  In this paper we study the effects of rotation in low-mass, low-metallicity
RGB stars. We present the first evolutionary models taking into account
self-consistently the latest prescriptions for the transport of angular
momentum by meridional circulation and shear turbulence in stellar interiors as
well as the associated mixing processes for chemicals computed from the ZAMS to
the upper RGB. We discuss in details the uncertainties associated with the
physical description of the rotational mixing and study carefully their effects
on the rotation profile, diffusion coefficients, structural evolution,
lifetimes and chemical signatures at the stellar surface. We focus in
particular on the various assumptions concerning the rotation law in the
convective envelope, the initial rotation velocity distribution, the presence
of mean molecular weight gradients and the treatment of the horizontal and
vertical turbulence. This exploration leads to two main conclusions : (1) After
the completion of the first dredge-up, the degree of differential rotation (and
hence mixing) is maximised in the case of a differentially rotating convective
envelope (i.e., j_CE(r) = cst), as anticipated in previous studies. (2) Even
with this assumption, and contrary to some previous claims, the present
treatment for the evolution of the rotation profile and associated meridional
circulation and shear turbulence does not lead to enough mixing of chemicals to
explain the abundance anomalies in low-metallicity field and globular cluster
RGB stars observed around the bump luminosity. This study raises questions that
need to be addressed in a near future. These include for example the
interaction between rotation and convection and the trigger of additional
hydrodynamical instabilities.
