  Using a (2+1)-d viscous hydrodynamical model, we study the dependence of flow
observables on the collision energy ranging from sqrt(s)=7.7 A GeV at the
Relativistic Heavy Ion Collider (RHIC) to sqrt(s)=2760 A GeV at the Large
Hadron Collider (LHC). With a realistic equation of state, Glauber model
initial conditions and a small specific shear viscosity eta/s = 0.08, the
differential charged hadron elliptic flow v_2^{ch}(p_T,sqrt(s)) is found to
exhibit a very broad maximum as a function of sqrt(s) around top RHIC energy,
rendering it almost independent of collision energy for 39 < sqrt(s) < 2760 A
GeV. Compared to ideal fluid dynamical simulations, this "saturation" of
elliptic flow is shifted to higher collision energies by shear viscous effects.
For color-glass motivated MC-KLN initial conditions, which require a larger
shear viscosity eta/s = 0.2 to reproduce the measured elliptic flow, a similar
"saturation" is not observed up to LHC energies, except for very low p_T. We
emphasize that this "saturation" of the elliptic flow is not associated with
the QCD phase transition, but arises from the interplay between radial and
elliptic flow which shifts with sqrt(s) depending on the fluid's viscosity and
leads to a subtle cancellation between increasing contributions from light and
decreasing contributions from heavy particles to v_2 in the sqrt(s) range where
v_2^{ch}(p_T,sqrt(s)) at fixed p_T is maximal. By generalizing the definition
of spatial eccentricity epsilon_x to isothermal hyper-surfaces, we calculate
epsilon_x on the kinetic freeze-out surface at different collision energies. Up
to top RHIC energy, sqrt(s)=200 A GeV, the fireball is still out-of-plane
deformed at freeze out, while at LHC energy the final spatial eccentricity is
predicted to approach zero.
