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Viscosity of a moving fluid (e.g. a gas) is commonly related to the exchange of momentum between layers of such fluid moving at different velocities.

Intuitively, one could think that in a gas such transfer happens either via molecules migration across the boundary of the moving layers (e.g. molecules from the faster layer move into the slower) or via collisions among molecules.

Conduction instead is related to kinetic energy transfer within a body. Again, this energy transfer in the above considered gas could be imagined taking act via migration of molecules at different energies, or via collisions.

Some authors agree with this depiction (e.g. Jacobson MZ, "Fundamentals of Atmospheric Modeling"), some instead say that viscosity is due only to momentum exchange via molecules migration, and conduction is due only to kinetic energy transfer due to collisions (e.g. Spurs JH & Aksel N, "Fluid Mechanics").

Which one is correct?

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As you mentioned in your post, viscosity is related to the exchange of momentum in a fluid. More specifically, viscosity is the diffusion coefficient of momentum that tells us how fast momentum diffuses within a fluid. Viscosity can be measured using experiments like the falling sphere method.

Conduction is the diffusion of energy in a medium, typically referring to thermal energy. Unlike viscosity, conduction is not a physical coefficient. In terms of temperature, the equivalent of viscosity would be thermal conductivity.

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