Linear stability analysis of bubble-induced convection in a horizontal liquid layer

We investigate with a linear analysis the stability of a horizontal liquid layer subjected to injection of gas bubbles through a bottom wall. The injection is assumed uniform in space and constant in time. Injected bubbles ascend in the liquid layer due to the Archimedean buoyancy force and are ejected from the top free surface of the liquid layer. Modeling this two-phase flow system as two interpenetrating liquid and gas continua, we show that homogeneous upward gas flows become unstable at large gas fluxes. We determine the critical conditions of this homogeneous-heterogeneous regime transition and show that the critical modes are made of stationary convection rolls, either multi- or whole-layered depending on liquid viscosity, the radius of bubbles, and the thickness of liquid layer. By examining the energy transfer from base to perturbation flows, we indicate that liquid convective motion is driven by the buoyancy on heterogeneously distributed bubbles. We also reveal that the lift forces on bubbles have significant stabilizing effects by homogenizing bubble distribution close to the bottom wall.