Particle motions induced by spherical convective elements in Stokes flow

The motions of fluid particles within and around a mass of hot, buoyant material (a thermal) rising through an extremely viscous, unbounded environment are computed using a simple kinematic model. The model is based on a similarity solution by Griffiths (1986a) and allows for growth of thermals due to outward diffusion of heat. Particle motions are also computed for the case of a non-expanding, isothermal sphere, such as a bubble of relatively low-viscosity fluid, in Stokes flow. Motions induced in the surroundings lead to large vertical displacements: the ‘total drift’ function and hydrodynamic mass corresponding to those defined for the inviscid case by Darwin (1953) and Lighthill (1956) are infinite in this unbounded geometry. Rotation of initially horizontal fluid elements (strain) in the surroundings is discussed. All material lying within an expanding thermal becomes confined at later times to a torus (dye ring) if the Rayleigh number for the thermal is large, to a central tapered blob if Ra < 50, or to an umbrella-shaped cap with narrow stem if Ra takes intermediate values. The ‘mushroom’ shape widely observed for tracers within laminar elements in thermal convection is predicted for intermediate-to-large Rayleigh numbers. Buoyancy and heat, on the other hand, are assumed to remain evenly distributed throughout an enlarging sphere. Laboratory experiments illustrate and confirm the predictions of the model.