Is the superconducting current made up of Cooper pairs? Inside the superconductor it should be $\mu_0\mathbf{j} = \mathbf{\nabla} \times  \mathbf{B} = 0$, since B is 0 due to the Messner effect.


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*This means that the current is carried by the surface. 

*But if the surface is not composed of Cooper pairs some scattering can cause them to slow down, which we know doesn't occur experimentally. So the current carried by the surface should be made up of Cooper pairs.

*On the other hand, all Cooper pairs should be in the same state, so if they are moving at the surface they should be moving also in the inside, so $\mu_0\mathbf{j} = \mathbf{\nabla} \times  \mathbf{B} \neq 0$.


What did I miss out?
 A: You miss first of all that Cooper pairs do not exist as some physical quantities. If you prefer, they are not particles as electrons. They are just correlations. 
The current is a collective response to a gradient of phase. You can generate such a gradient by a magnetic field, a voltage, a break of the condensate (like in Josephson system), with different phenomenologies. In the case of a magnetic field, you produce the Meißner effect as well, and the current flows at the edge. But practically it has nothing to do with a flow of Cooper pairs seen as particles.
What you could see as moving are the Bogoliubov excitations if you want. Those are the excitations of the condensate made of Cooper pairs. In short, the Cooper pairs are suitable entity to schematically describe the condensate at rest (in equilibrium in fact), and the Bogoliubov quasi-particles (called sometimes Bogolons, Boboliubon, or something more exotic, but most of the people just use quasi-particles) are suitable quantity to schematically describes non-equilibrium situations. As such, the Cooper pairs can not really flow, or move or do whatever else, but the quasi-particle can allowed to do so.
You can have a look on the BCS paper to understand this point more mathematically speaking, if that's your concern. In particular, the section about the Meißner effect might be useful.
