Consider a short cylinder whose top and bottom surfaces are insulated. The cylinder is initially at a uniform temperature and is subjected to convection from its side surface to a medium at temperature T. Is the heat transfer in this short cylinder one- or two- dimensional? Explain.

two-dimensional heat transfer

Explanation:

In the first scenario , the side area is parallel to the convection flow, then the heat transfer is two-dimensional since we have the heat transfer in the radial direction ( cylindrical coordinates= r) driven by the temperature difference between the temperature of the cylinder  and the medium. But also we have heat transfer in vertical direction  ( cylindrical coordinates= z) due to convection driven by temperature differences between the temperature of the medium next to the side surface at the bottom and the temperature of of the medium next to the side surface at the top.

the heat transfer would be one dimensional only if the temperature of the medium along the side wall would be uniform , then there would be no convection , thus the heat flow is 2 dimensional

In the second scenario,  ,the side area is perpendicular to the convection flow, then the heat transfer in radial direction would be asymmetrical , since one part is driven by convection and other not . Then we can split the radial heat flow into 2 different heat flows , one in vertical direction ( that would have convection ) and one in horizontal direction ( that would not have convection ) making a 2 dimensional heat transfer.