Linear Stability of Blasius Boundary Layers Containing Gas-Vapor Bubbles

In this paper the effects of the dynamics of bubbles with various gas-vapor contents on the stability of two-dimensional viscous bubbly flows of low void fraction are examined. Using a small perturbation approach, the equations of motion for the two-phase bubbly mixture are linearized and a system equivalent to a modified Orr-Sommerfeld equation is obtained; the latter is then used for the study of the spatial stability analysis of Blasius boundary layers. The effect of compressibility, inertia and energy dissipation due to the viscosity of the liquid and the transfer of heat and mass between the two phases are included in the bubble dynamic model. The numerical solution of the inherently stiff and diverging eigenvalue problem is obtained by means of the combination of a shooting method and the O'Drury-Davey's (1983) orthogonalization algorithm. The present analysis confirms the stabilizing effects due to the dispersed phase in the liquid. Results show that the presence of pure air bubbles in water has a moderate influence on the behavior of the marginal stability curve, the coupling of the perturbation field with the dynamics of the bubbles being effectively prevented by the presence of surface tension. When the latter is neglected, resonant bubble oscillations occur, leading to strong deviations from both the compressible and incompressible limits as a consequence of the increased importance of inertial and dissipative effects in the bubble dynamics. If the presence of vapor in the bubble is explicitly considered, the stabilizing effects are remarkably more evident, in particular when the mass fraction of the vapor approaches the critical value for the stability of bubble oscillations.