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Consider a perfect sphere of fluid which is not rotating, is under the influence of its own gravity only, and is heated from the centre. What does the flow of fluid in the body look like?

In the analogous situation for a perfect cylinder under a downward gravitational force, heated from the bottom, we get a Bénard cell. In reality we get several smaller cells, but this breaking of cylindrical symmetry can't occur in our idealized system. In the real-life spherical case (stars) we also get these smaller convection cells. But when we have no way of breaking the spherical symmetry, what happens? A single cell (as in the perfect cylinder) can't occur; this requires a preferred axis.

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  • $\begingroup$ 1.) As in reality no system is perfectly symmetric, convection will always occur. 2.) if you still want to suppress convection just for the sake of the argument, then outwards expansion will occur until a new hydrostatic equilibrium is found. If no new equilibrium can be found, then at some point the system will loose mass into its boundary condition/ infinity, akin to a stellar wind. $\endgroup$
    – AtmosphericPrisonEscape
    Commented Feb 22, 2023 at 13:16
  • $\begingroup$ @atmosphericprisonescape The Rayleigh-Benard instability happens between the plates despite the complete symmetry of the setup, and then the slightest perturbation will cause the build-up of the convection cells. My reading of rake's question is if the same kind of instability can evolve in a spherically symmetric geometry with a sufficiently high thermal and gravitational gradient, and I do not think you are answering that question. $\endgroup$
    – hyportnex
    Commented Feb 22, 2023 at 14:37
  • $\begingroup$ @hyportnex Read again "But when we have no way of breaking the spherical symmetry, what happens?" OP wants to keep symmetry by magic and asks what happens then. $\endgroup$
    – AtmosphericPrisonEscape
    Commented Feb 22, 2023 at 14:48
  • $\begingroup$ @AtmosphericPrisonEscape thank you for the suggestion (2); I had not thought of expansion. Presumably at some degree of expansion, we will lose as much heat to radiation as we gain from the heating, and so there is no energy source for further expansion, which would store gravitational potential energy. At this point, then, we are faced with the original question. Perhaps this results in a spherically symmetric temperature gradient with no convection, and this system is in an unstable equilibrium? Where, as hypnortex suggests, a perturbation would lead to the formation of cells. $\endgroup$
    – rake
    Commented Feb 22, 2023 at 15:19
  • $\begingroup$ @hyportnex The situation you suggest (an unstable equilibrium analogous to the Rayleigh-Bénard instability), if it would result from the setup I described, would indeed answer my question. Do you suggest that there would be an unstable, no-flow situation? $\endgroup$
    – rake
    Commented Feb 23, 2023 at 15:49

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