Initial Stage of Natural Convection Over a Hot Aerosol Sphere

Background: Analytical study is presented on the transient problem of buoyancy-induced motion due to the presence of a hot aerosol sphere in unbounded quiescent fluid. Method of Approach: Because the initial flow field is identically zero, the initial stage of the process is governed by viscous and buoyancy forces alone where the convective inertial terms in the momentum and energy balances are negligible, i.e., the initial development of the field is a linear process. The previous statement is examined by analyzing the scales of the various terms in the Navier-Stokes and energy equations. This scale analysis gives qualitative limitations on the validity of the linear approximation. A formal integral solution is obtained for arbitrary Prandtl number and for transient temperature field. Results: We consider, in detail, the idealized case of vanishing Prandtl number for which the thermal field is developed much faster than momentum. In this case, analytical treatment is feasible and explicit expressions for the field variables and the drag acting on the particle are derived. Detailed quantitative analysis of the spatial and temporal validity of the solution is also presented. Conclusions: The linear solution is valid throughout space for $t<10$ diffusion times. For $t>10$⁠, an island in space appears in which inertial effects become dominant. The transient process is characterized by two different time scales: for short times, the development of the field is linear, while for small distances from the sphere and finite times, it is proportional to the square root of time. The resultant drag force acting on the sphere is proportional to the square root of time throughout the process.