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I am not a physicist so bear with me. I'm trying to understand the mechanism through which energy is dissipated in a fluid or solid. Often the explanation is that it happens due to viscosity or friction and that velocity dependent forces have to be added that reduce the total energy. However at molecular and thermodynamic level energy should be conserved. I need to understand better the nature of the dissipated heat and where it goes.

This question How is viscosity described on the molecular level? gave me some good insights but I'm still puzzled by a few things. Let's take a scenario of molecules interacting via Lennard-Jones potentials and let them fall under gravity (or even in zero g). In the real world this bunch of particles will always reach equilibrium, i.e. zero velocity and kinetic energy relative to its inertial frame. Highly viscous materials will just do it quicker. As far as I understand this is because energy gets transformed to heat, although by energy we mean here actually "useful" energy which I'm not sure what it means.

My question is: what happens to the energy of the flow in a viscous fluid (at molecular level)? Does it get transformed to increased temperature, increased entropy, kinetic energy of hidden degrees of freedom, stored potential energy, heat dissipation into the environment or something else?

Community
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Total energy of a closed system is always conserved (time translation, Noether theorem). However, this energy might go from macroscopic motion (fluid flow) into random vibrations of the molecules.

I am not 100% sure of the thermodynamics, but I will try.

The total energy $U$ is still there but it has changed to heat. The velocity $v$ of the molecules also has not changed, however the directions got randomized!

What has also changed is the entropy $S$. There are only a few possible realizations of perfect laminar flow with a certain velocity. For random vibrations, you have way more possible configurations. So although the internal energy $U$ has not changed, the “useful” macroscopic energy $F$ (Helmholtz free energy) has changed: $$ F = U - TS \,.$$ In a closed system we know that $U$ is fixed. So let us rewrite it as $$ U = F + TS \,.$$ So we start with a configuration with low entropy $S$. This means that the usable energy is high. Then due to various relaxation processes the entropy $S$ will increase. Assuming that the temperature $T$ does not change a lot, we will then have a decrease in useful energy $F$.

So heat is still a form of energy, albeit not directly usable as kinetic energy of a macroscopic object (say a pendulum swinging).

Martin Ueding
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Part of the answer is, at low Reynolds number, in streamline flow, there's little if any drag. At higher speeds, though, the wake of a moving object creates vortices (and does so in random fashion, a spontaneous symmetry breaking occurs). Those random vortices cause a velocity-squared retarding force, which is energy-losing whether your motion is northerly or southerly... it is a definitely non conservative force field. Drag can be decreased by some shapes that shed vortices (so the vortex doesn't impinge on the moving object).

In any case, vortices decay to thermal energy in a fluid. The water at the base of Niagara falls is slightly warmer than the water at the top, so the energy isn't only lost to solid walls.

Whit3rd
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