If not by a viscous stress, how then does friction arise, and how is it properly described?

In very general terms, the fluid's viscosity is the relation between the strain rate and the viscous stress.

The magnitude of the viscous stresses depends upon the rate of deformation.

In particular, the local strain rate is the only property of the velocity flow that directly affects the viscous stress at a given point.

Where is the viscous stress and is the elastic stress.

Elastic and viscous stresses are usually combined under the name mechanical stress.

However, if the deformation is changing with time, even in fluids there will usually be some viscous stress, opposing that change.

Away from the wall, on the other hand, the turbulence generates a Reynolds stress, large compared with the viscous stress, in the usual way.

To generate this distribution of viscous stress, the mean velocity profile must rise steeply at the wall and then become comparatively flat.

In these regions the inertia force becomes less important and the flow is determined by the balance of viscous stresses and the pressure gradient.