  Comparing hydrodynamic simulations to heavy-ion data inevitably requires the
conversion of the fluid to particles. This conversion, typically done in the
Cooper-Frye formalism, is ambiguous for viscous fluids. We compute
self-consistent phase space corrections by solving the linearized Boltzmann
equation and contrast the solutions to those obtained using the ad-hoc
"democratic Grad" ansatz typically employed in the literature where
coefficients are independent of particle dynamics. Solutions are calculated
analytically for a massless gas and numerically for both a pion-nucleon gas and
for the general case of a hadron resonance gas. We find that the momentum
dependence of the corrections in all systems investigated is best fit by a
power close to 3/2 rather than the typically used quadratic ansatz. The effects
on harmonic flow coefficients $v_2$ and $v_4$ are substantial, and should be
taken into account when extracting medium properties from experimental data.
