=u_z-w_x$. Applying $\partial_z$ to the streamwise equation minus $\partial_x$ to the spanwise equation gives $\mathcal L\eta=-U'v_z$.
For the assumed <normal modes>, set $D=d/dy$ and $\kappa^2=\alpha^2+\beta^2$. The two equations become the <Orr-Sommerfeld equation> and <Squire equation>:
$$
\boxed{[(-i\omega+i\alpha U)(D^2-\kappa^2)-i\alpha U''-Re^{-1}(D^2-\kappa^2)^2]\widehat v=0,}
$$
$$
\boxed{[(-i\omega+i\alpha U)-Re^{-1}(D^2-\kappa^2)]\widehat\eta=-i\beta U'\widehat v.}
$$
At rigid no-slip walls the conditions are $\widehat v=D\widehat v=\widehat\eta=0$ for nonzero horizontal wavenumber.
Back to article page
