Vorticity generation behind curved shock waves The generation of a curved shock wave can take place in many applications. For instance, with high speed blunt bodies, the bow shock in front of the body will display a general curvature. Below is an example, which is a shadowgraph image of the Project Mercury reentry capsule during a reentry wind tunnel test.

Other examples include the curved shock produced during planar shock diffraction around convex corners. Below is an example of a schlieren image where an initially planar shock wave traveling at approximately Mach 2 diffracts over a 45$^{\circ}$ convex wall, resulting in the curved shock shape. 

Now, because the shock waves are curved, each streamline that passes through the shock at a given incidence angle will experience a different change in entropy. Hence, the flowfield behind the curved shock will have an entropy gradient such that $\nabla s \neq 0$. As a result, from the well known Crocco theorem,
$$ T \nabla s = \nabla h_0 - \vec{V} \times \left(\nabla \times \vec{V} \right) + \frac{\partial \vec{V}}{\partial t} $$
where $s$ is the entropy, $h_0$ is the total enthalpy, and $\vec{V}$ is the velocity, the non-zero entropy gradient would imply vorticity in the flowfield behind the curved shock. Hence, behind any curved shock wave, we will have the condition,
$$ \nabla \times \vec{V} \neq 0 $$
This is all well established and is relatively easy to follow and understand. However, I have a difficult time trying to justify the satisfaction of the Helmholtz vortex theorems to this particular class of problems. The one in question is his third theorem. The third theorem gas as follows: 
"In the absence of rotational external forces, a fluid that is initially irrotational remains irrotational."
Now the Helmholtz vortex theorems apply to inviscid flows where viscous forces are neglected. However, the curved shock problems described above can easily be established from an invsicid framework, such that the Helmholtz theorems should still apply. So my question is, if the flow starts irrotational ahead of a curved shock wave ($ \nabla \times \vec{V} = 0$), yet becomes rotational behind the curved shock wave ($ \nabla \times \vec{V} \neq 0$) because of the entropy gradient, then are we in some way violating Helmholtz's third vortex theorem? Is a curved shock wave somehow introducing a rotational external forces on the fluid elements? The common phrase regarding the third vortex theorem states that fluid elements initially free of vorticity remain free of vorticity, but this doesn't seem to be the case here. Anyways, I was hoping to get some clarification or perhaps I am missing something very fundamental. 
 A: The generation of vorticity via a curved shock front indeed comes directly from Crocco's theorem, since curved shocks have variable shock strength and thus variable entropy. However, the entropy generation resulting from "inviscid" shocks are itself induced by viscous diffusion and thermal conductivity (see Liepmann & Roshko or related text). These are neglected in theoretical frameworks, but is by far the most  important thing in numerics (numerical viscosity, see for example Toro's book on Riemann solvers or von Neumann & Richtmyer's paper) and even understanding what happens physically.
The Helmholtz vortex theorems are derived from an inviscid framework by consideration of vortex lines and tubes, if vortex filaments follow material lines within a closed curve (irreducible or reducible), then the vortex strength does not change with time (i.e. a flow that is irrotational will remain irrotational). The entropy gradient generated by a nonlinear shock front can only be understood by considering it as a local activation of viscosity and heat transfer, this then makes the Helmholtz theorems invalid in the proximity of shocks.
