Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity

Abstract

The flow induced by injection of a given amount of buoyancy or hot fluid from a localized source in a viscous fluid is investigated for conditions under which the Reynolds number Re is small compared with one, and the dimensionless buoyancy or Rayleigh number Ra is large compared with one. Laboratory experiments show that the buoyant fluid rises in the form of an extremely viscous ‘thermal’ which enlarges with time as a result of entrainment of surrounding fluid. The formation of a stable ‘chemical ring’ or torus of passive tracer similar in appearance to high Reynolds-number vortex rings is a notable feature of the creeping flow for high Rayleigh numbers. The possibility of large variations of viscosity due to temperature differences is included. A self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes’ flow. Experiments at 2.5 × 102 < Ra < 2.5 × 104 and Re < 10−2 demonstrate the nature of extremely viscous thermals, support the similarity solution and enable evaluation of a proportionality constant. Possible applications of the results to dispersion by viscous drops and particularly to thermal convection in the Earth's solid mantle are mentioned.