  In this paper, we construct growing modes of the linearized Navier-Stokes
equations about generic stationary shear flows of the boundary layer type in a
regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the
shear profiles are allowed to be linearly stable at the infinite Reynolds
number limit, and so the instability presented is purely due to the presence of
viscosity. The formal construction of approximate modes is well-documented in
physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien,
Drazin and Reid, but a rigorous construction requires delicate mathematical
details, involving for instance a treatment of primitive Airy functions and
singular solutions. Our analysis gives exact unstable eigenvalues and
eigenfunctions, showing that the solution could grow slowly at the rate of
$e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding to
deal with matching inner and outer asymptotic expansions, but instead involving
a careful study of singularity in the critical layers by deriving pointwise
bounds on the Green function of the corresponding Rayleigh and Airy operators.
