Surely the heat would just swirl in between the two walls?
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Convection occurs in both fluids and gases but "swirling between the walls" is pretty much exactly right. Take air between a hot wall and a cold wall. Air right against the hot wall picks up heat, expands making it less dense, it rises, more air comes in behind it, picking up heat, expanding, rising, etc. The air at the cold wall looses heat, shrinks, becomes more dense and sinks, bringing more hot air in behind it. This sets of a convection "cell" of air, up at the hot wall, down at the cold wall.
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convenction is only the heat transfer mode takes through fluid, from particle to particle with movement of particle. if consider any fluid flowing over or through an enclosed walls, its velocity increases away from wall. thus convection becomes more important. and also thermal resistance decreases with increases in velocity. so that heat from solid or to solid can given by thermal resistance. since thermal resistance is a unknown, heat transfer given as Q = h A (temperature difference)