# Natural convection and buoyancy: What is a better description of $g$?

In their derivation of the Grashof number (Gr), Çengel and Ghajar make the following comment:

Note that there is no noticeable gravity in space, and thus there can be no natural convection heat transfer in a spacecraft, even if the spacecraft is filled with atmospheric air.

Furthermore, one of their "concept questions" is

Consider a hot boiled egg in a spacecraft that is filled with air at atmospheric pressure and temperature at all times. Will the egg cool faster or slower when the spacecraft is in space instead of on the ground? Explain.

To which the official solution is

The hot boiled egg in a spacecraft will cool faster when the spacecraft is on the ground since there is no gravity in space, and thus there will be no natural convection currents which is due to the buoyancy force. [sic]

The implication is that in a spacecraft $\mathrm{Gr}=0$ because $g=0$. However, it is known by any high school–level physics student that $g\neq0$ in a spacecraft and that there is gravity in space. Furthermore, "no noticeable gravity" doesn't seem particularly rigorous to me.

So my question is about correctly describing $g$. This question discusses proper acceleration, which looks like the correct interpretation of $g$ for this case, but I'm not 100% sure. So I wonder this: If one were to succinctly but rigorously describe the variable $g$ in words as it relates to natural convection and buoyancy, what would that description be? Here are some thoughts I have:

• "The normal force on the body of fluid surrounding the object"
• "The proper acceleration of the object and that of the fluid surrounding it"
• It is worth commenting that there are two ways to define "weight" in basic Newtonian physics. The 'obvious' (but less useful) way is as the force of gravity acting on an object; the less obvious way is as the net reaction force acting on a object. Using the former saying that objects in orbit are weightless is incorrect, while it is correct under the latter definition. If we use the latter convention and also define the buoyant force as the weight of displaced fluid this is cleared up by the magic of good conventions at the cost of using a less naive definition of weight. – dmckee --- ex-moderator kitten Jul 12 '17 at 14:08
• Bouyancy forces seem to be external force fields on an object that causes a displacement of a fluid boundary, based on the images on wikipedia. And the Gr number seems to be derivable from Buckinghams pi theorem. – Emil Dec 27 '17 at 22:24
• I don't see how gravity is needed - unless the argument is since the egg won't float in water in micro-gravity, you can't "boil" it. The Sun facing side of space craft will be heated by radiation pressure to roughly $250\ C^{\circ}$ - and the temperature on the dark side will be cooled to roughly $-160\ C^{\circ}$. With those temperature you should't need water. It's hard to imagine there is no air conditioning or air movement inside - especially when people haven't taken a proper bath for an extended period of time. Maybe they just don't serve beans. – Cinaed Simson May 21 '19 at 22:05
• In the reference frame of the spacecraft, g actually does equal zero because the spacecraft is in continuous free fall around the earth. – David White Jan 29 at 16:22

When you say there is gravity in space what you mean is that, as perceived by an observer on earth, things in space fall towards earth with acceleration $g$.

However for an observer inside a freely falling spacecraft, objects inside the spacecraft do not accelerate. To him, his reference frame is inertial (within the extent of spacetime in which tidal effects may be ignored, which to be precise is true only in the infinitesimal limit). To this observer inside the spacecraft, there is no gravity.

Now convection flow, like any other physical phenomenon, is guided by local conditions (no action-at-a-distance). In a freely falling spacecraft it sees no gravity locally, and hence there will be no convection. What is perceived by an observer located on Earth or Andromeda galaxy or wherever-else is of no concern.

On the surface of the earth, we can measure an acceleration of objects with respect to the surface of the earth with a value something close to $9.8 \ {\rm m/s^2}$. Because most of the items we interact with (laboratories, atmosphere, etc.) are constrained by contact with the surface of the earth, this is a useful frame for measurement.

In a spacecraft, we can similarly calculate a value for gravitational acceleration with respect to the earth. But because our environment is no longer constrained by the surface (but is instead in free-fall), this value is not as locally relevant.

For the purpose of convection, measuring gravitational acceleration with respect to the craft (and the atmosphere within) is more useful. This acceleration will be measured to be nearly zero.

A useful value for $g$ is normally clear from context. If you're calculating the orbit time for a satellite, then the acceleration with respect to the planet below is necessary. If you're trying to figure out how much force you'll need to lift a box (or investigate convection), then acceleration with respect to the floor of your room is more useful.