Stability of a dusty-gas laminar boundary layer on a flat plate

Abstract

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.