  A model problem of the form -i\epsilon y''+q(x)y=\lambda y, y(-1)=y(1)=0, is
associated with well-known in hydrodynamics Orr--Sommerfeld operator. Here
(\lambda) is the spectral parameter, (\epsilon) is the small parameter which is
proportional to the viscocity of the liquid and to the reciprocal of the
Reynolds number, and (q(x)) is the velocity of the stationary flow of the
liquid in the channel (|x|\leqslant 1). We study the behaviour of the spectrum
of the corresponding model operator as (\epsilon\to 0) with monotonous analytic
functions. We assert that the sets of the accumulation points of the spectra
(the limit spectral graphs) of the model and the corresponding Orr--Sommerfeld
operators coincide as well as the main terms of the counting eigenvalue
functions along the curves of the graphs. We prove the estimate from below for
the resolvent of the operator (L(\epsilon)) associated with the model problem.
It turns out that the resolvent grows exponentially with respect to (\epsilon)
provided that the spectral parameter varies at some compacts belonging to the
numerical range of the operator.
