Equation of motion of a gas's center of mass due to heating So here's a pretty basic question that I don't know how to solve within standard thermodynamics.
Let's say I have a container with a gas in it. I transfer heat, $Q$, to the gas from the bottom of the container. Let us assume the local equilibrium hypothesis is true for such a system. We know that, since hot air is lighter, the lighter air will move up, causing the center of mass to move to the bottom.
Let the center of mass undergo a displacement, $\vec s_{CM}$. Is there a way to find the equation of motion for the velocity $\vec v_{CM}$ of the center of mass of the gas? (Assume any required parameters such as density $\rho$, etc.)
 A: The natural convection problem you are envisioning can be solved for the time- and spatial variations of the temperature, the velocity, and  density using the basic spatially differentiated transport equations (Navier-Stokes equations {with typically a linearized buoyancy term} in conjunction with the continuity {mass balance} equation and the differential energy balance equation {first law of thermodynamics}).
For approaches to setting this up, see Transport Phenomena by Bird, Stewart, and Lightfoot.  Depending on the specific geometry, an analytic solution is usually not possible, and one would have to resort to a numerical solution, typically involving computational fluid dynamics (CFD).
A: Let's consider a simpler problem: we'll heat the gas from the top of the container, and remove the heat from the bottom.  That way we can possibly avoid the complications of convection.
The exponential atmosphere with linear temperature lapse has the steady-state pressure profile
$$
P(z) = P_0 \left(
\frac{T_0}{T(z)}
\right)^{ Mg / R L}
$$
where $M$ is the molar mass of the gas, $g$ is the local gravitational acceleration, $R = N_A k$ is the ideal gas constant, and $L = dT/dz$ is the temperature lapse rate.
To find the height of the center of mass, you would compute the density from the ideal gas equation, then integrate over the density distribution in the usual way:
$$
z_\text{c.o.m.} = \frac{\int dz\,\rho(z) z}{\int dz\,\rho(z)}
$$
(though you'll want to be less sloppy about linear versus volume densities than I've been.) The density result is (from the same link)
$$
\rho(z) = \rho_0 \left(
\frac{T_0}{T_0 + Lz}
\right)^{\left(
1 + \frac{Mg}{RL}
\right)}
$$
which is clearly related the pressure result, since $T(z) = T_0 + Lz$ is the mathematical statement of the linear temperature lapse. This reduces to the more familiar
$$
\rho(z) \underset{L\to0}\longrightarrow \rho_0 \exp \left({-Mg z}/{R T_0}\right)
$$
in the constant-temperature limit.  You have said you're not interested in containers where the gravitational scale height matters, so you might take the more naïve approximation that $\lim_{L\to0}\rho(z) = \rho_0$, and the center of mass for a constant-temperature cylinder is just in the middle.
If you were to introduce heat-at-the-top and cool-at-the-bottom to your system slowly, so that the linear temperature lapse were a good approximation the entire time, you could compute
$$
\frac{d}{dt} z_\text{c.o.m.}
= \frac{dz_\text{c.o.m.}}{dL}  \frac{dL}{dt}
$$
But if you're neglecting convection, the heat is going to diffuse through the tube in a nonlinear, Newton-cooling, exponential sort of way, so there's no point in that algebra unless you just think it's fun. Better is to take the arguments that went into this setup and turn them into a computational model, where heat diffuses between adjacent slices based on the thermal conductivity of the gas, and mass flows between segments because of the pressure differences.  You might consider a model with constant-mass slices which grow and shrink, rather than a model with constant-height slices with mass exchange.
If you are not neglecting convection, you still have to do all of this work, but in three dimensions.
