I am little confused about the connection between these two concepts in Physics:

First, in classical mechanics, a basic result is the work-energy theorem, which states $$\Delta T = -W.$$ This equation is true whether or not forces are conservative.

Then for any conservative system, we can define the potential energy function U such that the force on each particle is given by

$$ \textbf{F}_i = -\nabla_i U$$

Now even if the system is not conservative, we can still decompose the forces into a conservative and nonconservative part and then define $U$ such that the above equation gives you the conservative part of the forces.

In principle then we can always write the energy of the system E as being:

$$E = T + U$$

Furthermore, a system is said to be conservative if and only if:

$$ \Delta E = 0 $$

and by the work energy theorem this is true if and only if $U = W$

Then the first law of thermodynamics comes in and states that for a closed system:

$$\Delta U= Q + W$$

I am using the convention that $Q$ and $W$ are the energy flow done ON the system to be consistent with classical mechanics. In thermodynamics heat is specifically defined as the energy transfer due to difference between the temperature of the system and its surroundings.

Then from here , one derives :

$$\Delta E = \Delta T + \Delta U = -W + W + Q = Q$$

So the change in the total energy of the system can only be due to heat.

But I don’t think it is quite right. For instance consider a closed system of a viscous fluid such that it’s temperature is equal to the surroundings.

By the above we have $$\Delta E = 0$$, but this cannot be possibly true because as we know friction dissipates energy by work. So where did I go wrong here ? I know the work energy theorem has to be correct, but maybe the first Law of thermodynamics the way I stated it, is not actually:

$$\Delta U= Q + W$$.

Edit: Could I answer my own question?

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    $\begingroup$ Does this answer your question? How does the work-energy theorem relate to the first law of thermodynamics? $\endgroup$ Commented Nov 17, 2022 at 5:53
  • $\begingroup$ No its really unclear. Why should the work energy theorem only apply in the context of nondeformable bodies? Anyway, I think I have figured out the answer to my own question and I'd like to answer it to make it more clear for future readers, if you would reopen the question. Basically, I concluded that the $W$ in the first law of thermodynamics is not the net work done on the body but only the contribution from the nonconservative forces. When you assume that, newtonian mechanics becomes consistent with thermodynamics. $\endgroup$ Commented Nov 17, 2022 at 23:54
  • $\begingroup$ Perhaps the duplicate link should point here. Please take a look at that answer. $\endgroup$ Commented Nov 18, 2022 at 0:24