The linear stability of incompressible boundary-layer flow of dusty
gas on a semi-infinite
flat plate is considered. The particles are assumed to be under the action
of the Stokes drag only. The problem is reduced to the solution of the
modified
Orr–Sommerfeld equation (Saffman 1962). This is solved numerically
using two
approaches: directly by orthonormalization method, and by perturbation
method at
small particle mass content. The stability characteristics are calculated
for both mono-
and polydisperse particles.
The dust suppresses the instability waves for a wide range of the particle
size.
The most efficient suppression takes place when the relaxation length of
the particle
velocity is close to the wavelength of the Tollmien–Schlichting (TS)
wave. The reduction
in growth rate per unit dust content is approximately ten times greater
than the
characteristic value of the growth rate for a clean gas.
For monosized dust the complex frequency of the TS wave changes in a
discontinuous way. As a result a domain in the space of independent parameters
arises
where two discrete TS modes exist and a domain where no TS mode may exist.
For
polydisperse dust with a discrete distribution in particle size the number
of breaks in
the dependence equals the number of particle sizes. For the continuous
distribution
in particle size the dependence of the complex-frequency on Reynolds number
and
wavenumber is continuous. The eigenfunction becomes a non-smooth function
of the
normal coordinate in this case.
Some comments are made about the role of the lift force acting on the
particles for
the problem in question.