Laminar natural convection about an isothermally heated sphere at small Grashof number

摘要: The flow induced by gravity about a very small heated isothermal sphere introduced into fluid in hydrostatic equilibrium is studied. natural-convection taken to be steady and laminar. conditions under which the Boussinesq model good approximation full conservation laws are described. For concentric finite cold outer with radius, ratio roughly less than Grashof number minus one-half power, recirculating occurs; rises near inner falls sphere. an unbounded medium ordinary perturbation expansion essentially leads velocities far from sphere; this singularity analogue of Whitehead paradox arising three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal importance convective transport away sphere, although diffusive dominant Approximate solution given nonlinear equations, first seeking similarity (in paraboloidal co-ordinates) for point heat source valid source, then linearization manner Oseen. Oseen solution. Both solutions describe wake above within enthalpy decays slowly relative rapid decay outside wake. updraft reduced growth distance constant magnitude restoration accelerations. Finally, role vertical stratification ambient density eventually stagnating updrafts predicted on basis constant-density atmosphere discussed.