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Using the Rankine-Hugoniot relations for a shockwave, one can show that entropy jumps across the shock, so that the entropy difference between upstream and downstream conditions is given by

$$s_2 - s_1 = c_p\ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{p_2}{p_1}\right).$$

In Chapter 3 of the book "Modern Compressible Flow" by John Anderson, the author attributes this jump in entropy to the irreversible effects of viscosity and thermal conduction in the inner structure of the shock. However, in his derivation of the Rankine-Hugonoit relations, the author used Euler's equations from the very beginning, which do not include viscous and conductive terms (in contrast to the more complete Navier-Stokes), and should therefore exclude all viscous and thermal conduction effects from the physics.

Thus, I don't find the author's explanation satisfying, and I am having a hard time understanding what's really going on. Why does the entropy increase across the shock?

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However, in his derivation of the Rankine-Hugonoit relations, the author used Euler's equations from the very beginning, which do not include viscous and conductive terms (in contrast to the more complete Navier-Stokes), and should therefore exclude all viscous and thermal conduction effects from the physics.

There are some subtle assumptions that some authors fail to explicitly point out when deriving the Rankine–Hugoniot relations. One of the first is that the asymptotic states, far from the shock ramp, must be in thermodynamic equilibrium. However, the fluid states near the ramp are a succession of two states NOT in equilibrium. So one typically assumes that the transition is abrupt/fast and then they effectively "sweep the details under the rug" leaving the reader confused and frustrated. Sadly, this sweeping of the details under the rug as it were is partly where entropy enters the derivation. As I explained at https://physics.stackexchange.com/a/177972/59023, the source of irreversibility is not at all trivial and is still a source of debate. An additional complication is that formally the Rankine–Hugoniot (RH) relations are time-reversible.

So what do we know and what can we say?

Well a shock wave is a nonlinearly steepened wave that has reached a stable balance between steepening (e.g., the $\mathbf{V} \cdot \nabla \mathbf{V}$ term) and energy dissipation/irreversibility. We can say this because without some form of irreversible energy dissipation, a nonlinearly steepening wave will reach a gradient catastrophe and undergo wave breaking.

So what really happens in the shock?

There is a change in kinetic energy across the shock and total energy must be conserved. So where does the extra energy go? The shock itself is an abrupt/fast compression, so we can assume there is no heat flux across the shock ramp, i.e., the process is adiabatic. However, the typical adiabatic equation of state cannot be used because the fluid on either side of the ramp is not in a succession of equilibrium states. This implies some form of energy dissipation is occurring, which we also can safely assume takes place in the ramp where the fluid is rapidly compressed and decelerated. In a collisionally mediated fluid like Earth's atmosphere, energy dissipation occurs through binary particle collisions. Thus, the bulk kinetic energy in the incident upstream flow is rapidly and irreversibly converted into random kinetic energy (i.e., heat) in the outgoing downstream flow.

Wait, how are the Rankine–Hugoniot relations time-reversible but we use them to describe an irreversible process?

The RH relations describe the asymptotic conditions, far from the shock ramp. They are merely conservation relations, nothing more. They can rely upon irreversible processes so long as the effects of the irreversibility are accounted for in the equations (e.g., increase in downstream temperature accounts for irreversible randomization of particles at shock ramp). Frank H. Shu has a book entitled The Physics of Astrophysics Volume II: Gas Dynamics that has some good discussions about the fundamentals of shocks and entropy generation.

Why does the entropy increase across the shock?

There will be no overarching answer that will fully satisfy you, I fear. The problem is that for the structure to be a shock, i.e., a stable discontinuity, energy dissipation must balance nonlinear steepening. If this is occurring, then in a collisionally mediated fluid, entropy must increase across the shock. In a collisionless medium like some plasmas, entropy need not increase but the processes leading to the change in kinetic energy must remain irreversible. That is, irreversibility is the fundamental concern here and it can by synonymous with entropy generation in some media like those mediated by collisions.

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A comment that should have been an answer:

Probably, the significant entropy change occurs right at the shock, which, in the author's approximation occurs at a plane, while the actual shock, involving viscous dissipation, occurs over a narrow finite region.

This sentence is helpful for an outsider in framing the other, longer answer which has so far appeared.

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