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Viscous hydrodynamics is commonly used to model the evolution of the mattercreated in an ultra-relativistic heavy-ion collision. It provides a gooddescription of transverse momentum spectra and anisotropic flow. Theseobservables, however, cannot be consistently derived using viscoushydrodynamics alone, because they depend on the microscopic interactions atfreeze-out. We derive the ideal hydrodynamic limit and the first-order viscouscorrection to anisotropic flow ($v_2$, $v_3$ and $v_4$) and momentum spectrumusing a transport calculation. The linear response coefficient to the initialanisotropy, $v_n(p_T)/\varepsilon_n$, depends little on $n$ in the idealhydrodynamic limit. The viscous correction to the spectrum depends not only onthe differential cross section, but also on the initial momentum distribution.This dependence is not captured by standard second-order viscous hydrodynamics.The viscous correction to anisotropic flow increases with $p_T$, but thisincrease is slower than usually assumed in viscous hydrodynamic calculations.In particular, it is too slow to explain the observed maximum of $v_n$ at$p_T\sim 3$ GeV/c.

Viscous corrections to anisotropic flow and transverse momentum spectra from transport theory