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Does it stay still wrt the axis? Does it rotate along with the tire? Is it drawn away from the center of rotation?

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Model the air in a tire as a viscuous fluid, and you should get a pretty good answer. The viscosity is low, but so is the mass, so inertia effects will be relevant. Check the Reynolds number, but I suspect you can largely ignore turbulance if you model the inside of the tire as a smooth surface and ignore the deformation where the tire flattens against the road.

When the tire has been at rest for a while, everything will be "motionless". When the tire first spins, the center of the air will stay where it was. The boundary layer by the tire wall will move with the tire, which then makes the next layer move, with makes the next layer move, etc. There is a overall time constant. Eventually in steady state all the air moves with the tire.

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    $\begingroup$ A agree with this completely, but would add that in the limit as $t\rightarrow\infty$, the air-tire system will be in rigid body rotation, and the curved trajectory of the air particles will demand a radial pressure gradient. Thus, the highest pressures will be near the tire and the lowest pressures will be near the wheel rim. $\endgroup$
    – Bryson S.
    May 26 '14 at 22:51
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At a slow rotation speed, I would think air would be (almost) still at the inner surfaces of the tire (with respect to the surfaces), but would slip with respect to the tire elsewhere. The centrifugal force will be greater farther from the axis of the wheel, so air will have higher pressure and density there. Turbulence can be a complicating factor at higher rotation speeds.

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