Modal and non-modal stability of dusty-gas boundary layer flow

The linear hydrodynamic stability of dusty-gas flow in a boundary layer on a flat plate is considered in the framework of a two-fluid model. The phase interaction is described by the Stokes and Saffman forces. The particle volume fraction is negligibly small, but the feedback effect on the carrier phase is taken into account due to the finiteness of the dispersed-phase mass concentration. In the main flow, the phase velocities coincide and the particles are distributed non-uniformly, namely, in the form of a localized dust layer. The equations of two-phase flow, linearized with respect to small three-dimensional disturbances, are reduced to five differential equations for the disturbance amplitudes of the normal components of the carrier-phase velocity and vorticity and three velocity components of the dispersed phase. In the framework of the classical (modal) approach, the analysis of two-dimensional disturbances is performed and the dependence of the critical Reynolds number on the dimensionless parameters is found. It is shown that the flow is most stable when the maximum of the particle concentration is located in the vicinity of the so-called "critical layer". The analysis of non-modal (algebraic) instability demonstrated that the maximal kinetic energy of the optimal disturbances is attained when the narrow dust layer is located in the vicinity of the displacement thickness of the boundary layer.