Model °ames in the Boussinesq limit: Rising bubbles

Abstract. Using the Boussinesq buoyancy approximation, we study a bubble of reaction products rising in the reactant °uid under the in°uence of gravity. Reaction on the surface of the bubble (the °ame) results in an increase of the volume of the bubble. We consider °uids with low Prandtl and high Froude numbers (heat di®usion dominates over viscous dissipation, and burning dominates over gravitational e®ects). We show that, under these conditions, all initially small bubbles follow the same growth pattern, regardless of the °ame speed, the reaction type, the gravity, the viscosity, the initial size, and, to some extent, the initial shape of the bubble. In the initial stage of this similarity solution a bubble grows radially in an essentially motionless °uid until it reaches some critical size, which is determined by the laminar °ame speed, the gravitational acceleration, and the Atwood number. Once the bubble reaches the critical size, convection becomes signi¯cant and the bubble evolves into a more complicated, mushroom-like shape. The similarity solution is expressed using the critical bubble size for the unit length and the critical size divided by the laminar °ame speed as the unit time. (18 August 2006) 1