Conduction, by definition, is heat transfer through a still body, be it solid, liquid or gas. Fine, I understand this easily for a solid. But I don’t see how conduction for a liquid or gas works without turning into convection… Surely the parts of the fluid that have more thermal energy are going to have changes in density, which would then of course result in the fluid having buoyant forces and start to move around. Wouldn’t that change it to convection? Where do you draw the line between conduction and convection for fluids? I guess what I’m asking is, is it possible for conduction of liquids/gases to exist without the subsequent convection caused by the fluid moving around due to temperature differences? Wouldn’t that mean conduction doesn’t exist for liquids/gases (or should I say, “pure conduction”?)? I know the last statement is false but I’m trying to see where I’m going wrong in my understanding here… 
In microgravity, convection doesn’t exist, so any heat flow would have to be via conduction and/or radiation. Fire burns spherically under these conditions.
One obvious case where conduction occurs without convection is where the top surface of the fluid (assuming a density decreasing with temperature) is heated instead of the bottom surface. Also, the convective (Rayleigh-Benard) instability does not happen for arbitrarily small temperature gradients, so a slowly-heated fluid will transfer heat by conduction only.
Lots of nice vids at YouTube under “flame in microgravity”.
If there was a liquid with a very small coefficient of expansion, there would be no convective currents. Maybe with very small heat flows, water close to 4C might work.
>Maybe with very small heat flows, water close to 4C might work.
Intriguingly, water around 2 C transferring heat within a couple °C temperature range would exhibit buoyant convection in the “wrong direction”, if the dimensions were large enough that these small buoyancies could still overcome viscous forces.
I don’t think conduction excludes movement. The point is that conduction is not facilitated by the movement. You can drag paper over a hotplate, for example, and heat the paper as it goes by. The paper is moving, but the transfer of heat into the paper and through its thickness is by conduction. Now, the transfer of heat sideways, in the direction you’re dragging the paper, THAT is convection.
In this example you could solve the following differential equation iteratively at all points in a mesh, using a numerical finite element solver:
kdel2(temp) - cpvvx*dx(temp) + heat=0
In this equation, the vx is the velocity of the paper, and the entire cpvvxdx(temp) term is the convective term. The k*del2(temp) term is the conduction, and the heat term is a body heat (such as what happens inside a heating element or inside a radioactive solid), which in our paper example would be zero.